5806 lines
160 KiB
C
5806 lines
160 KiB
C
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/*
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* Tiny arbitrary precision floating point library
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*
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* Copyright (c) 2017-2018 Fabrice Bellard
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*
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* Permission is hereby granted, free of charge, to any person obtaining a copy
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* of this software and associated documentation files (the "Software"), to deal
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* in the Software without restriction, including without limitation the rights
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* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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* copies of the Software, and to permit persons to whom the Software is
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* furnished to do so, subject to the following conditions:
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*
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* The above copyright notice and this permission notice shall be included in
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* all copies or substantial portions of the Software.
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*
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* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
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* THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
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* THE SOFTWARE.
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*/
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#include <stdlib.h>
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#include <stdio.h>
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#include <inttypes.h>
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#include <math.h>
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#include <string.h>
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#include <assert.h>
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#ifdef __AVX2__
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#include <immintrin.h>
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#endif
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#include "cutils.h"
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#include "libbf.h"
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/* enable it to check the multiplication result */
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//#define USE_MUL_CHECK
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/* enable it to use FFT/NTT multiplication */
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#define USE_FFT_MUL
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//#define inline __attribute__((always_inline))
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#define unused __attribute__((unused))
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#ifdef __AVX2__
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#define FFT_MUL_THRESHOLD 100 /* in limbs of the smallest factor */
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#else
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#define FFT_MUL_THRESHOLD 100 /* in limbs of the smallest factor */
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#endif
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/* XXX: adjust */
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#define BASECASE_DIV_THRESHOLD_B 300
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#define BASECASE_DIV_THRESHOLD_Q 300
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#if LIMB_BITS == 64
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#define FMT_LIMB1 "%" PRIx64
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#define FMT_LIMB "%016" PRIx64
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#define PRId_LIMB PRId64
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#define PRIu_LIMB PRIu64
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#else
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#define FMT_LIMB1 "%x"
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#define FMT_LIMB "%08x"
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#define PRId_LIMB "d"
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#define PRIu_LIMB "u"
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#endif
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typedef int bf_op2_func_t(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
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bf_flags_t flags);
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#ifdef USE_FFT_MUL
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#define FFT_MUL_R_OVERLAP_A (1 << 0)
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#define FFT_MUL_R_OVERLAP_B (1 << 1)
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static no_inline void fft_mul(bf_t *res, limb_t *a_tab, limb_t a_len,
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limb_t *b_tab, limb_t b_len, int mul_flags);
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static void fft_clear_cache(bf_context_t *s);
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#endif
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/* could leading zeros */
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static inline int clz(limb_t a)
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{
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if (a == 0) {
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return LIMB_BITS;
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} else {
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#if LIMB_BITS == 64
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return clz64(a);
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#else
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return clz32(a);
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#endif
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}
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}
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static inline int ctz(limb_t a)
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{
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if (a == 0) {
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return LIMB_BITS;
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} else {
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#if LIMB_BITS == 64
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return ctz64(a);
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#else
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return ctz32(a);
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#endif
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}
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}
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static inline int ceil_log2(limb_t a)
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{
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if (a <= 1)
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return 0;
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else
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return LIMB_BITS - clz(a - 1);
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}
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#if 0
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static inline slimb_t ceil_div(slimb_t a, limb_t b)
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{
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if (a >= 0)
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return (a + b - 1) / b;
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else
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return a / (slimb_t)b;
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}
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static inline slimb_t floor_div(slimb_t a, limb_t b)
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{
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if (a >= 0) {
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return a / b;
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} else {
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return (a - b + 1) / (slimb_t)b;
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}
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}
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#endif
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#define malloc(s) malloc_is_forbidden(s)
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#define free(p) free_is_forbidden(p)
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#define realloc(p, s) realloc_is_forbidden(p, s)
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void bf_context_init(bf_context_t *s, bf_realloc_func_t *realloc_func,
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void *realloc_opaque)
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{
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memset(s, 0, sizeof(*s));
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s->realloc_func = realloc_func;
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s->realloc_opaque = realloc_opaque;
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}
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void bf_context_end(bf_context_t *s)
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{
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bf_clear_cache(s);
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}
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/* 'size' must be > 0 */
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static void *bf_malloc(bf_context_t *s, size_t size)
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{
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return bf_realloc(s, NULL, size);
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}
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static void bf_free(bf_context_t *s, void *ptr)
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{
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bf_realloc(s, ptr, 0);
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}
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void bf_init(bf_context_t *s, bf_t *r)
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{
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r->ctx = s;
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r->sign = 0;
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r->expn = BF_EXP_ZERO;
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r->len = 0;
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r->tab = NULL;
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}
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void bf_resize(bf_t *r, limb_t len)
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{
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if (len != r->len) {
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r->tab = bf_realloc(r->ctx, r->tab, len * sizeof(limb_t));
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r->len = len;
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}
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}
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void bf_set_ui(bf_t *r, uint64_t a)
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{
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r->sign = 0;
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if (a == 0) {
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r->expn = BF_EXP_ZERO;
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bf_resize(r, 0);
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}
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#if LIMB_BITS == 32
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else if (a <= 0xffffffff)
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#else
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else
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#endif
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{
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int shift;
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bf_resize(r, 1);
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shift = clz(a);
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r->tab[0] = a << shift;
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r->expn = LIMB_BITS - shift;
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}
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#if LIMB_BITS == 32
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else {
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uint32_t a1, a0;
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int shift;
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bf_resize(r, 2);
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a0 = a;
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a1 = a >> 32;
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shift = clz(a1);
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r->tab[0] = a0 << shift;
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r->tab[1] = (a1 << shift) | (a0 >> (LIMB_BITS - shift));
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r->expn = 2 * LIMB_BITS - shift;
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}
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#endif
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}
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void bf_set_si(bf_t *r, int64_t a)
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{
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if (a < 0) {
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bf_set_ui(r, -a);
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r->sign = 1;
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} else {
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bf_set_ui(r, a);
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}
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}
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void bf_set_nan(bf_t *r)
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{
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bf_resize(r, 0);
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r->expn = BF_EXP_NAN;
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r->sign = 0;
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}
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void bf_set_zero(bf_t *r, int is_neg)
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{
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bf_resize(r, 0);
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r->expn = BF_EXP_ZERO;
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r->sign = is_neg;
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}
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void bf_set_inf(bf_t *r, int is_neg)
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{
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bf_resize(r, 0);
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r->expn = BF_EXP_INF;
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r->sign = is_neg;
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}
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void bf_set(bf_t *r, const bf_t *a)
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{
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if (r == a)
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return;
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r->sign = a->sign;
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r->expn = a->expn;
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bf_resize(r, a->len);
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memcpy(r->tab, a->tab, a->len * sizeof(limb_t));
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}
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/* equivalent to bf_set(r, a); bf_delete(a) */
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void bf_move(bf_t *r, bf_t *a)
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{
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bf_context_t *s = r->ctx;
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if (r == a)
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return;
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bf_free(s, r->tab);
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*r = *a;
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}
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static limb_t get_limbz(const bf_t *a, limb_t idx)
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{
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if (idx >= a->len)
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return 0;
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else
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return a->tab[idx];
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}
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/* get LIMB_BITS at bit position 'pos' in tab */
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static inline limb_t get_bits(const limb_t *tab, limb_t len, slimb_t pos)
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{
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limb_t i, a0, a1;
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int p;
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i = pos >> LIMB_LOG2_BITS;
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p = pos & (LIMB_BITS - 1);
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if (i < len)
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a0 = tab[i];
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else
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a0 = 0;
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if (p == 0) {
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return a0;
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} else {
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i++;
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if (i < len)
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a1 = tab[i];
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else
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a1 = 0;
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return (a0 >> p) | (a1 << (LIMB_BITS - p));
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}
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}
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static inline limb_t get_bit(const limb_t *tab, limb_t len, slimb_t pos)
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{
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slimb_t i;
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i = pos >> LIMB_LOG2_BITS;
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if (i < 0 || i >= len)
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return 0;
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return (tab[i] >> (pos & (LIMB_BITS - 1))) & 1;
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}
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static inline limb_t limb_mask(int start, int last)
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{
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limb_t v;
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int n;
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n = last - start + 1;
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if (n == LIMB_BITS)
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v = -1;
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else
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v = (((limb_t)1 << n) - 1) << start;
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return v;
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}
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|
|
|
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/* return != 0 if one bit between 0 and bit_pos inclusive is not zero. */
|
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|
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static inline limb_t scan_bit_nz(const bf_t *r, slimb_t bit_pos)
|
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{
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slimb_t pos;
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limb_t v;
|
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|
|
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pos = bit_pos >> LIMB_LOG2_BITS;
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if (pos < 0)
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return 0;
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||
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v = r->tab[pos] & limb_mask(0, bit_pos & (LIMB_BITS - 1));
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if (v != 0)
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return 1;
|
||
|
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pos--;
|
||
|
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while (pos >= 0) {
|
||
|
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if (r->tab[pos] != 0)
|
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|
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return 1;
|
||
|
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pos--;
|
||
|
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}
|
||
|
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return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* return the addend for rounding. Note that prec can be <= 0 for bf_rint() */
|
||
|
|
static int bf_get_rnd_add(int *pret, const bf_t *r, limb_t l,
|
||
|
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slimb_t prec, int rnd_mode)
|
||
|
|
{
|
||
|
|
int add_one, inexact;
|
||
|
|
limb_t bit1, bit0;
|
||
|
|
|
||
|
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if (rnd_mode == BF_RNDF) {
|
||
|
|
bit0 = 1; /* faithful rounding does not honor the INEXACT flag */
|
||
|
|
} else {
|
||
|
|
/* starting limb for bit 'prec + 1' */
|
||
|
|
bit0 = scan_bit_nz(r, l * LIMB_BITS - 1 - bf_max(0, prec + 1));
|
||
|
|
}
|
||
|
|
|
||
|
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/* get the bit at 'prec' */
|
||
|
|
bit1 = get_bit(r->tab, l, l * LIMB_BITS - 1 - prec);
|
||
|
|
inexact = (bit1 | bit0) != 0;
|
||
|
|
|
||
|
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add_one = 0;
|
||
|
|
switch(rnd_mode) {
|
||
|
|
case BF_RNDZ:
|
||
|
|
break;
|
||
|
|
case BF_RNDN:
|
||
|
|
if (bit1) {
|
||
|
|
if (bit0) {
|
||
|
|
add_one = 1;
|
||
|
|
} else {
|
||
|
|
/* round to even */
|
||
|
|
add_one =
|
||
|
|
get_bit(r->tab, l, l * LIMB_BITS - 1 - (prec - 1));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
break;
|
||
|
|
case BF_RNDD:
|
||
|
|
case BF_RNDU:
|
||
|
|
if (r->sign == (rnd_mode == BF_RNDD))
|
||
|
|
add_one = inexact;
|
||
|
|
break;
|
||
|
|
case BF_RNDNA:
|
||
|
|
case BF_RNDF:
|
||
|
|
add_one = bit1;
|
||
|
|
break;
|
||
|
|
case BF_RNDNU:
|
||
|
|
if (bit1) {
|
||
|
|
if (r->sign)
|
||
|
|
add_one = bit0;
|
||
|
|
else
|
||
|
|
add_one = 1;
|
||
|
|
}
|
||
|
|
break;
|
||
|
|
default:
|
||
|
|
abort();
|
||
|
|
}
|
||
|
|
|
||
|
|
if (inexact)
|
||
|
|
*pret |= BF_ST_INEXACT;
|
||
|
|
return add_one;
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_set_overflow(bf_t *r, int sign, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
slimb_t i, l, e_max;
|
||
|
|
int rnd_mode;
|
||
|
|
|
||
|
|
rnd_mode = flags & BF_RND_MASK;
|
||
|
|
if (prec == BF_PREC_INF ||
|
||
|
|
rnd_mode == BF_RNDN ||
|
||
|
|
rnd_mode == BF_RNDNA ||
|
||
|
|
rnd_mode == BF_RNDNU ||
|
||
|
|
(rnd_mode == BF_RNDD && sign == 1) ||
|
||
|
|
(rnd_mode == BF_RNDU && sign == 0)) {
|
||
|
|
bf_set_inf(r, sign);
|
||
|
|
} else {
|
||
|
|
/* set to maximum finite number */
|
||
|
|
l = (prec + LIMB_BITS - 1) / LIMB_BITS;
|
||
|
|
bf_resize(r, l);
|
||
|
|
r->tab[0] = limb_mask((-prec) & (LIMB_BITS - 1),
|
||
|
|
LIMB_BITS - 1);
|
||
|
|
for(i = 1; i < l; i++)
|
||
|
|
r->tab[i] = (limb_t)-1;
|
||
|
|
e_max = (limb_t)1 << (bf_get_exp_bits(flags) - 1);
|
||
|
|
r->expn = e_max;
|
||
|
|
r->sign = sign;
|
||
|
|
}
|
||
|
|
return BF_ST_OVERFLOW | BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* round to prec1 bits assuming 'r' is non zero and finite. 'r' is
|
||
|
|
assumed to have length 'l'. Note: 'prec1' can be negative or
|
||
|
|
infinite (BF_PREC_INF). */
|
||
|
|
static int __bf_round(bf_t *r, limb_t prec1, bf_flags_t flags, limb_t l)
|
||
|
|
{
|
||
|
|
limb_t v, a;
|
||
|
|
int shift, add_one, ret, rnd_mode;
|
||
|
|
slimb_t i, bit_pos, pos, e_min, e_max, e_range, prec;
|
||
|
|
|
||
|
|
/* e_min and e_max are computed to match the IEEE 754 conventions */
|
||
|
|
e_range = (limb_t)1 << (bf_get_exp_bits(flags) - 1);
|
||
|
|
e_min = -e_range + 3;
|
||
|
|
e_max = e_range;
|
||
|
|
|
||
|
|
if (unlikely(r->expn < e_min) && (flags & BF_FLAG_SUBNORMAL)) {
|
||
|
|
/* restrict the precision in case of potentially subnormal
|
||
|
|
result */
|
||
|
|
prec = prec1 - (e_min - r->expn);
|
||
|
|
} else {
|
||
|
|
prec = prec1;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* round to prec bits */
|
||
|
|
rnd_mode = flags & BF_RND_MASK;
|
||
|
|
ret = 0;
|
||
|
|
add_one = bf_get_rnd_add(&ret, r, l, prec, rnd_mode);
|
||
|
|
|
||
|
|
if (prec <= 0) {
|
||
|
|
if (add_one) {
|
||
|
|
bf_resize(r, 1);
|
||
|
|
r->tab[0] = (limb_t)1 << (LIMB_BITS - 1);
|
||
|
|
r->expn += 1 - prec;
|
||
|
|
ret |= BF_ST_UNDERFLOW | BF_ST_INEXACT;
|
||
|
|
return ret;
|
||
|
|
} else {
|
||
|
|
goto underflow;
|
||
|
|
}
|
||
|
|
} else if (add_one) {
|
||
|
|
limb_t carry;
|
||
|
|
|
||
|
|
/* add one starting at digit 'prec - 1' */
|
||
|
|
bit_pos = l * LIMB_BITS - 1 - (prec - 1);
|
||
|
|
pos = bit_pos >> LIMB_LOG2_BITS;
|
||
|
|
carry = (limb_t)1 << (bit_pos & (LIMB_BITS - 1));
|
||
|
|
|
||
|
|
for(i = pos; i < l; i++) {
|
||
|
|
v = r->tab[i] + carry;
|
||
|
|
carry = (v < carry);
|
||
|
|
r->tab[i] = v;
|
||
|
|
if (carry == 0)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if (carry) {
|
||
|
|
/* shift right by one digit */
|
||
|
|
v = 1;
|
||
|
|
for(i = l - 1; i >= pos; i--) {
|
||
|
|
a = r->tab[i];
|
||
|
|
r->tab[i] = (a >> 1) | (v << (LIMB_BITS - 1));
|
||
|
|
v = a;
|
||
|
|
}
|
||
|
|
r->expn++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* check underflow */
|
||
|
|
if (unlikely(r->expn < e_min)) {
|
||
|
|
if (flags & BF_FLAG_SUBNORMAL) {
|
||
|
|
/* if inexact, also set the underflow flag */
|
||
|
|
if (ret & BF_ST_INEXACT)
|
||
|
|
ret |= BF_ST_UNDERFLOW;
|
||
|
|
} else {
|
||
|
|
underflow:
|
||
|
|
ret |= BF_ST_UNDERFLOW | BF_ST_INEXACT;
|
||
|
|
bf_set_zero(r, r->sign);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* check overflow */
|
||
|
|
if (unlikely(r->expn > e_max))
|
||
|
|
return bf_set_overflow(r, r->sign, prec1, flags);
|
||
|
|
|
||
|
|
/* keep the bits starting at 'prec - 1' */
|
||
|
|
bit_pos = l * LIMB_BITS - 1 - (prec - 1);
|
||
|
|
i = bit_pos >> LIMB_LOG2_BITS;
|
||
|
|
if (i >= 0) {
|
||
|
|
shift = bit_pos & (LIMB_BITS - 1);
|
||
|
|
if (shift != 0)
|
||
|
|
r->tab[i] &= limb_mask(shift, LIMB_BITS - 1);
|
||
|
|
} else {
|
||
|
|
i = 0;
|
||
|
|
}
|
||
|
|
/* remove trailing zeros */
|
||
|
|
while (r->tab[i] == 0)
|
||
|
|
i++;
|
||
|
|
if (i > 0) {
|
||
|
|
l -= i;
|
||
|
|
memmove(r->tab, r->tab + i, l * sizeof(limb_t));
|
||
|
|
}
|
||
|
|
bf_resize(r, l);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* 'r' must be a finite number */
|
||
|
|
int bf_normalize_and_round(bf_t *r, limb_t prec1, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
limb_t l, v, a;
|
||
|
|
int shift, ret;
|
||
|
|
slimb_t i;
|
||
|
|
|
||
|
|
// bf_print_str("bf_renorm", r);
|
||
|
|
l = r->len;
|
||
|
|
while (l > 0 && r->tab[l - 1] == 0)
|
||
|
|
l--;
|
||
|
|
if (l == 0) {
|
||
|
|
/* zero */
|
||
|
|
r->expn = BF_EXP_ZERO;
|
||
|
|
bf_resize(r, 0);
|
||
|
|
ret = 0;
|
||
|
|
} else {
|
||
|
|
r->expn -= (r->len - l) * LIMB_BITS;
|
||
|
|
/* shift to have the MSB set to '1' */
|
||
|
|
v = r->tab[l - 1];
|
||
|
|
shift = clz(v);
|
||
|
|
if (shift != 0) {
|
||
|
|
v = 0;
|
||
|
|
for(i = 0; i < l; i++) {
|
||
|
|
a = r->tab[i];
|
||
|
|
r->tab[i] = (a << shift) | (v >> (LIMB_BITS - shift));
|
||
|
|
v = a;
|
||
|
|
}
|
||
|
|
r->expn -= shift;
|
||
|
|
}
|
||
|
|
ret = __bf_round(r, prec1, flags, l);
|
||
|
|
}
|
||
|
|
// bf_print_str("r_final", r);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* return true if rounding can be done at precision 'prec' assuming
|
||
|
|
the exact result r is such that |r-a| <= 2^(EXP(a)-k). */
|
||
|
|
/* XXX: check the case where the exponent would be incremented by the
|
||
|
|
rounding */
|
||
|
|
int bf_can_round(const bf_t *a, slimb_t prec, bf_rnd_t rnd_mode, slimb_t k)
|
||
|
|
{
|
||
|
|
BOOL is_rndn;
|
||
|
|
slimb_t bit_pos, n;
|
||
|
|
limb_t bit;
|
||
|
|
|
||
|
|
if (a->expn == BF_EXP_INF || a->expn == BF_EXP_NAN)
|
||
|
|
return FALSE;
|
||
|
|
if (rnd_mode == BF_RNDF) {
|
||
|
|
return (k >= (prec + 1));
|
||
|
|
}
|
||
|
|
if (a->expn == BF_EXP_ZERO)
|
||
|
|
return FALSE;
|
||
|
|
is_rndn = (rnd_mode == BF_RNDN || rnd_mode == BF_RNDNA ||
|
||
|
|
rnd_mode == BF_RNDNU);
|
||
|
|
if (k < (prec + 2))
|
||
|
|
return FALSE;
|
||
|
|
bit_pos = a->len * LIMB_BITS - 1 - prec;
|
||
|
|
n = k - prec;
|
||
|
|
/* bit pattern for RNDN or RNDNA: 0111.. or 1000...
|
||
|
|
for other rounding modes: 000... or 111...
|
||
|
|
*/
|
||
|
|
bit = get_bit(a->tab, a->len, bit_pos);
|
||
|
|
bit_pos--;
|
||
|
|
n--;
|
||
|
|
bit ^= is_rndn;
|
||
|
|
/* XXX: slow, but a few iterations on average */
|
||
|
|
while (n != 0) {
|
||
|
|
if (get_bit(a->tab, a->len, bit_pos) != bit)
|
||
|
|
return TRUE;
|
||
|
|
bit_pos--;
|
||
|
|
n--;
|
||
|
|
}
|
||
|
|
return FALSE;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_round(bf_t *r, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
if (r->len == 0)
|
||
|
|
return 0;
|
||
|
|
return __bf_round(r, prec, flags, r->len);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* for debugging */
|
||
|
|
static unused void dump_limbs(const char *str, const limb_t *tab, limb_t n)
|
||
|
|
{
|
||
|
|
limb_t i;
|
||
|
|
printf("%s: len=%" PRId_LIMB "\n", str, n);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
printf("%" PRId_LIMB ": " FMT_LIMB "\n",
|
||
|
|
i, tab[i]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* for debugging */
|
||
|
|
void bf_print_str(const char *str, const bf_t *a)
|
||
|
|
{
|
||
|
|
slimb_t i;
|
||
|
|
printf("%s=", str);
|
||
|
|
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
printf("NaN");
|
||
|
|
} else {
|
||
|
|
if (a->sign)
|
||
|
|
putchar('-');
|
||
|
|
if (a->expn == BF_EXP_ZERO) {
|
||
|
|
putchar('0');
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
printf("Inf");
|
||
|
|
} else {
|
||
|
|
printf("0x0.");
|
||
|
|
for(i = a->len - 1; i >= 0; i--)
|
||
|
|
printf(FMT_LIMB, a->tab[i]);
|
||
|
|
printf("p%" PRId_LIMB, a->expn);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
printf("\n");
|
||
|
|
}
|
||
|
|
|
||
|
|
/* compare the absolute value of 'a' and 'b'. Return < 0 if a < b, 0
|
||
|
|
if a = b and > 0 otherwise. */
|
||
|
|
int bf_cmpu(const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
slimb_t i;
|
||
|
|
limb_t len, v1, v2;
|
||
|
|
|
||
|
|
if (a->expn != b->expn) {
|
||
|
|
if (a->expn < b->expn)
|
||
|
|
return -1;
|
||
|
|
else
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
len = bf_max(a->len, b->len);
|
||
|
|
for(i = len - 1; i >= 0; i--) {
|
||
|
|
v1 = get_limbz(a, a->len - len + i);
|
||
|
|
v2 = get_limbz(b, b->len - len + i);
|
||
|
|
if (v1 != v2) {
|
||
|
|
if (v1 < v2)
|
||
|
|
return -1;
|
||
|
|
else
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Full order: -0 < 0, NaN == NaN and NaN is larger than all other numbers */
|
||
|
|
int bf_cmp_full(const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
int res;
|
||
|
|
|
||
|
|
if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) {
|
||
|
|
if (a->expn == b->expn)
|
||
|
|
res = 0;
|
||
|
|
else if (a->expn == BF_EXP_NAN)
|
||
|
|
res = 1;
|
||
|
|
else
|
||
|
|
res = -1;
|
||
|
|
} else if (a->sign != b->sign) {
|
||
|
|
res = 1 - 2 * a->sign;
|
||
|
|
} else {
|
||
|
|
res = bf_cmpu(a, b);
|
||
|
|
if (a->sign)
|
||
|
|
res = -res;
|
||
|
|
}
|
||
|
|
return res;
|
||
|
|
}
|
||
|
|
|
||
|
|
#define BF_CMP_EQ 1
|
||
|
|
#define BF_CMP_LT 2
|
||
|
|
#define BF_CMP_LE 3
|
||
|
|
|
||
|
|
static int bf_cmp(const bf_t *a, const bf_t *b, int op)
|
||
|
|
{
|
||
|
|
BOOL is_both_zero;
|
||
|
|
int res;
|
||
|
|
|
||
|
|
if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN)
|
||
|
|
return 0;
|
||
|
|
if (a->sign != b->sign) {
|
||
|
|
is_both_zero = (a->expn == BF_EXP_ZERO && b->expn == BF_EXP_ZERO);
|
||
|
|
if (is_both_zero) {
|
||
|
|
return op & BF_CMP_EQ;
|
||
|
|
} else if (op & BF_CMP_LT) {
|
||
|
|
return a->sign;
|
||
|
|
} else {
|
||
|
|
return FALSE;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
res = bf_cmpu(a, b);
|
||
|
|
if (res == 0) {
|
||
|
|
return op & BF_CMP_EQ;
|
||
|
|
} else if (op & BF_CMP_LT) {
|
||
|
|
return (res < 0) ^ a->sign;
|
||
|
|
} else {
|
||
|
|
return FALSE;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_cmp_eq(const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
return bf_cmp(a, b, BF_CMP_EQ);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_cmp_le(const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
return bf_cmp(a, b, BF_CMP_LE);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_cmp_lt(const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
return bf_cmp(a, b, BF_CMP_LT);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Compute the number of bits 'n' matching the pattern:
|
||
|
|
a= X1000..0
|
||
|
|
b= X0111..1
|
||
|
|
|
||
|
|
When computing a-b, the result will have at least n leading zero
|
||
|
|
bits.
|
||
|
|
|
||
|
|
Precondition: a > b and a.expn - b.expn = 0 or 1
|
||
|
|
*/
|
||
|
|
static limb_t count_cancelled_bits(const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
slimb_t bit_offset, b_offset, n;
|
||
|
|
int p, p1;
|
||
|
|
limb_t v1, v2, mask;
|
||
|
|
|
||
|
|
bit_offset = a->len * LIMB_BITS - 1;
|
||
|
|
b_offset = (b->len - a->len) * LIMB_BITS - (LIMB_BITS - 1) +
|
||
|
|
a->expn - b->expn;
|
||
|
|
n = 0;
|
||
|
|
|
||
|
|
/* first search the equals bits */
|
||
|
|
for(;;) {
|
||
|
|
v1 = get_limbz(a, bit_offset >> LIMB_LOG2_BITS);
|
||
|
|
v2 = get_bits(b->tab, b->len, bit_offset + b_offset);
|
||
|
|
// printf("v1=" FMT_LIMB " v2=" FMT_LIMB "\n", v1, v2);
|
||
|
|
if (v1 != v2)
|
||
|
|
break;
|
||
|
|
n += LIMB_BITS;
|
||
|
|
bit_offset -= LIMB_BITS;
|
||
|
|
}
|
||
|
|
/* find the position of the first different bit */
|
||
|
|
p = clz(v1 ^ v2) + 1;
|
||
|
|
n += p;
|
||
|
|
/* then search for '0' in a and '1' in b */
|
||
|
|
p = LIMB_BITS - p;
|
||
|
|
if (p > 0) {
|
||
|
|
/* search in the trailing p bits of v1 and v2 */
|
||
|
|
mask = limb_mask(0, p - 1);
|
||
|
|
p1 = bf_min(clz(v1 & mask), clz((~v2) & mask)) - (LIMB_BITS - p);
|
||
|
|
n += p1;
|
||
|
|
if (p1 != p)
|
||
|
|
goto done;
|
||
|
|
}
|
||
|
|
bit_offset -= LIMB_BITS;
|
||
|
|
for(;;) {
|
||
|
|
v1 = get_limbz(a, bit_offset >> LIMB_LOG2_BITS);
|
||
|
|
v2 = get_bits(b->tab, b->len, bit_offset + b_offset);
|
||
|
|
// printf("v1=" FMT_LIMB " v2=" FMT_LIMB "\n", v1, v2);
|
||
|
|
if (v1 != 0 || v2 != -1) {
|
||
|
|
/* different: count the matching bits */
|
||
|
|
p1 = bf_min(clz(v1), clz(~v2));
|
||
|
|
n += p1;
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
n += LIMB_BITS;
|
||
|
|
bit_offset -= LIMB_BITS;
|
||
|
|
}
|
||
|
|
done:
|
||
|
|
return n;
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_add_internal(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags, int b_neg)
|
||
|
|
{
|
||
|
|
const bf_t *tmp;
|
||
|
|
int is_sub, ret, cmp_res, a_sign, b_sign;
|
||
|
|
|
||
|
|
a_sign = a->sign;
|
||
|
|
b_sign = b->sign ^ b_neg;
|
||
|
|
is_sub = a_sign ^ b_sign;
|
||
|
|
cmp_res = bf_cmpu(a, b);
|
||
|
|
if (cmp_res < 0) {
|
||
|
|
tmp = a;
|
||
|
|
a = b;
|
||
|
|
b = tmp;
|
||
|
|
a_sign = b_sign; /* b_sign is never used later */
|
||
|
|
}
|
||
|
|
/* abs(a) >= abs(b) */
|
||
|
|
if (cmp_res == 0 && is_sub && a->expn < BF_EXP_INF) {
|
||
|
|
/* zero result */
|
||
|
|
bf_set_zero(r, (flags & BF_RND_MASK) == BF_RNDD);
|
||
|
|
ret = 0;
|
||
|
|
} else if (a->len == 0 || b->len == 0) {
|
||
|
|
ret = 0;
|
||
|
|
if (a->expn >= BF_EXP_INF) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
/* at least one operand is NaN */
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (b->expn == BF_EXP_INF && is_sub) {
|
||
|
|
/* infinities with different signs */
|
||
|
|
bf_set_nan(r);
|
||
|
|
ret = BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, a_sign);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
/* at least one zero and not subtract */
|
||
|
|
bf_set(r, a);
|
||
|
|
r->sign = a_sign;
|
||
|
|
goto renorm;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
slimb_t d, a_offset, b_bit_offset, i, cancelled_bits;
|
||
|
|
limb_t carry, v1, v2, u, r_len, carry1, precl, tot_len, z, sub_mask;
|
||
|
|
|
||
|
|
r->sign = a_sign;
|
||
|
|
r->expn = a->expn;
|
||
|
|
d = a->expn - b->expn;
|
||
|
|
/* must add more precision for the leading cancelled bits in
|
||
|
|
subtraction */
|
||
|
|
if (is_sub) {
|
||
|
|
if (d <= 1)
|
||
|
|
cancelled_bits = count_cancelled_bits(a, b);
|
||
|
|
else
|
||
|
|
cancelled_bits = 1;
|
||
|
|
} else {
|
||
|
|
cancelled_bits = 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* add two extra bits for rounding */
|
||
|
|
precl = (cancelled_bits + prec + 2 + LIMB_BITS - 1) / LIMB_BITS;
|
||
|
|
tot_len = bf_max(a->len, b->len + (d + LIMB_BITS - 1) / LIMB_BITS);
|
||
|
|
r_len = bf_min(precl, tot_len);
|
||
|
|
bf_resize(r, r_len);
|
||
|
|
a_offset = a->len - r_len;
|
||
|
|
b_bit_offset = (b->len - r_len) * LIMB_BITS + d;
|
||
|
|
|
||
|
|
/* compute the bits before for the rounding */
|
||
|
|
carry = is_sub;
|
||
|
|
z = 0;
|
||
|
|
sub_mask = -is_sub;
|
||
|
|
i = r_len - tot_len;
|
||
|
|
while (i < 0) {
|
||
|
|
slimb_t ap, bp;
|
||
|
|
BOOL inflag;
|
||
|
|
|
||
|
|
ap = a_offset + i;
|
||
|
|
bp = b_bit_offset + i * LIMB_BITS;
|
||
|
|
inflag = FALSE;
|
||
|
|
if (ap >= 0 && ap < a->len) {
|
||
|
|
v1 = a->tab[ap];
|
||
|
|
inflag = TRUE;
|
||
|
|
} else {
|
||
|
|
v1 = 0;
|
||
|
|
}
|
||
|
|
if (bp + LIMB_BITS > 0 && bp < (slimb_t)(b->len * LIMB_BITS)) {
|
||
|
|
v2 = get_bits(b->tab, b->len, bp);
|
||
|
|
inflag = TRUE;
|
||
|
|
} else {
|
||
|
|
v2 = 0;
|
||
|
|
}
|
||
|
|
if (!inflag) {
|
||
|
|
/* outside 'a' and 'b': go directly to the next value
|
||
|
|
inside a or b so that the running time does not
|
||
|
|
depend on the exponent difference */
|
||
|
|
i = 0;
|
||
|
|
if (ap < 0)
|
||
|
|
i = bf_min(i, -a_offset);
|
||
|
|
/* b_bit_offset + i * LIMB_BITS + LIMB_BITS >= 1
|
||
|
|
equivalent to
|
||
|
|
i >= ceil(-b_bit_offset + 1 - LIMB_BITS) / LIMB_BITS)
|
||
|
|
*/
|
||
|
|
if (bp + LIMB_BITS <= 0)
|
||
|
|
i = bf_min(i, (-b_bit_offset) >> LIMB_LOG2_BITS);
|
||
|
|
} else {
|
||
|
|
i++;
|
||
|
|
}
|
||
|
|
v2 ^= sub_mask;
|
||
|
|
u = v1 + v2;
|
||
|
|
carry1 = u < v1;
|
||
|
|
u += carry;
|
||
|
|
carry = (u < carry) | carry1;
|
||
|
|
z |= u;
|
||
|
|
}
|
||
|
|
/* and the result */
|
||
|
|
for(i = 0; i < r_len; i++) {
|
||
|
|
v1 = get_limbz(a, a_offset + i);
|
||
|
|
v2 = get_bits(b->tab, b->len, b_bit_offset + i * LIMB_BITS);
|
||
|
|
v2 ^= sub_mask;
|
||
|
|
u = v1 + v2;
|
||
|
|
carry1 = u < v1;
|
||
|
|
u += carry;
|
||
|
|
carry = (u < carry) | carry1;
|
||
|
|
r->tab[i] = u;
|
||
|
|
}
|
||
|
|
/* set the extra bits for the rounding */
|
||
|
|
r->tab[0] |= (z != 0);
|
||
|
|
|
||
|
|
/* carry is only possible in add case */
|
||
|
|
if (!is_sub && carry) {
|
||
|
|
bf_resize(r, r_len + 1);
|
||
|
|
r->tab[r_len] = 1;
|
||
|
|
r->expn += LIMB_BITS;
|
||
|
|
}
|
||
|
|
renorm:
|
||
|
|
ret = bf_normalize_and_round(r, prec, flags);
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
static int __bf_add(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_add_internal(r, a, b, prec, flags, 0);
|
||
|
|
}
|
||
|
|
|
||
|
|
static int __bf_sub(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_add_internal(r, a, b, prec, flags, 1);
|
||
|
|
}
|
||
|
|
|
||
|
|
static limb_t mp_add(limb_t *res, const limb_t *op1, const limb_t *op2,
|
||
|
|
limb_t n, limb_t carry)
|
||
|
|
{
|
||
|
|
slimb_t i;
|
||
|
|
limb_t k, a, v, k1;
|
||
|
|
|
||
|
|
k = carry;
|
||
|
|
for(i=0;i<n;i++) {
|
||
|
|
v = op1[i];
|
||
|
|
a = v + op2[i];
|
||
|
|
k1 = a < v;
|
||
|
|
a = a + k;
|
||
|
|
k = (a < k) | k1;
|
||
|
|
res[i] = a;
|
||
|
|
}
|
||
|
|
return k;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* tabr[] += taba[] * b, return the high word. */
|
||
|
|
static limb_t mp_add_mul1(limb_t *tabr, const limb_t *taba, limb_t n,
|
||
|
|
limb_t b)
|
||
|
|
{
|
||
|
|
limb_t i, l;
|
||
|
|
dlimb_t t;
|
||
|
|
|
||
|
|
l = 0;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
t = (dlimb_t)taba[i] * (dlimb_t)b + l + tabr[i];
|
||
|
|
tabr[i] = t;
|
||
|
|
l = t >> LIMB_BITS;
|
||
|
|
}
|
||
|
|
return l;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* tabr[] -= taba[] * b. Return the value to substract to the high
|
||
|
|
word. */
|
||
|
|
static limb_t mp_sub_mul1(limb_t *tabr, const limb_t *taba, limb_t n,
|
||
|
|
limb_t b)
|
||
|
|
{
|
||
|
|
limb_t i, l;
|
||
|
|
dlimb_t t;
|
||
|
|
|
||
|
|
l = 0;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
t = tabr[i] - (dlimb_t)taba[i] * (dlimb_t)b - l;
|
||
|
|
tabr[i] = t;
|
||
|
|
l = -(t >> LIMB_BITS);
|
||
|
|
}
|
||
|
|
return l;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* WARNING: d must be >= 2^(LIMB_BITS-1) */
|
||
|
|
static inline limb_t udiv1norm_init(limb_t d)
|
||
|
|
{
|
||
|
|
limb_t a0, a1;
|
||
|
|
a1 = -d - 1;
|
||
|
|
a0 = -1;
|
||
|
|
return (((dlimb_t)a1 << LIMB_BITS) | a0) / d;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* return the quotient and the remainder in '*pr'of 'a1*2^LIMB_BITS+a0
|
||
|
|
/ d' with 0 <= a1 < d. */
|
||
|
|
static inline limb_t udiv1norm(limb_t *pr, limb_t a1, limb_t a0,
|
||
|
|
limb_t d, limb_t d_inv)
|
||
|
|
{
|
||
|
|
limb_t n1m, n_adj, q, r, ah;
|
||
|
|
dlimb_t a;
|
||
|
|
n1m = ((slimb_t)a0 >> (LIMB_BITS - 1));
|
||
|
|
n_adj = a0 + (n1m & d);
|
||
|
|
a = (dlimb_t)d_inv * (a1 - n1m) + n_adj;
|
||
|
|
q = (a >> LIMB_BITS) + a1;
|
||
|
|
/* compute a - q * r and update q so that the remainder is\
|
||
|
|
between 0 and d - 1 */
|
||
|
|
a = ((dlimb_t)a1 << LIMB_BITS) | a0;
|
||
|
|
a = a - (dlimb_t)q * d - d;
|
||
|
|
ah = a >> LIMB_BITS;
|
||
|
|
q += 1 + ah;
|
||
|
|
r = (limb_t)a + (ah & d);
|
||
|
|
*pr = r;
|
||
|
|
return q;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* b must be >= 1 << (LIMB_BITS - 1) */
|
||
|
|
static limb_t mp_div1norm(limb_t *tabr, const limb_t *taba, limb_t n,
|
||
|
|
limb_t b, limb_t r)
|
||
|
|
{
|
||
|
|
slimb_t i;
|
||
|
|
|
||
|
|
if (n >= 3) {
|
||
|
|
limb_t b_inv;
|
||
|
|
b_inv = udiv1norm_init(b);
|
||
|
|
for(i = n - 1; i >= 0; i--) {
|
||
|
|
tabr[i] = udiv1norm(&r, r, taba[i], b, b_inv);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
dlimb_t a1;
|
||
|
|
for(i = n - 1; i >= 0; i--) {
|
||
|
|
a1 = ((dlimb_t)r << LIMB_BITS) | taba[i];
|
||
|
|
tabr[i] = a1 / b;
|
||
|
|
r = a1 % b;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* base case division: divides taba[0..na-1] by tabb[0..nb-1]. tabb[nb
|
||
|
|
- 1] must be >= 1 << (LIMB_BITS - 1). na - nb must be >= 0. 'taba'
|
||
|
|
is modified and contains the remainder (nb limbs). tabq[0..na-nb]
|
||
|
|
contains the quotient. taba[na] is modified. */
|
||
|
|
static void mp_divnorm(limb_t *tabq,
|
||
|
|
limb_t *taba, limb_t na,
|
||
|
|
const limb_t *tabb, limb_t nb)
|
||
|
|
{
|
||
|
|
limb_t r, a, c, q, v, b1, b1_inv, n, dummy_r;
|
||
|
|
slimb_t i;
|
||
|
|
|
||
|
|
b1 = tabb[nb - 1];
|
||
|
|
if (nb == 1) {
|
||
|
|
taba[0] = mp_div1norm(tabq, taba, na, b1, 0);
|
||
|
|
return;
|
||
|
|
}
|
||
|
|
taba[na] = 0;
|
||
|
|
n = na - nb;
|
||
|
|
if (n >= 3)
|
||
|
|
b1_inv = udiv1norm_init(b1);
|
||
|
|
else
|
||
|
|
b1_inv = 0;
|
||
|
|
|
||
|
|
/* XXX: could simplify the first iteration */
|
||
|
|
for(i = n; i >= 0; i--) {
|
||
|
|
if (unlikely(taba[i + nb] >= b1)) {
|
||
|
|
q = -1;
|
||
|
|
} else if (b1_inv) {
|
||
|
|
q = udiv1norm(&dummy_r, taba[i + nb], taba[i + nb - 1], b1, b1_inv);
|
||
|
|
} else {
|
||
|
|
dlimb_t al;
|
||
|
|
al = ((dlimb_t)taba[i + nb] << LIMB_BITS) | taba[i + nb - 1];
|
||
|
|
q = al / b1;
|
||
|
|
r = al % b1;
|
||
|
|
}
|
||
|
|
r = mp_sub_mul1(taba + i, tabb, nb, q);
|
||
|
|
|
||
|
|
v = taba[i + nb];
|
||
|
|
a = v - r;
|
||
|
|
c = (a > v);
|
||
|
|
taba[i + nb] = a;
|
||
|
|
|
||
|
|
if (c != 0) {
|
||
|
|
/* negative result */
|
||
|
|
for(;;) {
|
||
|
|
q--;
|
||
|
|
c = mp_add(taba + i, taba + i, tabb, nb, 0);
|
||
|
|
/* propagate carry and test if positive result */
|
||
|
|
if (c != 0) {
|
||
|
|
if (++taba[i + nb] == 0) {
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
tabq[i] = q;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_mul(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
limb_t i;
|
||
|
|
int ret, r_sign;
|
||
|
|
|
||
|
|
if (a->len < b->len) {
|
||
|
|
const bf_t *tmp = a;
|
||
|
|
a = b;
|
||
|
|
b = tmp;
|
||
|
|
}
|
||
|
|
r_sign = a->sign ^ b->sign;
|
||
|
|
/* here b->len <= a->len */
|
||
|
|
if (b->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
ret = 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF || b->expn == BF_EXP_INF) {
|
||
|
|
if ((a->expn == BF_EXP_INF && b->expn == BF_EXP_ZERO) ||
|
||
|
|
(a->expn == BF_EXP_ZERO && b->expn == BF_EXP_INF)) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
ret = BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, r_sign);
|
||
|
|
ret = 0;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, r_sign);
|
||
|
|
ret = 0;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
bf_t tmp, *r1 = NULL;
|
||
|
|
limb_t a_len, b_len, precl;
|
||
|
|
limb_t *a_tab, *b_tab;
|
||
|
|
|
||
|
|
a_len = a->len;
|
||
|
|
b_len = b->len;
|
||
|
|
|
||
|
|
if ((flags & BF_RND_MASK) == BF_RNDF) {
|
||
|
|
/* faithful rounding does not require using the full inputs */
|
||
|
|
precl = (prec + 2 + LIMB_BITS - 1) / LIMB_BITS;
|
||
|
|
a_len = bf_min(a_len, precl);
|
||
|
|
b_len = bf_min(b_len, precl);
|
||
|
|
}
|
||
|
|
a_tab = a->tab + a->len - a_len;
|
||
|
|
b_tab = b->tab + b->len - b_len;
|
||
|
|
|
||
|
|
#ifdef USE_FFT_MUL
|
||
|
|
if (b_len >= FFT_MUL_THRESHOLD) {
|
||
|
|
int mul_flags = 0;
|
||
|
|
if (r == a)
|
||
|
|
mul_flags |= FFT_MUL_R_OVERLAP_A;
|
||
|
|
if (r == b)
|
||
|
|
mul_flags |= FFT_MUL_R_OVERLAP_B;
|
||
|
|
fft_mul(r, a_tab, a_len, b_tab, b_len, mul_flags);
|
||
|
|
} else
|
||
|
|
#endif
|
||
|
|
{
|
||
|
|
if (r == a || r == b) {
|
||
|
|
bf_init(r->ctx, &tmp);
|
||
|
|
r1 = r;
|
||
|
|
r = &tmp;
|
||
|
|
}
|
||
|
|
bf_resize(r, a_len + b_len);
|
||
|
|
memset(r->tab, 0, sizeof(limb_t) * a_len);
|
||
|
|
for(i = 0; i < b_len; i++) {
|
||
|
|
r->tab[i + a_len] = mp_add_mul1(r->tab + i, a_tab, a_len,
|
||
|
|
b_tab[i]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
r->sign = r_sign;
|
||
|
|
r->expn = a->expn + b->expn;
|
||
|
|
ret = bf_normalize_and_round(r, prec, flags);
|
||
|
|
if (r == &tmp)
|
||
|
|
bf_move(r1, &tmp);
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* multiply 'r' by 2^e */
|
||
|
|
int bf_mul_2exp(bf_t *r, slimb_t e, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
slimb_t e_max;
|
||
|
|
if (r->len == 0)
|
||
|
|
return 0;
|
||
|
|
e_max = ((limb_t)1 << BF_EXP_BITS_MAX) - 1;
|
||
|
|
e = bf_max(e, -e_max);
|
||
|
|
e = bf_min(e, e_max);
|
||
|
|
r->expn += e;
|
||
|
|
return __bf_round(r, prec, flags, r->len);
|
||
|
|
}
|
||
|
|
|
||
|
|
static void bf_recip_rec(bf_t *a, const bf_t *x, limb_t prec1)
|
||
|
|
{
|
||
|
|
bf_t t0;
|
||
|
|
limb_t prec;
|
||
|
|
|
||
|
|
bf_init(a->ctx, &t0);
|
||
|
|
|
||
|
|
if (prec1 <= LIMB_BITS - 3) {
|
||
|
|
limb_t v;
|
||
|
|
/* initial approximation */
|
||
|
|
v = x->tab[x->len - 1];
|
||
|
|
/* 2^(L-1) <= v <= 2^L-1 (L=LIMB_BITS) */
|
||
|
|
v = ((dlimb_t)1 << (2 * LIMB_BITS - 2)) / v;
|
||
|
|
/* 2^(L-2) <= v <= 2^(L-1) */
|
||
|
|
bf_resize(a, 1);
|
||
|
|
a->sign = x->sign;
|
||
|
|
a->expn = 2 - x->expn;
|
||
|
|
if (v == ((limb_t)1 << (LIMB_BITS - 1))) {
|
||
|
|
a->tab[0] = v;
|
||
|
|
} else {
|
||
|
|
a->tab[0] = v << 1;
|
||
|
|
a->expn--;
|
||
|
|
}
|
||
|
|
a->tab[0] &= limb_mask(LIMB_BITS - prec1, LIMB_BITS - 1);
|
||
|
|
} else {
|
||
|
|
/* XXX: prove the added precision */
|
||
|
|
bf_recip_rec(a, x, (prec1 / 2) + 8);
|
||
|
|
prec = prec1 + 8;
|
||
|
|
|
||
|
|
/* a = a + a * (1 - x * a) */
|
||
|
|
bf_mul(&t0, x, a, prec, BF_RNDF);
|
||
|
|
t0.sign ^= 1;
|
||
|
|
bf_add_si(&t0, &t0, 1, prec, BF_RNDF);
|
||
|
|
bf_mul(&t0, &t0, a, prec, BF_RNDF);
|
||
|
|
bf_add(a, a, &t0, prec1, BF_RNDF);
|
||
|
|
}
|
||
|
|
// bf_print_str("r", a);
|
||
|
|
bf_delete(&t0);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Note: only faithful rounding is supported */
|
||
|
|
void bf_recip(bf_t *r, const bf_t *a, limb_t prec)
|
||
|
|
{
|
||
|
|
assert(r != a);
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
bf_set_zero(r, a->sign);
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, a->sign);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
// bf_print_str("a", a);
|
||
|
|
bf_recip_rec(r, a, prec);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* add zero limbs if necessary to have at least precl limbs */
|
||
|
|
static void bf_add_zero_limbs(bf_t *r, limb_t precl)
|
||
|
|
{
|
||
|
|
limb_t l = r->len;
|
||
|
|
if (l < precl) {
|
||
|
|
bf_resize(r, precl);
|
||
|
|
memmove(r->tab + precl - l, r->tab,
|
||
|
|
l * sizeof(limb_t));
|
||
|
|
memset(r->tab, 0, (precl - l) * sizeof(limb_t));
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* set a bit to 1 at bit position >= (precl * LIMB_BITS - 1) */
|
||
|
|
static void bf_or_one(bf_t *r, limb_t precl)
|
||
|
|
{
|
||
|
|
bf_add_zero_limbs(r, precl);
|
||
|
|
r->tab[0] |= 1;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Return e such as a=m*2^e with m odd integer. return 0 if a is zero,
|
||
|
|
Infinite or Nan. */
|
||
|
|
slimb_t bf_get_exp_min(const bf_t *a)
|
||
|
|
{
|
||
|
|
slimb_t i;
|
||
|
|
limb_t v;
|
||
|
|
int k;
|
||
|
|
|
||
|
|
for(i = 0; i < a->len; i++) {
|
||
|
|
v = a->tab[i];
|
||
|
|
if (v != 0) {
|
||
|
|
k = ctz(v);
|
||
|
|
return a->expn - (a->len - i) * LIMB_BITS + k;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* a and b must be finite numbers with a >= 0 and b > 0. 'q' is the
|
||
|
|
integer defined as floor(a/b) and r = a - q * b. */
|
||
|
|
static void bf_tdivremu(bf_t *q, bf_t *r,
|
||
|
|
const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
if (a->expn < b->expn) {
|
||
|
|
bf_set_ui(q, 0);
|
||
|
|
bf_set(r, a);
|
||
|
|
} else {
|
||
|
|
/* for large numbers, use the floating point division in
|
||
|
|
faithful mode */
|
||
|
|
bf_div(q, a, b, bf_max(a->expn - b->expn + 1, 2), BF_RNDF);
|
||
|
|
bf_rint(q, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_mul(r, q, b, BF_PREC_INF, BF_RNDN);
|
||
|
|
bf_sub(r, a, r, BF_PREC_INF, BF_RNDN);
|
||
|
|
if (r->len != 0 && r->sign) {
|
||
|
|
bf_add_si(q, q, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_add(r, r, b, BF_PREC_INF, BF_RNDZ);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static int __bf_div(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
int ret, r_sign;
|
||
|
|
limb_t n, nb, precl;
|
||
|
|
|
||
|
|
r_sign = a->sign ^ b->sign;
|
||
|
|
if (a->expn >= BF_EXP_INF || b->expn >= BF_EXP_INF) {
|
||
|
|
if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF && b->expn == BF_EXP_INF) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
bf_set_inf(r, r_sign);
|
||
|
|
return 0;
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, r_sign);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
} else if (a->expn == BF_EXP_ZERO) {
|
||
|
|
if (b->expn == BF_EXP_ZERO) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, r_sign);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
} else if (b->expn == BF_EXP_ZERO) {
|
||
|
|
bf_set_inf(r, r_sign);
|
||
|
|
return BF_ST_DIVIDE_ZERO;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* number of limbs of the quotient (2 extra bits for rounding) */
|
||
|
|
precl = (prec + 2 + LIMB_BITS - 1) / LIMB_BITS;
|
||
|
|
nb = b->len;
|
||
|
|
n = bf_max(a->len, precl);
|
||
|
|
|
||
|
|
if (nb <= BASECASE_DIV_THRESHOLD_B ||
|
||
|
|
(slimb_t)n <= (BASECASE_DIV_THRESHOLD_Q - 1)) {
|
||
|
|
limb_t *taba, na, i;
|
||
|
|
slimb_t d;
|
||
|
|
|
||
|
|
na = n + nb;
|
||
|
|
taba = bf_malloc(r->ctx, (na + 1) * sizeof(limb_t));
|
||
|
|
d = na - a->len;
|
||
|
|
memset(taba, 0, d * sizeof(limb_t));
|
||
|
|
memcpy(taba + d, a->tab, a->len * sizeof(limb_t));
|
||
|
|
bf_resize(r, n + 1);
|
||
|
|
mp_divnorm(r->tab, taba, na, b->tab, nb);
|
||
|
|
|
||
|
|
/* see if non zero remainder */
|
||
|
|
for(i = 0; i < nb; i++) {
|
||
|
|
if (taba[i] != 0) {
|
||
|
|
r->tab[0] |= 1;
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
bf_free(r->ctx, taba);
|
||
|
|
r->expn = a->expn - b->expn + LIMB_BITS;
|
||
|
|
r->sign = r_sign;
|
||
|
|
ret = bf_normalize_and_round(r, prec, flags);
|
||
|
|
} else if ((flags & BF_RND_MASK) == BF_RNDF) {
|
||
|
|
bf_t b_inv;
|
||
|
|
bf_init(r->ctx, &b_inv);
|
||
|
|
bf_recip(&b_inv, b, prec + 3);
|
||
|
|
ret = bf_mul(r, a, &b_inv, prec, flags);
|
||
|
|
bf_delete(&b_inv);
|
||
|
|
} else {
|
||
|
|
bf_t a1_s, *a1 = &a1_s;
|
||
|
|
bf_t b1_s, *b1 = &b1_s;
|
||
|
|
bf_t rem_s, *rem = &rem_s;
|
||
|
|
|
||
|
|
/* convert the mantissa of 'a' and 'b' to integers and generate
|
||
|
|
a quotient with at least prec + 2 bits */
|
||
|
|
a1->expn = (n + nb) * LIMB_BITS;
|
||
|
|
a1->tab = a->tab;
|
||
|
|
a1->len = a->len;
|
||
|
|
a1->sign = 0;
|
||
|
|
|
||
|
|
b1->expn = nb * LIMB_BITS;
|
||
|
|
b1->tab = b->tab;
|
||
|
|
b1->len = nb;
|
||
|
|
b1->sign = 0;
|
||
|
|
|
||
|
|
bf_init(r->ctx, rem);
|
||
|
|
bf_tdivremu(r, rem, a1, b1);
|
||
|
|
/* the remainder is not zero: put it in the rounding bits */
|
||
|
|
if (rem->len != 0) {
|
||
|
|
bf_or_one(r, precl);
|
||
|
|
}
|
||
|
|
bf_delete(rem);
|
||
|
|
r->expn += a->expn - b->expn - n * LIMB_BITS;
|
||
|
|
r->sign = r_sign;
|
||
|
|
ret = bf_round(r, prec, flags);
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* division and remainder.
|
||
|
|
|
||
|
|
rnd_mode is the rounding mode for the quotient. The additional
|
||
|
|
rounding mode BF_RND_EUCLIDIAN is supported.
|
||
|
|
|
||
|
|
'q' is an integer. 'r' is rounded with prec and flags (prec can be
|
||
|
|
BF_PREC_INF).
|
||
|
|
*/
|
||
|
|
int bf_divrem(bf_t *q, bf_t *r, const bf_t *a, const bf_t *b,
|
||
|
|
limb_t prec, bf_flags_t flags, int rnd_mode)
|
||
|
|
{
|
||
|
|
bf_t a1_s, *a1 = &a1_s;
|
||
|
|
bf_t b1_s, *b1 = &b1_s;
|
||
|
|
int q_sign;
|
||
|
|
BOOL is_ceil, is_rndn;
|
||
|
|
|
||
|
|
assert(q != a && q != b);
|
||
|
|
assert(r != a && r != b);
|
||
|
|
assert(q != r);
|
||
|
|
|
||
|
|
if (a->len == 0 || b->len == 0) {
|
||
|
|
bf_set_zero(q, 0);
|
||
|
|
if (a->expn == BF_EXP_NAN || b->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF || b->expn == BF_EXP_ZERO) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
bf_set(r, a);
|
||
|
|
return bf_round(r, prec, flags);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
q_sign = a->sign ^ b->sign;
|
||
|
|
is_rndn = (rnd_mode == BF_RNDN || rnd_mode == BF_RNDNA ||
|
||
|
|
rnd_mode == BF_RNDNU);
|
||
|
|
switch(rnd_mode) {
|
||
|
|
default:
|
||
|
|
case BF_RNDZ:
|
||
|
|
case BF_RNDN:
|
||
|
|
case BF_RNDNA:
|
||
|
|
is_ceil = FALSE;
|
||
|
|
break;
|
||
|
|
case BF_RNDD:
|
||
|
|
is_ceil = q_sign;
|
||
|
|
break;
|
||
|
|
case BF_RNDU:
|
||
|
|
is_ceil = q_sign ^ 1;
|
||
|
|
break;
|
||
|
|
case BF_DIVREM_EUCLIDIAN:
|
||
|
|
is_ceil = a->sign;
|
||
|
|
break;
|
||
|
|
case BF_RNDNU:
|
||
|
|
/* XXX: unsupported yet */
|
||
|
|
abort();
|
||
|
|
}
|
||
|
|
|
||
|
|
a1->expn = a->expn;
|
||
|
|
a1->tab = a->tab;
|
||
|
|
a1->len = a->len;
|
||
|
|
a1->sign = 0;
|
||
|
|
|
||
|
|
b1->expn = b->expn;
|
||
|
|
b1->tab = b->tab;
|
||
|
|
b1->len = b->len;
|
||
|
|
b1->sign = 0;
|
||
|
|
|
||
|
|
/* XXX: could improve to avoid having a large 'q' */
|
||
|
|
bf_tdivremu(q, r, a1, b1);
|
||
|
|
|
||
|
|
if (r->len != 0) {
|
||
|
|
if (is_rndn) {
|
||
|
|
int res;
|
||
|
|
b1->expn--;
|
||
|
|
res = bf_cmpu(r, b1);
|
||
|
|
b1->expn++;
|
||
|
|
if (res > 0 ||
|
||
|
|
(res == 0 &&
|
||
|
|
(rnd_mode == BF_RNDNA ||
|
||
|
|
get_bit(q->tab, q->len, q->len * LIMB_BITS - q->expn)))) {
|
||
|
|
goto do_sub_r;
|
||
|
|
}
|
||
|
|
} else if (is_ceil) {
|
||
|
|
do_sub_r:
|
||
|
|
bf_add_si(q, q, 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_sub(r, r, b1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
r->sign ^= a->sign;
|
||
|
|
q->sign = q_sign;
|
||
|
|
return bf_round(r, prec, flags);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_fmod(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t q_s, *q = &q_s;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
bf_init(r->ctx, q);
|
||
|
|
ret = bf_divrem(q, r, a, b, prec, flags, BF_RNDZ);
|
||
|
|
bf_delete(q);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_remainder(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t q_s, *q = &q_s;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
bf_init(r->ctx, q);
|
||
|
|
ret = bf_divrem(q, r, a, b, prec, flags, BF_RNDN);
|
||
|
|
bf_delete(q);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline int bf_get_limb(slimb_t *pres, const bf_t *a, int flags)
|
||
|
|
{
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
return bf_get_int32(pres, a, flags);
|
||
|
|
#else
|
||
|
|
return bf_get_int64(pres, a, flags);
|
||
|
|
#endif
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_remquo(slimb_t *pq, bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t q_s, *q = &q_s;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
bf_init(r->ctx, q);
|
||
|
|
ret = bf_divrem(q, r, a, b, prec, flags, BF_RNDN);
|
||
|
|
bf_get_limb(pq, q, BF_GET_INT_MOD);
|
||
|
|
bf_delete(q);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
static unused inline limb_t mul_mod(limb_t a, limb_t b, limb_t m)
|
||
|
|
{
|
||
|
|
dlimb_t t;
|
||
|
|
t = (dlimb_t)a * (dlimb_t)b;
|
||
|
|
return t % m;
|
||
|
|
}
|
||
|
|
|
||
|
|
#if defined(USE_MUL_CHECK)
|
||
|
|
static limb_t mp_mod1(const limb_t *tab, limb_t n, limb_t m, limb_t r)
|
||
|
|
{
|
||
|
|
slimb_t i;
|
||
|
|
dlimb_t t;
|
||
|
|
|
||
|
|
for(i = n - 1; i >= 0; i--) {
|
||
|
|
t = ((dlimb_t)r << LIMB_BITS) | tab[i];
|
||
|
|
r = t % m;
|
||
|
|
}
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
|
||
|
|
/* (128.0 / sqrt((i + 64) / 256)) & 0xff */
|
||
|
|
static const uint8_t rsqrt_table[192] = {
|
||
|
|
0,254,252,250,248,247,245,243,241,240,238,236,235,233,232,230,
|
||
|
|
229,228,226,225,223,222,221,220,218,217,216,215,214,212,211,210,
|
||
|
|
209,208,207,206,205,204,203,202,201,200,199,198,197,196,195,194,
|
||
|
|
194,193,192,191,190,189,189,188,187,186,185,185,184,183,182,182,
|
||
|
|
181,180,180,179,178,178,177,176,176,175,174,174,173,172,172,171,
|
||
|
|
171,170,169,169,168,168,167,167,166,166,165,164,164,163,163,162,
|
||
|
|
162,161,161,160,160,159,159,158,158,158,157,157,156,156,155,155,
|
||
|
|
154,154,154,153,153,152,152,151,151,151,150,150,149,149,149,148,
|
||
|
|
148,147,147,147,146,146,146,145,145,144,144,144,143,143,143,142,
|
||
|
|
142,142,141,141,141,140,140,140,139,139,139,138,138,138,137,137,
|
||
|
|
137,137,136,136,136,135,135,135,134,134,134,134,133,133,133,132,
|
||
|
|
132,132,132,131,131,131,131,130,130,130,130,129,129,129,129,128,
|
||
|
|
};
|
||
|
|
|
||
|
|
static void __bf_rsqrt(bf_t *a, const bf_t *x, limb_t prec1)
|
||
|
|
{
|
||
|
|
bf_t t0;
|
||
|
|
limb_t prec;
|
||
|
|
|
||
|
|
if (prec1 <= 7) {
|
||
|
|
slimb_t e;
|
||
|
|
limb_t v;
|
||
|
|
/* initial approximation using 8 mantissa bits */
|
||
|
|
v = x->tab[x->len - 1];
|
||
|
|
e = x->expn;
|
||
|
|
if (e & 1) {
|
||
|
|
v >>= 1;
|
||
|
|
e++;
|
||
|
|
}
|
||
|
|
v = rsqrt_table[(v >> (LIMB_BITS - 8)) - 64];
|
||
|
|
e = 1 - (e / 2);
|
||
|
|
if (v == 0) {
|
||
|
|
v = 128; /* real table value is 256 */
|
||
|
|
e++;
|
||
|
|
}
|
||
|
|
bf_resize(a, 1);
|
||
|
|
a->tab[0] = (v << (LIMB_BITS - 8)) &
|
||
|
|
limb_mask(LIMB_BITS - prec1, LIMB_BITS - 1);
|
||
|
|
a->expn = e;
|
||
|
|
a->sign = 0;
|
||
|
|
} else {
|
||
|
|
/* XXX: prove rounding */
|
||
|
|
__bf_rsqrt(a, x, (prec1 / 2) + 2);
|
||
|
|
|
||
|
|
prec = prec1 + 3;
|
||
|
|
|
||
|
|
/* x' = x + (x/2) * (1 - a * x^2) */
|
||
|
|
bf_init(a->ctx, &t0);
|
||
|
|
|
||
|
|
bf_mul(&t0, a, a, prec, BF_RNDF);
|
||
|
|
bf_mul(&t0, &t0, x, prec, BF_RNDF);
|
||
|
|
t0.sign ^= 1;
|
||
|
|
bf_add_si(&t0, &t0, 1, prec, BF_RNDF);
|
||
|
|
bf_mul(&t0, &t0, a, prec, BF_RNDF);
|
||
|
|
if (t0.len != 0)
|
||
|
|
t0.expn--;
|
||
|
|
bf_add(a, a, &t0, prec1, BF_RNDF);
|
||
|
|
|
||
|
|
bf_delete(&t0);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static int __bf_sqrt(bf_t *x, const bf_t *a, limb_t prec1, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t t0, t1;
|
||
|
|
limb_t prec;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
/* XXX: prove rounding */
|
||
|
|
__bf_rsqrt(x, a, (prec1 / 2) + 2);
|
||
|
|
prec = prec1 + 3;
|
||
|
|
|
||
|
|
/* x' = a * x + (x/2) * (a - (a * x)^2) */
|
||
|
|
|
||
|
|
bf_init(x->ctx, &t0);
|
||
|
|
bf_init(x->ctx, &t1);
|
||
|
|
bf_mul(&t1, x, a, prec, BF_RNDF);
|
||
|
|
bf_mul(&t0, &t1, &t1, prec, BF_RNDF);
|
||
|
|
t0.sign ^= 1;
|
||
|
|
bf_add(&t0, &t0, a, prec, BF_RNDF);
|
||
|
|
bf_mul(&t0, &t0, x, prec, BF_RNDF);
|
||
|
|
if (t0.len != 0)
|
||
|
|
t0.expn--;
|
||
|
|
ret = bf_add(x, &t1, &t0, prec1, flags);
|
||
|
|
|
||
|
|
bf_delete(&t0);
|
||
|
|
bf_delete(&t1);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Note: only faithful rounding is supported */
|
||
|
|
void bf_rsqrt(bf_t *r, const bf_t *a, limb_t prec)
|
||
|
|
{
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN ||
|
||
|
|
(a->sign && a->expn != BF_EXP_ZERO)) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
bf_set_zero(r, a->sign);
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, 0);
|
||
|
|
}
|
||
|
|
} else if (a->sign) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else {
|
||
|
|
// bf_print_str("a", a);
|
||
|
|
__bf_rsqrt(r, a, prec);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* return floor(sqrt(a)) */
|
||
|
|
static limb_t bf_isqrt(limb_t a)
|
||
|
|
{
|
||
|
|
unsigned int l;
|
||
|
|
limb_t u, s;
|
||
|
|
|
||
|
|
if (a == 0)
|
||
|
|
return 0;
|
||
|
|
l = ceil_log2(a);
|
||
|
|
u = (limb_t)1 << ((l + 1) / 2);
|
||
|
|
/* u >= floor(sqrt(a)) */
|
||
|
|
for(;;) {
|
||
|
|
s = u;
|
||
|
|
u = ((a / s) + s) / 2;
|
||
|
|
if (u >= s)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
return s;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Integer square root with remainder. 'a' must be an integer. r =
|
||
|
|
floor(sqrt(a)) and rem = a - r^2. BF_ST_INEXACT is set if the result
|
||
|
|
is inexact. 'rem' can be NULL if the remainder is not needed. */
|
||
|
|
int bf_sqrtrem(bf_t *r, bf_t *rem1, const bf_t *a)
|
||
|
|
{
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (a->expn == BF_EXP_INF && a->sign) {
|
||
|
|
goto invalid_op;
|
||
|
|
} else {
|
||
|
|
bf_set(r, a);
|
||
|
|
}
|
||
|
|
if (rem1)
|
||
|
|
bf_set_ui(rem1, 0);
|
||
|
|
ret = 0;
|
||
|
|
} else if (a->sign) {
|
||
|
|
invalid_op:
|
||
|
|
bf_set_nan(r);
|
||
|
|
if (rem1)
|
||
|
|
bf_set_ui(rem1, 0);
|
||
|
|
ret = BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
bf_t rem_s, *rem;
|
||
|
|
int res;
|
||
|
|
|
||
|
|
bf_sqrt(r, a, (a->expn + 1) / 2, BF_RNDF);
|
||
|
|
bf_rint(r, BF_PREC_INF, BF_RNDZ);
|
||
|
|
/* see if the result is exact by computing the remainder */
|
||
|
|
if (rem1) {
|
||
|
|
rem = rem1;
|
||
|
|
} else {
|
||
|
|
rem = &rem_s;
|
||
|
|
bf_init(r->ctx, rem);
|
||
|
|
}
|
||
|
|
bf_mul(rem, r, r, BF_PREC_INF, BF_RNDZ);
|
||
|
|
ret = 0;
|
||
|
|
if (rem1) {
|
||
|
|
bf_neg(rem);
|
||
|
|
bf_add(rem, rem, a, BF_PREC_INF, BF_RNDZ);
|
||
|
|
if (rem->len != 0) {
|
||
|
|
ret = BF_ST_INEXACT;
|
||
|
|
if (rem->sign) {
|
||
|
|
bf_t a1_s, *a1 = &a1_s;
|
||
|
|
bf_add_si(r, r, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
a1->tab = a->tab;
|
||
|
|
a1->len = a->len;
|
||
|
|
a1->sign = a->sign;
|
||
|
|
a1->expn = a->expn + 1;
|
||
|
|
bf_add(rem, rem, r, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_add_si(rem, rem, 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
ret = 0;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
res = bf_cmpu(rem, a);
|
||
|
|
bf_delete(rem);
|
||
|
|
// printf("res2=%d\n", res2);
|
||
|
|
if (res > 0) {
|
||
|
|
/* need to correct the result */
|
||
|
|
bf_add_si(r, r, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
}
|
||
|
|
ret = (res != 0 ? BF_ST_INEXACT : 0);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_sqrt(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
int rnd_mode, ret;
|
||
|
|
|
||
|
|
assert(r != a);
|
||
|
|
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (a->expn == BF_EXP_INF && a->sign) {
|
||
|
|
goto invalid_op;
|
||
|
|
} else {
|
||
|
|
bf_set(r, a);
|
||
|
|
}
|
||
|
|
ret = 0;
|
||
|
|
} else if (a->sign) {
|
||
|
|
invalid_op:
|
||
|
|
bf_set_nan(r);
|
||
|
|
ret = BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
rnd_mode = flags & BF_RND_MASK;
|
||
|
|
if (rnd_mode == BF_RNDF) {
|
||
|
|
ret = __bf_sqrt(r, a, prec, flags);
|
||
|
|
} else {
|
||
|
|
bf_t a1_s, *a1 = &a1_s;
|
||
|
|
slimb_t d, prec2;
|
||
|
|
int res1, res2;
|
||
|
|
|
||
|
|
bf_init(r->ctx, a1);
|
||
|
|
bf_set(a1, a);
|
||
|
|
/* convert the mantissa to an integer with at most 2 *
|
||
|
|
prec + 4 bits */
|
||
|
|
prec2 = prec + 2;
|
||
|
|
/* make '-a->expn + d' divisible by two */
|
||
|
|
d = prec2 * 2 - (a->expn & 1);
|
||
|
|
a1->expn = d;
|
||
|
|
res1 = bf_rint(a1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
res2 = bf_sqrtrem(r, NULL, a1);
|
||
|
|
bf_delete(a1);
|
||
|
|
if ((res2 | res1) != 0) {
|
||
|
|
bf_or_one(r, (prec2 + LIMB_BITS - 1) / LIMB_BITS);
|
||
|
|
}
|
||
|
|
/* update the exponent */
|
||
|
|
r->expn -= (-a->expn + d) >> 1;
|
||
|
|
ret = bf_round(r, prec, flags);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
static no_inline int bf_op2(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags, bf_op2_func_t *func)
|
||
|
|
{
|
||
|
|
bf_t tmp;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
if (r == a || r == b) {
|
||
|
|
bf_init(r->ctx, &tmp);
|
||
|
|
ret = func(&tmp, a, b, prec, flags);
|
||
|
|
bf_move(r, &tmp);
|
||
|
|
} else {
|
||
|
|
ret = func(r, a, b, prec, flags);
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_add(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_op2(r, a, b, prec, flags, __bf_add);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_sub(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_op2(r, a, b, prec, flags, __bf_sub);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_div(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_op2(r, a, b, prec, flags, __bf_div);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_mul_ui(bf_t *r, const bf_t *a, uint64_t b1, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t b;
|
||
|
|
int ret;
|
||
|
|
bf_init(r->ctx, &b);
|
||
|
|
bf_set_ui(&b, b1);
|
||
|
|
ret = bf_mul(r, a, &b, prec, flags);
|
||
|
|
bf_delete(&b);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_mul_si(bf_t *r, const bf_t *a, int64_t b1, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t b;
|
||
|
|
int ret;
|
||
|
|
bf_init(r->ctx, &b);
|
||
|
|
bf_set_si(&b, b1);
|
||
|
|
ret = bf_mul(r, a, &b, prec, flags);
|
||
|
|
bf_delete(&b);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_add_si(bf_t *r, const bf_t *a, int64_t b1, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t b;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
bf_init(r->ctx, &b);
|
||
|
|
bf_set_si(&b, b1);
|
||
|
|
ret = bf_add(r, a, &b, prec, flags);
|
||
|
|
bf_delete(&b);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_pow_ui(bf_t *r, const bf_t *a, limb_t b, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
int ret, n_bits, i;
|
||
|
|
|
||
|
|
assert(r != a);
|
||
|
|
if (b == 0) {
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
bf_set(r, a);
|
||
|
|
ret = 0;
|
||
|
|
n_bits = LIMB_BITS - clz(b);
|
||
|
|
for(i = n_bits - 2; i >= 0; i--) {
|
||
|
|
ret |= bf_mul(r, r, r, prec, flags);
|
||
|
|
if ((b >> i) & 1)
|
||
|
|
ret |= bf_mul(r, r, a, prec, flags);
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_pow_ui_ui(bf_t *r, limb_t a1, limb_t b, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_t a;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
bf_init(r->ctx, &a);
|
||
|
|
bf_set_ui(&a, a1);
|
||
|
|
ret = bf_pow_ui(r, &a, b, prec, flags);
|
||
|
|
bf_delete(&a);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* convert to integer (single rounding) */
|
||
|
|
int bf_rint(bf_t *r, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
int ret;
|
||
|
|
if (r->len == 0)
|
||
|
|
return 0;
|
||
|
|
if (r->expn <= 0) {
|
||
|
|
ret = __bf_round(r, r->expn, flags & ~BF_FLAG_SUBNORMAL, r->len) &
|
||
|
|
~BF_ST_UNDERFLOW;
|
||
|
|
} else {
|
||
|
|
ret = __bf_round(r, bf_min(r->expn, prec), flags, r->len);
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* logical operations */
|
||
|
|
#define BF_LOGIC_OR 0
|
||
|
|
#define BF_LOGIC_XOR 1
|
||
|
|
#define BF_LOGIC_AND 2
|
||
|
|
|
||
|
|
static inline limb_t bf_logic_op1(limb_t a, limb_t b, int op)
|
||
|
|
{
|
||
|
|
switch(op) {
|
||
|
|
case BF_LOGIC_OR:
|
||
|
|
return a | b;
|
||
|
|
case BF_LOGIC_XOR:
|
||
|
|
return a ^ b;
|
||
|
|
default:
|
||
|
|
case BF_LOGIC_AND:
|
||
|
|
return a & b;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static void bf_logic_op(bf_t *r, const bf_t *a1, const bf_t *b1, int op)
|
||
|
|
{
|
||
|
|
bf_t b1_s, a1_s, *a, *b;
|
||
|
|
limb_t a_sign, b_sign, r_sign;
|
||
|
|
slimb_t l, i, a_bit_offset, b_bit_offset;
|
||
|
|
limb_t v1, v2, v1_mask, v2_mask, r_mask;
|
||
|
|
|
||
|
|
assert(r != a1 && r != b1);
|
||
|
|
|
||
|
|
if (a1->expn <= 0)
|
||
|
|
a_sign = 0; /* minus zero is considered as positive */
|
||
|
|
else
|
||
|
|
a_sign = a1->sign;
|
||
|
|
|
||
|
|
if (b1->expn <= 0)
|
||
|
|
b_sign = 0; /* minus zero is considered as positive */
|
||
|
|
else
|
||
|
|
b_sign = b1->sign;
|
||
|
|
|
||
|
|
if (a_sign) {
|
||
|
|
a = &a1_s;
|
||
|
|
bf_init(r->ctx, a);
|
||
|
|
bf_add_si(a, a1, 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
} else {
|
||
|
|
a = (bf_t *)a1;
|
||
|
|
}
|
||
|
|
|
||
|
|
if (b_sign) {
|
||
|
|
b = &b1_s;
|
||
|
|
bf_init(r->ctx, b);
|
||
|
|
bf_add_si(b, b1, 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
} else {
|
||
|
|
b = (bf_t *)b1;
|
||
|
|
}
|
||
|
|
|
||
|
|
r_sign = bf_logic_op1(a_sign, b_sign, op);
|
||
|
|
if (op == BF_LOGIC_AND && r_sign == 0) {
|
||
|
|
/* no need to compute extra zeros for and */
|
||
|
|
if (a_sign == 0 && b_sign == 0)
|
||
|
|
l = bf_min(a->expn, b->expn);
|
||
|
|
else if (a_sign == 0)
|
||
|
|
l = a->expn;
|
||
|
|
else
|
||
|
|
l = b->expn;
|
||
|
|
} else {
|
||
|
|
l = bf_max(a->expn, b->expn);
|
||
|
|
}
|
||
|
|
/* Note: a or b can be zero */
|
||
|
|
l = (bf_max(l, 1) + LIMB_BITS - 1) / LIMB_BITS;
|
||
|
|
bf_resize(r, l);
|
||
|
|
a_bit_offset = a->len * LIMB_BITS - a->expn;
|
||
|
|
b_bit_offset = b->len * LIMB_BITS - b->expn;
|
||
|
|
v1_mask = -a_sign;
|
||
|
|
v2_mask = -b_sign;
|
||
|
|
r_mask = -r_sign;
|
||
|
|
for(i = 0; i < l; i++) {
|
||
|
|
v1 = get_bits(a->tab, a->len, a_bit_offset + i * LIMB_BITS) ^ v1_mask;
|
||
|
|
v2 = get_bits(b->tab, b->len, b_bit_offset + i * LIMB_BITS) ^ v2_mask;
|
||
|
|
r->tab[i] = bf_logic_op1(v1, v2, op) ^ r_mask;
|
||
|
|
}
|
||
|
|
r->expn = l * LIMB_BITS;
|
||
|
|
r->sign = r_sign;
|
||
|
|
bf_normalize_and_round(r, BF_PREC_INF, BF_RNDZ);
|
||
|
|
if (r_sign)
|
||
|
|
bf_add_si(r, r, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
if (a == &a1_s)
|
||
|
|
bf_delete(a);
|
||
|
|
if (b == &b1_s)
|
||
|
|
bf_delete(b);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* 'a' and 'b' must be integers */
|
||
|
|
void bf_logic_or(bf_t *r, const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
bf_logic_op(r, a, b, BF_LOGIC_OR);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* 'a' and 'b' must be integers */
|
||
|
|
void bf_logic_xor(bf_t *r, const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
bf_logic_op(r, a, b, BF_LOGIC_XOR);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* 'a' and 'b' must be integers */
|
||
|
|
void bf_logic_and(bf_t *r, const bf_t *a, const bf_t *b)
|
||
|
|
{
|
||
|
|
bf_logic_op(r, a, b, BF_LOGIC_AND);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* conversion between fixed size types */
|
||
|
|
|
||
|
|
typedef union {
|
||
|
|
double d;
|
||
|
|
uint64_t u;
|
||
|
|
} Float64Union;
|
||
|
|
|
||
|
|
int bf_get_float64(const bf_t *a, double *pres, bf_rnd_t rnd_mode)
|
||
|
|
{
|
||
|
|
Float64Union u;
|
||
|
|
int e, ret;
|
||
|
|
uint64_t m;
|
||
|
|
|
||
|
|
ret = 0;
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
u.u = 0x7ff8000000000000; /* quiet nan */
|
||
|
|
} else {
|
||
|
|
bf_t b_s, *b = &b_s;
|
||
|
|
|
||
|
|
bf_init(a->ctx, b);
|
||
|
|
bf_set(b, a);
|
||
|
|
if (bf_is_finite(b)) {
|
||
|
|
ret = bf_round(b, 53, rnd_mode | BF_FLAG_SUBNORMAL | bf_set_exp_bits(11));
|
||
|
|
}
|
||
|
|
if (b->expn == BF_EXP_INF) {
|
||
|
|
e = (1 << 11) - 1;
|
||
|
|
m = 0;
|
||
|
|
} else if (b->expn == BF_EXP_ZERO) {
|
||
|
|
e = 0;
|
||
|
|
m = 0;
|
||
|
|
} else {
|
||
|
|
e = b->expn + 1023 - 1;
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
if (b->len == 2) {
|
||
|
|
m = ((uint64_t)b->tab[1] << 32) | b->tab[0];
|
||
|
|
} else {
|
||
|
|
m = ((uint64_t)b->tab[0] << 32);
|
||
|
|
}
|
||
|
|
#else
|
||
|
|
m = b->tab[0];
|
||
|
|
#endif
|
||
|
|
if (e <= 0) {
|
||
|
|
/* subnormal */
|
||
|
|
m = m >> (12 - e);
|
||
|
|
e = 0;
|
||
|
|
} else {
|
||
|
|
m = (m << 1) >> 12;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
u.u = m | ((uint64_t)e << 52) | ((uint64_t)b->sign << 63);
|
||
|
|
bf_delete(b);
|
||
|
|
}
|
||
|
|
*pres = u.d;
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
void bf_set_float64(bf_t *a, double d)
|
||
|
|
{
|
||
|
|
Float64Union u;
|
||
|
|
uint64_t m;
|
||
|
|
int shift, e, sgn;
|
||
|
|
|
||
|
|
u.d = d;
|
||
|
|
sgn = u.u >> 63;
|
||
|
|
e = (u.u >> 52) & ((1 << 11) - 1);
|
||
|
|
m = u.u & (((uint64_t)1 << 52) - 1);
|
||
|
|
if (e == ((1 << 11) - 1)) {
|
||
|
|
if (m != 0) {
|
||
|
|
bf_set_nan(a);
|
||
|
|
} else {
|
||
|
|
bf_set_inf(a, sgn);
|
||
|
|
}
|
||
|
|
} else if (e == 0) {
|
||
|
|
if (m == 0) {
|
||
|
|
bf_set_zero(a, sgn);
|
||
|
|
} else {
|
||
|
|
/* subnormal number */
|
||
|
|
m <<= 12;
|
||
|
|
shift = clz64(m);
|
||
|
|
m <<= shift;
|
||
|
|
e = -shift;
|
||
|
|
goto norm;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
m = (m << 11) | ((uint64_t)1 << 63);
|
||
|
|
norm:
|
||
|
|
a->expn = e - 1023 + 1;
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
bf_resize(a, 2);
|
||
|
|
a->tab[0] = m;
|
||
|
|
a->tab[1] = m >> 32;
|
||
|
|
#else
|
||
|
|
bf_resize(a, 1);
|
||
|
|
a->tab[0] = m;
|
||
|
|
#endif
|
||
|
|
a->sign = sgn;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* The rounding mode is always BF_RNDZ. Return BF_ST_OVERFLOW if there
|
||
|
|
is an overflow and 0 otherwise. */
|
||
|
|
int bf_get_int32(int *pres, const bf_t *a, int flags)
|
||
|
|
{
|
||
|
|
uint32_t v;
|
||
|
|
int ret;
|
||
|
|
if (a->expn >= BF_EXP_INF) {
|
||
|
|
ret = 0;
|
||
|
|
if (flags & BF_GET_INT_MOD) {
|
||
|
|
v = 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
v = (uint32_t)INT32_MAX + a->sign;
|
||
|
|
} else {
|
||
|
|
v = INT32_MAX;
|
||
|
|
}
|
||
|
|
} else if (a->expn <= 0) {
|
||
|
|
v = 0;
|
||
|
|
ret = 0;
|
||
|
|
} else if (a->expn <= 31) {
|
||
|
|
v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn);
|
||
|
|
if (a->sign)
|
||
|
|
v = -v;
|
||
|
|
ret = 0;
|
||
|
|
} else if (!(flags & BF_GET_INT_MOD)) {
|
||
|
|
ret = BF_ST_OVERFLOW;
|
||
|
|
if (a->sign) {
|
||
|
|
v = (uint32_t)INT32_MAX + 1;
|
||
|
|
if (a->expn == 32 &&
|
||
|
|
(a->tab[a->len - 1] >> (LIMB_BITS - 32)) == v) {
|
||
|
|
ret = 0;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
v = INT32_MAX;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
v = get_bits(a->tab, a->len, a->len * LIMB_BITS - a->expn);
|
||
|
|
if (a->sign)
|
||
|
|
v = -v;
|
||
|
|
ret = 0;
|
||
|
|
}
|
||
|
|
*pres = v;
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* The rounding mode is always BF_RNDZ. Return BF_ST_OVERFLOW if there
|
||
|
|
is an overflow and 0 otherwise. */
|
||
|
|
int bf_get_int64(int64_t *pres, const bf_t *a, int flags)
|
||
|
|
{
|
||
|
|
uint64_t v;
|
||
|
|
int ret;
|
||
|
|
if (a->expn >= BF_EXP_INF) {
|
||
|
|
ret = 0;
|
||
|
|
if (flags & BF_GET_INT_MOD) {
|
||
|
|
v = 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
v = (uint64_t)INT64_MAX + a->sign;
|
||
|
|
} else {
|
||
|
|
v = INT64_MAX;
|
||
|
|
}
|
||
|
|
} else if (a->expn <= 0) {
|
||
|
|
v = 0;
|
||
|
|
ret = 0;
|
||
|
|
} else if (a->expn <= 63) {
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
if (a->expn <= 32)
|
||
|
|
v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn);
|
||
|
|
else
|
||
|
|
v = (((uint64_t)a->tab[a->len - 1] << 32) |
|
||
|
|
get_limbz(a, a->len - 2)) >> (64 - a->expn);
|
||
|
|
#else
|
||
|
|
v = a->tab[a->len - 1] >> (LIMB_BITS - a->expn);
|
||
|
|
#endif
|
||
|
|
if (a->sign)
|
||
|
|
v = -v;
|
||
|
|
ret = 0;
|
||
|
|
} else if (!(flags & BF_GET_INT_MOD)) {
|
||
|
|
ret = BF_ST_OVERFLOW;
|
||
|
|
if (a->sign) {
|
||
|
|
uint64_t v1;
|
||
|
|
v = (uint64_t)INT64_MAX + 1;
|
||
|
|
if (a->expn == 64) {
|
||
|
|
v1 = a->tab[a->len - 1];
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
v1 |= (v1 << 32) | get_limbz(a, a->len - 2);
|
||
|
|
#endif
|
||
|
|
if (v1 == v)
|
||
|
|
ret = 0;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
v = INT64_MAX;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
slimb_t bit_pos = a->len * LIMB_BITS - a->expn;
|
||
|
|
v = get_bits(a->tab, a->len, bit_pos);
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
v |= (uint64_t)get_bits(a->tab, a->len, bit_pos + 32) << 32;
|
||
|
|
#endif
|
||
|
|
if (a->sign)
|
||
|
|
v = -v;
|
||
|
|
ret = 0;
|
||
|
|
}
|
||
|
|
*pres = v;
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* base conversion from radix */
|
||
|
|
|
||
|
|
static const uint8_t digits_per_limb_table[BF_RADIX_MAX - 1] = {
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
32,20,16,13,12,11,10,10, 9, 9, 8, 8, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,
|
||
|
|
#else
|
||
|
|
64,40,32,27,24,22,21,20,19,18,17,17,16,16,16,15,15,15,14,14,14,14,13,13,13,13,13,13,13,12,12,12,12,12,12,
|
||
|
|
#endif
|
||
|
|
};
|
||
|
|
|
||
|
|
static limb_t get_limb_radix(int radix)
|
||
|
|
{
|
||
|
|
int i, k;
|
||
|
|
limb_t radixl;
|
||
|
|
|
||
|
|
k = digits_per_limb_table[radix - 2];
|
||
|
|
radixl = radix;
|
||
|
|
for(i = 1; i < k; i++)
|
||
|
|
radixl *= radix;
|
||
|
|
return radixl;
|
||
|
|
}
|
||
|
|
|
||
|
|
static void bf_integer_from_radix_rec(bf_t *r, const limb_t *tab,
|
||
|
|
limb_t n, int level, limb_t n0,
|
||
|
|
limb_t radix, bf_t *pow_tab)
|
||
|
|
{
|
||
|
|
if (n == 1) {
|
||
|
|
bf_set_ui(r, tab[0]);
|
||
|
|
} else {
|
||
|
|
bf_t T_s, *T = &T_s, *B;
|
||
|
|
limb_t n1, n2;
|
||
|
|
|
||
|
|
n2 = (((n0 * 2) >> (level + 1)) + 1) / 2;
|
||
|
|
n1 = n - n2;
|
||
|
|
// printf("level=%d n0=%ld n1=%ld n2=%ld\n", level, n0, n1, n2);
|
||
|
|
B = &pow_tab[level];
|
||
|
|
if (B->len == 0) {
|
||
|
|
bf_pow_ui_ui(B, radix, n2, BF_PREC_INF, BF_RNDZ);
|
||
|
|
}
|
||
|
|
bf_integer_from_radix_rec(r, tab + n2, n1, level + 1, n0,
|
||
|
|
radix, pow_tab);
|
||
|
|
bf_mul(r, r, B, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_init(r->ctx, T);
|
||
|
|
bf_integer_from_radix_rec(T, tab, n2, level + 1, n0,
|
||
|
|
radix, pow_tab);
|
||
|
|
bf_add(r, r, T, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_delete(T);
|
||
|
|
}
|
||
|
|
// bf_print_str(" r=", r);
|
||
|
|
}
|
||
|
|
|
||
|
|
static void bf_integer_from_radix(bf_t *r, const limb_t *tab,
|
||
|
|
limb_t n, limb_t radix)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
int pow_tab_len, i;
|
||
|
|
limb_t radixl;
|
||
|
|
bf_t *pow_tab;
|
||
|
|
|
||
|
|
radixl = get_limb_radix(radix);
|
||
|
|
pow_tab_len = ceil_log2(n) + 2; /* XXX: check */
|
||
|
|
pow_tab = bf_malloc(s, sizeof(pow_tab[0]) * pow_tab_len);
|
||
|
|
for(i = 0; i < pow_tab_len; i++)
|
||
|
|
bf_init(r->ctx, &pow_tab[i]);
|
||
|
|
bf_integer_from_radix_rec(r, tab, n, 0, n, radixl, pow_tab);
|
||
|
|
for(i = 0; i < pow_tab_len; i++) {
|
||
|
|
bf_delete(&pow_tab[i]);
|
||
|
|
}
|
||
|
|
bf_free(s, pow_tab);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* compute and round T * radix^expn. */
|
||
|
|
int bf_mul_pow_radix(bf_t *r, const bf_t *T, limb_t radix,
|
||
|
|
slimb_t expn, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
int ret, expn_sign, overflow;
|
||
|
|
slimb_t e, extra_bits, prec1, ziv_extra_bits;
|
||
|
|
bf_t B_s, *B = &B_s;
|
||
|
|
|
||
|
|
if (T->len == 0) {
|
||
|
|
bf_set(r, T);
|
||
|
|
return 0;
|
||
|
|
} else if (expn == 0) {
|
||
|
|
bf_set(r, T);
|
||
|
|
return bf_round(r, prec, flags);
|
||
|
|
}
|
||
|
|
|
||
|
|
e = expn;
|
||
|
|
expn_sign = 0;
|
||
|
|
if (e < 0) {
|
||
|
|
e = -e;
|
||
|
|
expn_sign = 1;
|
||
|
|
}
|
||
|
|
bf_init(r->ctx, B);
|
||
|
|
if (prec == BF_PREC_INF) {
|
||
|
|
/* infinite precision: only used if the result is known to be exact */
|
||
|
|
bf_pow_ui_ui(B, radix, e, BF_PREC_INF, BF_RNDN);
|
||
|
|
if (expn_sign) {
|
||
|
|
ret = bf_div(r, T, B, T->len * LIMB_BITS, BF_RNDN);
|
||
|
|
} else {
|
||
|
|
ret = bf_mul(r, T, B, BF_PREC_INF, BF_RNDN);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
ziv_extra_bits = 16;
|
||
|
|
for(;;) {
|
||
|
|
prec1 = prec + ziv_extra_bits;
|
||
|
|
/* XXX: correct overflow/underflow handling */
|
||
|
|
/* XXX: rigorous error analysis needed */
|
||
|
|
extra_bits = ceil_log2(e) * 2 + 1;
|
||
|
|
ret = bf_pow_ui_ui(B, radix, e, prec1 + extra_bits, BF_RNDN);
|
||
|
|
overflow = !bf_is_finite(B);
|
||
|
|
/* XXX: if bf_pow_ui_ui returns an exact result, can stop
|
||
|
|
after the next operation */
|
||
|
|
if (expn_sign)
|
||
|
|
ret |= bf_div(r, T, B, prec1 + extra_bits, BF_RNDN);
|
||
|
|
else
|
||
|
|
ret |= bf_mul(r, T, B, prec1 + extra_bits, BF_RNDN);
|
||
|
|
if ((ret & BF_ST_INEXACT) &&
|
||
|
|
!bf_can_round(r, prec, flags & BF_RND_MASK, prec1) &&
|
||
|
|
!overflow) {
|
||
|
|
/* and more precision and retry */
|
||
|
|
ziv_extra_bits = ziv_extra_bits + (ziv_extra_bits / 2);
|
||
|
|
} else {
|
||
|
|
ret = bf_round(r, prec, flags) | (ret & BF_ST_INEXACT);
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
bf_delete(B);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline int to_digit(int c)
|
||
|
|
{
|
||
|
|
if (c >= '0' && c <= '9')
|
||
|
|
return c - '0';
|
||
|
|
else if (c >= 'A' && c <= 'Z')
|
||
|
|
return c - 'A' + 10;
|
||
|
|
else if (c >= 'a' && c <= 'z')
|
||
|
|
return c - 'a' + 10;
|
||
|
|
else
|
||
|
|
return 36;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* add a limb at 'pos' and decrement pos. new space is created if needed */
|
||
|
|
static void bf_add_limb(bf_t *a, slimb_t *ppos, limb_t v)
|
||
|
|
{
|
||
|
|
slimb_t pos;
|
||
|
|
pos = *ppos;
|
||
|
|
if (unlikely(pos < 0)) {
|
||
|
|
limb_t new_size, d;
|
||
|
|
new_size = bf_max(a->len + 1, a->len * 3 / 2);
|
||
|
|
a->tab = bf_realloc(a->ctx, a->tab, sizeof(limb_t) * new_size);
|
||
|
|
d = new_size - a->len;
|
||
|
|
memmove(a->tab + d, a->tab, a->len * sizeof(limb_t));
|
||
|
|
a->len = new_size;
|
||
|
|
pos += d;
|
||
|
|
}
|
||
|
|
a->tab[pos--] = v;
|
||
|
|
*ppos = pos;
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_tolower(int c)
|
||
|
|
{
|
||
|
|
if (c >= 'A' && c <= 'Z')
|
||
|
|
c = c - 'A' + 'a';
|
||
|
|
return c;
|
||
|
|
}
|
||
|
|
|
||
|
|
static int strcasestart(const char *str, const char *val, const char **ptr)
|
||
|
|
{
|
||
|
|
const char *p, *q;
|
||
|
|
p = str;
|
||
|
|
q = val;
|
||
|
|
while (*q != '\0') {
|
||
|
|
if (bf_tolower(*p) != *q)
|
||
|
|
return 0;
|
||
|
|
p++;
|
||
|
|
q++;
|
||
|
|
}
|
||
|
|
if (ptr)
|
||
|
|
*ptr = p;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
|
||
|
|
/*
|
||
|
|
Return (status, n, exp). 'status' is the floating point status. 'n'
|
||
|
|
is the parsed number.
|
||
|
|
If prec = BF_PREC_INF:
|
||
|
|
If the number is an integer or if the radix is a power of two,
|
||
|
|
*pexponent = 0.
|
||
|
|
Otherwise, '*pexponent' is the exponent in radix 'radix'.
|
||
|
|
Otherwise
|
||
|
|
*pexponent = 0
|
||
|
|
*/
|
||
|
|
int bf_atof2(bf_t *r, slimb_t *pexponent,
|
||
|
|
const char *str, const char **pnext, int radix,
|
||
|
|
limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
const char *p, *p_start;
|
||
|
|
int is_neg, radix_bits, exp_is_neg, ret, digits_per_limb, shift, sep;
|
||
|
|
limb_t cur_limb;
|
||
|
|
slimb_t pos, expn, int_len, digit_count;
|
||
|
|
BOOL has_decpt, is_bin_exp, is_float;
|
||
|
|
bf_t a_s, *a;
|
||
|
|
|
||
|
|
/* optional separator between digits */
|
||
|
|
sep = (flags & BF_ATOF_UNDERSCORE_SEP) ? '_' : 256;
|
||
|
|
|
||
|
|
*pexponent = 0;
|
||
|
|
p = str;
|
||
|
|
if (!(flags & (BF_ATOF_INT_ONLY | BF_ATOF_JS_QUIRKS)) &&
|
||
|
|
radix <= 16 &&
|
||
|
|
strcasestart(p, "nan", &p)) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
ret = 0;
|
||
|
|
goto done;
|
||
|
|
}
|
||
|
|
is_neg = 0;
|
||
|
|
|
||
|
|
if (p[0] == '+') {
|
||
|
|
p++;
|
||
|
|
p_start = p;
|
||
|
|
if (flags & BF_ATOF_NO_PREFIX_AFTER_SIGN)
|
||
|
|
goto no_radix_prefix;
|
||
|
|
} else if (p[0] == '-') {
|
||
|
|
is_neg = 1;
|
||
|
|
p++;
|
||
|
|
p_start = p;
|
||
|
|
if (flags & BF_ATOF_NO_PREFIX_AFTER_SIGN)
|
||
|
|
goto no_radix_prefix;
|
||
|
|
} else {
|
||
|
|
p_start = p;
|
||
|
|
}
|
||
|
|
if (p[0] == '0') {
|
||
|
|
if ((p[1] == 'x' || p[1] == 'X') &&
|
||
|
|
(radix == 0 || radix == 16) &&
|
||
|
|
!(flags & BF_ATOF_NO_HEX)) {
|
||
|
|
radix = 16;
|
||
|
|
p += 2;
|
||
|
|
} else if ((p[1] == 'o' || p[1] == 'O') &&
|
||
|
|
radix == 0 && (flags & BF_ATOF_BIN_OCT)) {
|
||
|
|
p += 2;
|
||
|
|
radix = 8;
|
||
|
|
} else if ((p[1] == 'b' || p[1] == 'B') &&
|
||
|
|
radix == 0 && (flags & BF_ATOF_BIN_OCT)) {
|
||
|
|
p += 2;
|
||
|
|
radix = 2;
|
||
|
|
} else if ((p[1] >= '0' && p[1] <= '7') &&
|
||
|
|
radix == 0 && (flags & BF_ATOF_LEGACY_OCTAL)) {
|
||
|
|
int i;
|
||
|
|
/* the separator is not allowed in legacy octal literals */
|
||
|
|
for (i = 2; (p[i] >= '0' && p[i] <= '7'); i++)
|
||
|
|
continue;
|
||
|
|
if (p[i] == '8' || p[i] == '9')
|
||
|
|
goto no_prefix;
|
||
|
|
p += 1;
|
||
|
|
radix = 8;
|
||
|
|
} else {
|
||
|
|
/* 0 cannot be followed by a separator */
|
||
|
|
if (p[1] == sep) {
|
||
|
|
p++;
|
||
|
|
bf_set_zero(r, 0);
|
||
|
|
ret = 0;
|
||
|
|
if (flags & BF_ATOF_INT_PREC_INF)
|
||
|
|
ret |= BF_ATOF_ST_INTEGER;
|
||
|
|
goto done;
|
||
|
|
}
|
||
|
|
goto no_prefix;
|
||
|
|
}
|
||
|
|
/* there must be a digit after the prefix */
|
||
|
|
if (to_digit((uint8_t)*p) >= radix) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
ret = 0;
|
||
|
|
goto done;
|
||
|
|
}
|
||
|
|
no_prefix: ;
|
||
|
|
} else {
|
||
|
|
no_radix_prefix:
|
||
|
|
if (!(flags & BF_ATOF_INT_ONLY) && radix <= 16 &&
|
||
|
|
((!(flags & BF_ATOF_JS_QUIRKS) && strcasestart(p, "inf", &p)) ||
|
||
|
|
((flags & BF_ATOF_JS_QUIRKS) && strstart(p, "Infinity", &p)))) {
|
||
|
|
bf_set_inf(r, is_neg);
|
||
|
|
ret = 0;
|
||
|
|
goto done;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
if (radix == 0)
|
||
|
|
radix = 10;
|
||
|
|
if ((radix & (radix - 1)) != 0) {
|
||
|
|
radix_bits = 0; /* base is not a power of two */
|
||
|
|
a = &a_s;
|
||
|
|
bf_init(r->ctx, a);
|
||
|
|
} else {
|
||
|
|
radix_bits = ceil_log2(radix);
|
||
|
|
a = r;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* skip leading zeros */
|
||
|
|
/* XXX: could also skip zeros after the decimal point */
|
||
|
|
while (*p == '0' || (*p == sep && to_digit(p[1]) < radix))
|
||
|
|
p++;
|
||
|
|
|
||
|
|
if (radix_bits) {
|
||
|
|
shift = digits_per_limb = LIMB_BITS;
|
||
|
|
} else {
|
||
|
|
radix_bits = 0;
|
||
|
|
shift = digits_per_limb = digits_per_limb_table[radix - 2];
|
||
|
|
}
|
||
|
|
cur_limb = 0;
|
||
|
|
bf_resize(a, 1);
|
||
|
|
pos = 0;
|
||
|
|
has_decpt = FALSE;
|
||
|
|
int_len = digit_count = 0;
|
||
|
|
is_float = FALSE;
|
||
|
|
for(;;) {
|
||
|
|
limb_t c;
|
||
|
|
if (*p == '.' && (p > p_start || to_digit(p[1]) < radix)) {
|
||
|
|
if ((flags & BF_ATOF_INT_ONLY) ||
|
||
|
|
(radix != 10 && (flags & BF_ATOF_ONLY_DEC_FLOAT)))
|
||
|
|
break;
|
||
|
|
if (has_decpt)
|
||
|
|
break;
|
||
|
|
is_float = TRUE;
|
||
|
|
has_decpt = TRUE;
|
||
|
|
int_len = digit_count;
|
||
|
|
p++;
|
||
|
|
}
|
||
|
|
if (*p == sep && to_digit(p[1]) < radix)
|
||
|
|
p++;
|
||
|
|
c = to_digit(*p);
|
||
|
|
if (c >= radix)
|
||
|
|
break;
|
||
|
|
digit_count++;
|
||
|
|
p++;
|
||
|
|
if (radix_bits) {
|
||
|
|
shift -= radix_bits;
|
||
|
|
if (shift <= 0) {
|
||
|
|
cur_limb |= c >> (-shift);
|
||
|
|
bf_add_limb(a, &pos, cur_limb);
|
||
|
|
if (shift < 0)
|
||
|
|
cur_limb = c << (LIMB_BITS + shift);
|
||
|
|
else
|
||
|
|
cur_limb = 0;
|
||
|
|
shift += LIMB_BITS;
|
||
|
|
} else {
|
||
|
|
cur_limb |= c << shift;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
cur_limb = cur_limb * radix + c;
|
||
|
|
shift--;
|
||
|
|
if (shift == 0) {
|
||
|
|
bf_add_limb(a, &pos, cur_limb);
|
||
|
|
shift = digits_per_limb;
|
||
|
|
cur_limb = 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if (!has_decpt)
|
||
|
|
int_len = digit_count;
|
||
|
|
|
||
|
|
/* add the last limb and pad with zeros */
|
||
|
|
if (shift != digits_per_limb) {
|
||
|
|
if (radix_bits == 0) {
|
||
|
|
while (shift != 0) {
|
||
|
|
cur_limb *= radix;
|
||
|
|
shift--;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
bf_add_limb(a, &pos, cur_limb);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* reset the next limbs to zero (we prefer to reallocate in the
|
||
|
|
renormalization) */
|
||
|
|
memset(a->tab, 0, (pos + 1) * sizeof(limb_t));
|
||
|
|
|
||
|
|
if (p == p_start)
|
||
|
|
goto error;
|
||
|
|
|
||
|
|
/* parse the exponent, if any */
|
||
|
|
expn = 0;
|
||
|
|
is_bin_exp = FALSE;
|
||
|
|
if (!(flags & BF_ATOF_INT_ONLY) &&
|
||
|
|
!(radix != 10 && (flags & BF_ATOF_ONLY_DEC_FLOAT)) &&
|
||
|
|
((radix == 10 && (*p == 'e' || *p == 'E')) ||
|
||
|
|
(radix != 10 && (*p == '@' ||
|
||
|
|
(radix_bits && (*p == 'p' || *p == 'P'))))) &&
|
||
|
|
p > p_start) {
|
||
|
|
is_bin_exp = (*p == 'p' || *p == 'P');
|
||
|
|
p++;
|
||
|
|
is_float = TRUE;
|
||
|
|
exp_is_neg = 0;
|
||
|
|
if (*p == '+') {
|
||
|
|
p++;
|
||
|
|
} else if (*p == '-') {
|
||
|
|
exp_is_neg = 1;
|
||
|
|
p++;
|
||
|
|
}
|
||
|
|
for(;;) {
|
||
|
|
int c;
|
||
|
|
if (*p == sep && to_digit(p[1]) < 10)
|
||
|
|
p++;
|
||
|
|
c = to_digit(*p);
|
||
|
|
if (c >= 10)
|
||
|
|
break;
|
||
|
|
if (unlikely(expn > ((EXP_MAX - 2 - 9) / 10))) {
|
||
|
|
/* exponent overflow */
|
||
|
|
if (exp_is_neg) {
|
||
|
|
bf_set_zero(r, is_neg);
|
||
|
|
ret = BF_ST_UNDERFLOW | BF_ST_INEXACT;
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, is_neg);
|
||
|
|
ret = BF_ST_OVERFLOW | BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
goto done;
|
||
|
|
}
|
||
|
|
p++;
|
||
|
|
expn = expn * 10 + c;
|
||
|
|
}
|
||
|
|
if (exp_is_neg)
|
||
|
|
expn = -expn;
|
||
|
|
} else if (!is_float) {
|
||
|
|
if (*p == 'n' && (flags & BF_ATOF_INT_N_SUFFIX)) {
|
||
|
|
p++;
|
||
|
|
prec = BF_PREC_INF;
|
||
|
|
} else if (flags & BF_ATOF_INT_PREC_INF) {
|
||
|
|
prec = BF_PREC_INF;
|
||
|
|
} else {
|
||
|
|
is_float = TRUE;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if (radix_bits) {
|
||
|
|
/* XXX: may overflow */
|
||
|
|
if (!is_bin_exp)
|
||
|
|
expn *= radix_bits;
|
||
|
|
a->expn = expn + (int_len * radix_bits);
|
||
|
|
a->sign = is_neg;
|
||
|
|
ret = bf_normalize_and_round(a, prec, flags);
|
||
|
|
} else {
|
||
|
|
limb_t l;
|
||
|
|
pos++;
|
||
|
|
l = a->len - pos; /* number of limbs */
|
||
|
|
if (l == 0) {
|
||
|
|
bf_set_zero(r, is_neg);
|
||
|
|
ret = 0;
|
||
|
|
} else {
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
|
||
|
|
expn -= l * digits_per_limb - int_len;
|
||
|
|
bf_init(r->ctx, T);
|
||
|
|
bf_integer_from_radix(T, a->tab + pos, l, radix);
|
||
|
|
T->sign = is_neg;
|
||
|
|
if (prec == BF_PREC_INF && is_float) {
|
||
|
|
/* return the exponent */
|
||
|
|
*pexponent = expn;
|
||
|
|
bf_set(r, T);
|
||
|
|
ret = 0;
|
||
|
|
} else {
|
||
|
|
ret = bf_mul_pow_radix(r, T, radix, expn, prec, flags);
|
||
|
|
}
|
||
|
|
bf_delete(T);
|
||
|
|
}
|
||
|
|
bf_delete(a);
|
||
|
|
}
|
||
|
|
if (!is_float)
|
||
|
|
ret |= BF_ATOF_ST_INTEGER;
|
||
|
|
done:
|
||
|
|
if (pnext)
|
||
|
|
*pnext = p;
|
||
|
|
return ret;
|
||
|
|
error:
|
||
|
|
if (!radix_bits)
|
||
|
|
bf_delete(a);
|
||
|
|
ret = 0;
|
||
|
|
if (flags & BF_ATOF_NAN_IF_EMPTY) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, 0);
|
||
|
|
if (flags & BF_ATOF_INT_PREC_INF)
|
||
|
|
ret |= BF_ATOF_ST_INTEGER;
|
||
|
|
}
|
||
|
|
goto done;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_atof(bf_t *r, const char *str, const char **pnext, int radix,
|
||
|
|
limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
slimb_t dummy_exp;
|
||
|
|
return bf_atof2(r, &dummy_exp, str, pnext, radix, prec, flags);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* base conversion to radix */
|
||
|
|
|
||
|
|
#if LIMB_BITS == 64
|
||
|
|
#define RADIXL_10 UINT64_C(10000000000000000000)
|
||
|
|
#else
|
||
|
|
#define RADIXL_10 UINT64_C(1000000000)
|
||
|
|
#endif
|
||
|
|
|
||
|
|
static const uint32_t inv_log2_radix[BF_RADIX_MAX - 1][LIMB_BITS / 2 + 1] = {
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
{ 0x80000000, 0x00000000,},
|
||
|
|
{ 0x50c24e60, 0xd4d4f4a7,},
|
||
|
|
{ 0x40000000, 0x00000000,},
|
||
|
|
{ 0x372068d2, 0x0a1ee5ca,},
|
||
|
|
{ 0x3184648d, 0xb8153e7a,},
|
||
|
|
{ 0x2d983275, 0x9d5369c4,},
|
||
|
|
{ 0x2aaaaaaa, 0xaaaaaaab,},
|
||
|
|
{ 0x28612730, 0x6a6a7a54,},
|
||
|
|
{ 0x268826a1, 0x3ef3fde6,},
|
||
|
|
{ 0x25001383, 0xbac8a744,},
|
||
|
|
{ 0x23b46706, 0x82c0c709,},
|
||
|
|
{ 0x229729f1, 0xb2c83ded,},
|
||
|
|
{ 0x219e7ffd, 0xa5ad572b,},
|
||
|
|
{ 0x20c33b88, 0xda7c29ab,},
|
||
|
|
{ 0x20000000, 0x00000000,},
|
||
|
|
{ 0x1f50b57e, 0xac5884b3,},
|
||
|
|
{ 0x1eb22cc6, 0x8aa6e26f,},
|
||
|
|
{ 0x1e21e118, 0x0c5daab2,},
|
||
|
|
{ 0x1d9dcd21, 0x439834e4,},
|
||
|
|
{ 0x1d244c78, 0x367a0d65,},
|
||
|
|
{ 0x1cb40589, 0xac173e0c,},
|
||
|
|
{ 0x1c4bd95b, 0xa8d72b0d,},
|
||
|
|
{ 0x1bead768, 0x98f8ce4c,},
|
||
|
|
{ 0x1b903469, 0x050f72e5,},
|
||
|
|
{ 0x1b3b433f, 0x2eb06f15,},
|
||
|
|
{ 0x1aeb6f75, 0x9c46fc38,},
|
||
|
|
{ 0x1aa038eb, 0x0e3bfd17,},
|
||
|
|
{ 0x1a593062, 0xb38d8c56,},
|
||
|
|
{ 0x1a15f4c3, 0x2b95a2e6,},
|
||
|
|
{ 0x19d630dc, 0xcc7ddef9,},
|
||
|
|
{ 0x19999999, 0x9999999a,},
|
||
|
|
{ 0x195fec80, 0x8a609431,},
|
||
|
|
{ 0x1928ee7b, 0x0b4f22f9,},
|
||
|
|
{ 0x18f46acf, 0x8c06e318,},
|
||
|
|
{ 0x18c23246, 0xdc0a9f3d,},
|
||
|
|
#else
|
||
|
|
{ 0x80000000, 0x00000000, 0x00000000,},
|
||
|
|
{ 0x50c24e60, 0xd4d4f4a7, 0x021f57bc,},
|
||
|
|
{ 0x40000000, 0x00000000, 0x00000000,},
|
||
|
|
{ 0x372068d2, 0x0a1ee5ca, 0x19ea911b,},
|
||
|
|
{ 0x3184648d, 0xb8153e7a, 0x7fc2d2e1,},
|
||
|
|
{ 0x2d983275, 0x9d5369c4, 0x4dec1661,},
|
||
|
|
{ 0x2aaaaaaa, 0xaaaaaaaa, 0xaaaaaaab,},
|
||
|
|
{ 0x28612730, 0x6a6a7a53, 0x810fabde,},
|
||
|
|
{ 0x268826a1, 0x3ef3fde6, 0x23e2566b,},
|
||
|
|
{ 0x25001383, 0xbac8a744, 0x385a3349,},
|
||
|
|
{ 0x23b46706, 0x82c0c709, 0x3f891718,},
|
||
|
|
{ 0x229729f1, 0xb2c83ded, 0x15fba800,},
|
||
|
|
{ 0x219e7ffd, 0xa5ad572a, 0xe169744b,},
|
||
|
|
{ 0x20c33b88, 0xda7c29aa, 0x9bddee52,},
|
||
|
|
{ 0x20000000, 0x00000000, 0x00000000,},
|
||
|
|
{ 0x1f50b57e, 0xac5884b3, 0x70e28eee,},
|
||
|
|
{ 0x1eb22cc6, 0x8aa6e26f, 0x06d1a2a2,},
|
||
|
|
{ 0x1e21e118, 0x0c5daab1, 0x81b4f4bf,},
|
||
|
|
{ 0x1d9dcd21, 0x439834e3, 0x81667575,},
|
||
|
|
{ 0x1d244c78, 0x367a0d64, 0xc8204d6d,},
|
||
|
|
{ 0x1cb40589, 0xac173e0c, 0x3b7b16ba,},
|
||
|
|
{ 0x1c4bd95b, 0xa8d72b0d, 0x5879f25a,},
|
||
|
|
{ 0x1bead768, 0x98f8ce4c, 0x66cc2858,},
|
||
|
|
{ 0x1b903469, 0x050f72e5, 0x0cf5488e,},
|
||
|
|
{ 0x1b3b433f, 0x2eb06f14, 0x8c89719c,},
|
||
|
|
{ 0x1aeb6f75, 0x9c46fc37, 0xab5fc7e9,},
|
||
|
|
{ 0x1aa038eb, 0x0e3bfd17, 0x1bd62080,},
|
||
|
|
{ 0x1a593062, 0xb38d8c56, 0x7998ab45,},
|
||
|
|
{ 0x1a15f4c3, 0x2b95a2e6, 0x46aed6a0,},
|
||
|
|
{ 0x19d630dc, 0xcc7ddef9, 0x5aadd61b,},
|
||
|
|
{ 0x19999999, 0x99999999, 0x9999999a,},
|
||
|
|
{ 0x195fec80, 0x8a609430, 0xe1106014,},
|
||
|
|
{ 0x1928ee7b, 0x0b4f22f9, 0x5f69791d,},
|
||
|
|
{ 0x18f46acf, 0x8c06e318, 0x4d2aeb2c,},
|
||
|
|
{ 0x18c23246, 0xdc0a9f3d, 0x3fe16970,},
|
||
|
|
#endif
|
||
|
|
};
|
||
|
|
|
||
|
|
static const limb_t log2_radix[BF_RADIX_MAX - 1] = {
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
0x20000000,
|
||
|
|
0x32b80347,
|
||
|
|
0x40000000,
|
||
|
|
0x4a4d3c26,
|
||
|
|
0x52b80347,
|
||
|
|
0x59d5d9fd,
|
||
|
|
0x60000000,
|
||
|
|
0x6570068e,
|
||
|
|
0x6a4d3c26,
|
||
|
|
0x6eb3a9f0,
|
||
|
|
0x72b80347,
|
||
|
|
0x766a008e,
|
||
|
|
0x79d5d9fd,
|
||
|
|
0x7d053f6d,
|
||
|
|
0x80000000,
|
||
|
|
0x82cc7edf,
|
||
|
|
0x8570068e,
|
||
|
|
0x87ef05ae,
|
||
|
|
0x8a4d3c26,
|
||
|
|
0x8c8ddd45,
|
||
|
|
0x8eb3a9f0,
|
||
|
|
0x90c10501,
|
||
|
|
0x92b80347,
|
||
|
|
0x949a784c,
|
||
|
|
0x966a008e,
|
||
|
|
0x982809d6,
|
||
|
|
0x99d5d9fd,
|
||
|
|
0x9b74948f,
|
||
|
|
0x9d053f6d,
|
||
|
|
0x9e88c6b3,
|
||
|
|
0xa0000000,
|
||
|
|
0xa16bad37,
|
||
|
|
0xa2cc7edf,
|
||
|
|
0xa4231623,
|
||
|
|
0xa570068e,
|
||
|
|
#else
|
||
|
|
0x2000000000000000,
|
||
|
|
0x32b803473f7ad0f4,
|
||
|
|
0x4000000000000000,
|
||
|
|
0x4a4d3c25e68dc57f,
|
||
|
|
0x52b803473f7ad0f4,
|
||
|
|
0x59d5d9fd5010b366,
|
||
|
|
0x6000000000000000,
|
||
|
|
0x6570068e7ef5a1e8,
|
||
|
|
0x6a4d3c25e68dc57f,
|
||
|
|
0x6eb3a9f01975077f,
|
||
|
|
0x72b803473f7ad0f4,
|
||
|
|
0x766a008e4788cbcd,
|
||
|
|
0x79d5d9fd5010b366,
|
||
|
|
0x7d053f6d26089673,
|
||
|
|
0x8000000000000000,
|
||
|
|
0x82cc7edf592262d0,
|
||
|
|
0x8570068e7ef5a1e8,
|
||
|
|
0x87ef05ae409a0289,
|
||
|
|
0x8a4d3c25e68dc57f,
|
||
|
|
0x8c8ddd448f8b845a,
|
||
|
|
0x8eb3a9f01975077f,
|
||
|
|
0x90c10500d63aa659,
|
||
|
|
0x92b803473f7ad0f4,
|
||
|
|
0x949a784bcd1b8afe,
|
||
|
|
0x966a008e4788cbcd,
|
||
|
|
0x982809d5be7072dc,
|
||
|
|
0x99d5d9fd5010b366,
|
||
|
|
0x9b74948f5532da4b,
|
||
|
|
0x9d053f6d26089673,
|
||
|
|
0x9e88c6b3626a72aa,
|
||
|
|
0xa000000000000000,
|
||
|
|
0xa16bad3758efd873,
|
||
|
|
0xa2cc7edf592262d0,
|
||
|
|
0xa4231623369e78e6,
|
||
|
|
0xa570068e7ef5a1e8,
|
||
|
|
#endif
|
||
|
|
};
|
||
|
|
|
||
|
|
/* compute floor(a*b) or ceil(a*b) with b = log2(radix) or
|
||
|
|
b=1/log2(radix). For is_inv = 0, strict accuracy is not guaranteed
|
||
|
|
when radix is not a power of two. */
|
||
|
|
slimb_t bf_mul_log2_radix(slimb_t a1, unsigned int radix, int is_inv,
|
||
|
|
int is_ceil1)
|
||
|
|
{
|
||
|
|
int is_neg;
|
||
|
|
limb_t a;
|
||
|
|
BOOL is_ceil;
|
||
|
|
|
||
|
|
is_ceil = is_ceil1;
|
||
|
|
a = a1;
|
||
|
|
if (a1 < 0) {
|
||
|
|
a = -a;
|
||
|
|
is_neg = 1;
|
||
|
|
} else {
|
||
|
|
is_neg = 0;
|
||
|
|
}
|
||
|
|
is_ceil ^= is_neg;
|
||
|
|
if ((radix & (radix - 1)) == 0) {
|
||
|
|
int radix_bits;
|
||
|
|
/* radix is a power of two */
|
||
|
|
radix_bits = ceil_log2(radix);
|
||
|
|
if (is_inv) {
|
||
|
|
if (is_ceil)
|
||
|
|
a += radix_bits - 1;
|
||
|
|
a = a / radix_bits;
|
||
|
|
} else {
|
||
|
|
a = a * radix_bits;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
const uint32_t *tab;
|
||
|
|
limb_t b0, b1;
|
||
|
|
dlimb_t t;
|
||
|
|
|
||
|
|
if (is_inv) {
|
||
|
|
tab = inv_log2_radix[radix - 2];
|
||
|
|
#if LIMB_BITS == 32
|
||
|
|
b1 = tab[0];
|
||
|
|
b0 = tab[1];
|
||
|
|
#else
|
||
|
|
b1 = ((limb_t)tab[0] << 32) | tab[1];
|
||
|
|
b0 = (limb_t)tab[2] << 32;
|
||
|
|
#endif
|
||
|
|
t = (dlimb_t)b0 * (dlimb_t)a;
|
||
|
|
t = (dlimb_t)b1 * (dlimb_t)a + (t >> LIMB_BITS);
|
||
|
|
a = t >> (LIMB_BITS - 1);
|
||
|
|
} else {
|
||
|
|
b0 = log2_radix[radix - 2];
|
||
|
|
t = (dlimb_t)b0 * (dlimb_t)a;
|
||
|
|
a = t >> (LIMB_BITS - 3);
|
||
|
|
}
|
||
|
|
/* a = floor(result) and 'result' cannot be an integer */
|
||
|
|
a += is_ceil;
|
||
|
|
}
|
||
|
|
if (is_neg)
|
||
|
|
a = -a;
|
||
|
|
return a;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* 'n' is the number of output limbs */
|
||
|
|
static void bf_integer_to_radix_rec(bf_t *pow_tab,
|
||
|
|
limb_t *out, const bf_t *a, limb_t n,
|
||
|
|
int level, limb_t n0, limb_t radixl,
|
||
|
|
unsigned int radixl_bits)
|
||
|
|
{
|
||
|
|
limb_t n1, n2, q_prec;
|
||
|
|
assert(n >= 1);
|
||
|
|
if (n == 1) {
|
||
|
|
out[0] = get_bits(a->tab, a->len, a->len * LIMB_BITS - a->expn);
|
||
|
|
} else if (n == 2) {
|
||
|
|
dlimb_t t;
|
||
|
|
slimb_t pos;
|
||
|
|
pos = a->len * LIMB_BITS - a->expn;
|
||
|
|
t = ((dlimb_t)get_bits(a->tab, a->len, pos + LIMB_BITS) << LIMB_BITS) |
|
||
|
|
get_bits(a->tab, a->len, pos);
|
||
|
|
if (likely(radixl == RADIXL_10)) {
|
||
|
|
/* use division by a constant when possible */
|
||
|
|
out[0] = t % RADIXL_10;
|
||
|
|
out[1] = t / RADIXL_10;
|
||
|
|
} else {
|
||
|
|
out[0] = t % radixl;
|
||
|
|
out[1] = t / radixl;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
bf_t Q, R, *B, *B_inv;
|
||
|
|
int q_add;
|
||
|
|
bf_init(a->ctx, &Q);
|
||
|
|
bf_init(a->ctx, &R);
|
||
|
|
n2 = (((n0 * 2) >> (level + 1)) + 1) / 2;
|
||
|
|
n1 = n - n2;
|
||
|
|
B = &pow_tab[2 * level];
|
||
|
|
B_inv = &pow_tab[2 * level + 1];
|
||
|
|
if (B->len == 0) {
|
||
|
|
/* compute BASE^n2 */
|
||
|
|
bf_pow_ui_ui(B, radixl, n2, BF_PREC_INF, BF_RNDZ);
|
||
|
|
/* we use enough bits for the maximum possible 'n1' value,
|
||
|
|
i.e. n2 + 1 */
|
||
|
|
bf_recip(B_inv, B, (n2 + 1) * radixl_bits + 2);
|
||
|
|
}
|
||
|
|
// printf("%d: n1=% " PRId64 " n2=%" PRId64 "\n", level, n1, n2);
|
||
|
|
q_prec = n1 * radixl_bits;
|
||
|
|
bf_mul(&Q, a, B_inv, q_prec, BF_RNDN);
|
||
|
|
bf_rint(&Q, BF_PREC_INF, BF_RNDZ);
|
||
|
|
|
||
|
|
bf_mul(&R, &Q, B, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_sub(&R, a, &R, BF_PREC_INF, BF_RNDZ);
|
||
|
|
/* adjust if necessary */
|
||
|
|
q_add = 0;
|
||
|
|
while (R.sign && R.len != 0) {
|
||
|
|
bf_add(&R, &R, B, BF_PREC_INF, BF_RNDZ);
|
||
|
|
q_add--;
|
||
|
|
}
|
||
|
|
while (bf_cmpu(&R, B) >= 0) {
|
||
|
|
bf_sub(&R, &R, B, BF_PREC_INF, BF_RNDZ);
|
||
|
|
q_add++;
|
||
|
|
}
|
||
|
|
if (q_add != 0) {
|
||
|
|
bf_add_si(&Q, &Q, q_add, BF_PREC_INF, BF_RNDZ);
|
||
|
|
}
|
||
|
|
bf_integer_to_radix_rec(pow_tab, out + n2, &Q, n1, level + 1, n0,
|
||
|
|
radixl, radixl_bits);
|
||
|
|
bf_integer_to_radix_rec(pow_tab, out, &R, n2, level + 1, n0,
|
||
|
|
radixl, radixl_bits);
|
||
|
|
bf_delete(&Q);
|
||
|
|
bf_delete(&R);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static void bf_integer_to_radix(bf_t *r, const bf_t *a, limb_t radixl)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
limb_t r_len;
|
||
|
|
bf_t *pow_tab;
|
||
|
|
int i, pow_tab_len;
|
||
|
|
|
||
|
|
r_len = r->len;
|
||
|
|
pow_tab_len = (ceil_log2(r_len) + 2) * 2; /* XXX: check */
|
||
|
|
pow_tab = bf_malloc(s, sizeof(pow_tab[0]) * pow_tab_len);
|
||
|
|
for(i = 0; i < pow_tab_len; i++)
|
||
|
|
bf_init(r->ctx, &pow_tab[i]);
|
||
|
|
|
||
|
|
bf_integer_to_radix_rec(pow_tab, r->tab, a, r_len, 0, r_len, radixl,
|
||
|
|
ceil_log2(radixl));
|
||
|
|
|
||
|
|
for(i = 0; i < pow_tab_len; i++) {
|
||
|
|
bf_delete(&pow_tab[i]);
|
||
|
|
}
|
||
|
|
bf_free(s, pow_tab);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* a must be >= 0. 'P' is the wanted number of digits in radix
|
||
|
|
'radix'. 'r' is the mantissa represented as an integer. *pE
|
||
|
|
contains the exponent. */
|
||
|
|
static void bf_convert_to_radix(bf_t *r, slimb_t *pE,
|
||
|
|
const bf_t *a, int radix,
|
||
|
|
limb_t P, bf_rnd_t rnd_mode,
|
||
|
|
BOOL is_fixed_exponent)
|
||
|
|
{
|
||
|
|
slimb_t E, e, prec, extra_bits, ziv_extra_bits, prec0;
|
||
|
|
bf_t B_s, *B = &B_s;
|
||
|
|
int e_sign, ret, res;
|
||
|
|
|
||
|
|
if (a->len == 0) {
|
||
|
|
/* zero case */
|
||
|
|
*pE = 0;
|
||
|
|
bf_set(r, a);
|
||
|
|
return;
|
||
|
|
}
|
||
|
|
|
||
|
|
if (is_fixed_exponent) {
|
||
|
|
E = *pE;
|
||
|
|
} else {
|
||
|
|
/* compute the new exponent */
|
||
|
|
E = 1 + bf_mul_log2_radix(a->expn - 1, radix, TRUE, FALSE);
|
||
|
|
}
|
||
|
|
// bf_print_str("a", a);
|
||
|
|
// printf("E=%ld P=%ld radix=%d\n", E, P, radix);
|
||
|
|
|
||
|
|
for(;;) {
|
||
|
|
e = P - E;
|
||
|
|
e_sign = 0;
|
||
|
|
if (e < 0) {
|
||
|
|
e = -e;
|
||
|
|
e_sign = 1;
|
||
|
|
}
|
||
|
|
/* Note: precision for log2(radix) is not critical here */
|
||
|
|
prec0 = bf_mul_log2_radix(P, radix, FALSE, TRUE);
|
||
|
|
ziv_extra_bits = 16;
|
||
|
|
for(;;) {
|
||
|
|
prec = prec0 + ziv_extra_bits;
|
||
|
|
/* XXX: rigorous error analysis needed */
|
||
|
|
extra_bits = ceil_log2(e) * 2 + 1;
|
||
|
|
ret = bf_pow_ui_ui(r, radix, e, prec + extra_bits, BF_RNDN);
|
||
|
|
if (!e_sign)
|
||
|
|
ret |= bf_mul(r, r, a, prec + extra_bits, BF_RNDN);
|
||
|
|
else
|
||
|
|
ret |= bf_div(r, a, r, prec + extra_bits, BF_RNDN);
|
||
|
|
/* if the result is not exact, check that it can be safely
|
||
|
|
rounded to an integer */
|
||
|
|
if ((ret & BF_ST_INEXACT) &&
|
||
|
|
!bf_can_round(r, r->expn, rnd_mode, prec)) {
|
||
|
|
/* and more precision and retry */
|
||
|
|
ziv_extra_bits = ziv_extra_bits + (ziv_extra_bits / 2);
|
||
|
|
continue;
|
||
|
|
} else {
|
||
|
|
bf_rint(r, BF_PREC_INF, rnd_mode);
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if (is_fixed_exponent)
|
||
|
|
break;
|
||
|
|
/* check that the result is < B^P */
|
||
|
|
/* XXX: do an fast approximate test first ? */
|
||
|
|
bf_init(r->ctx, B);
|
||
|
|
bf_pow_ui_ui(B, radix, P, BF_PREC_INF, BF_RNDZ);
|
||
|
|
res = bf_cmpu(r, B);
|
||
|
|
bf_delete(B);
|
||
|
|
if (res < 0)
|
||
|
|
break;
|
||
|
|
/* try a larger exponent */
|
||
|
|
E++;
|
||
|
|
}
|
||
|
|
*pE = E;
|
||
|
|
}
|
||
|
|
|
||
|
|
static void limb_to_a(char *buf, limb_t n, unsigned int radix, int len)
|
||
|
|
{
|
||
|
|
int digit, i;
|
||
|
|
|
||
|
|
if (radix == 10) {
|
||
|
|
/* specific case with constant divisor */
|
||
|
|
for(i = len - 1; i >= 0; i--) {
|
||
|
|
digit = (limb_t)n % 10;
|
||
|
|
n = (limb_t)n / 10;
|
||
|
|
buf[i] = digit + '0';
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
for(i = len - 1; i >= 0; i--) {
|
||
|
|
digit = (limb_t)n % radix;
|
||
|
|
n = (limb_t)n / radix;
|
||
|
|
if (digit < 10)
|
||
|
|
digit += '0';
|
||
|
|
else
|
||
|
|
digit += 'a' - 10;
|
||
|
|
buf[i] = digit;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* for power of 2 radixes */
|
||
|
|
static void limb_to_a2(char *buf, limb_t n, unsigned int radix_bits, int len)
|
||
|
|
{
|
||
|
|
int digit, i;
|
||
|
|
unsigned int mask;
|
||
|
|
|
||
|
|
mask = (1 << radix_bits) - 1;
|
||
|
|
for(i = len - 1; i >= 0; i--) {
|
||
|
|
digit = n & mask;
|
||
|
|
n >>= radix_bits;
|
||
|
|
if (digit < 10)
|
||
|
|
digit += '0';
|
||
|
|
else
|
||
|
|
digit += 'a' - 10;
|
||
|
|
buf[i] = digit;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* 'a' must be an integer. A dot is added before the 'dot_pos'
|
||
|
|
digit. dot_pos = n_digits does not display the dot. 0 <= dot_pos <=
|
||
|
|
n_digits. n_digits >= 1. */
|
||
|
|
static void output_digits(DynBuf *s, const bf_t *a1, int radix, limb_t n_digits,
|
||
|
|
limb_t dot_pos)
|
||
|
|
{
|
||
|
|
limb_t i, v, l;
|
||
|
|
slimb_t pos, pos_incr;
|
||
|
|
int digits_per_limb, buf_pos, radix_bits, first_buf_pos;
|
||
|
|
char buf[65];
|
||
|
|
bf_t a_s, *a;
|
||
|
|
|
||
|
|
if ((radix & (radix - 1)) != 0) {
|
||
|
|
limb_t n, radixl;
|
||
|
|
|
||
|
|
digits_per_limb = digits_per_limb_table[radix - 2];
|
||
|
|
radixl = get_limb_radix(radix);
|
||
|
|
a = &a_s;
|
||
|
|
bf_init(a1->ctx, a);
|
||
|
|
n = (n_digits + digits_per_limb - 1) / digits_per_limb;
|
||
|
|
bf_resize(a, n);
|
||
|
|
bf_integer_to_radix(a, a1, radixl);
|
||
|
|
radix_bits = 0;
|
||
|
|
pos = n;
|
||
|
|
pos_incr = 1;
|
||
|
|
first_buf_pos = pos * digits_per_limb - n_digits;
|
||
|
|
} else {
|
||
|
|
a = (bf_t *)a1;
|
||
|
|
radix_bits = ceil_log2(radix);
|
||
|
|
digits_per_limb = LIMB_BITS / radix_bits;
|
||
|
|
pos_incr = digits_per_limb * radix_bits;
|
||
|
|
pos = a->len * LIMB_BITS - a->expn + n_digits * radix_bits;
|
||
|
|
first_buf_pos = 0;
|
||
|
|
}
|
||
|
|
buf_pos = digits_per_limb;
|
||
|
|
i = 0;
|
||
|
|
while (i < n_digits) {
|
||
|
|
if (buf_pos == digits_per_limb) {
|
||
|
|
pos -= pos_incr;
|
||
|
|
if (radix_bits == 0) {
|
||
|
|
v = get_limbz(a, pos);
|
||
|
|
limb_to_a(buf, v, radix, digits_per_limb);
|
||
|
|
} else {
|
||
|
|
v = get_bits(a->tab, a->len, pos);
|
||
|
|
limb_to_a2(buf, v, radix_bits, digits_per_limb);
|
||
|
|
}
|
||
|
|
buf_pos = first_buf_pos;
|
||
|
|
first_buf_pos = 0;
|
||
|
|
}
|
||
|
|
if (i < dot_pos) {
|
||
|
|
l = dot_pos;
|
||
|
|
} else {
|
||
|
|
if (i == dot_pos)
|
||
|
|
dbuf_putc(s, '.');
|
||
|
|
l = n_digits;
|
||
|
|
}
|
||
|
|
l = bf_min(digits_per_limb - buf_pos, l - i);
|
||
|
|
dbuf_put(s, (uint8_t *)(buf + buf_pos), l);
|
||
|
|
buf_pos += l;
|
||
|
|
i += l;
|
||
|
|
}
|
||
|
|
if (a != a1)
|
||
|
|
bf_delete(a);
|
||
|
|
}
|
||
|
|
|
||
|
|
static void *bf_dbuf_realloc(void *opaque, void *ptr, size_t size)
|
||
|
|
{
|
||
|
|
bf_context_t *s = opaque;
|
||
|
|
return bf_realloc(s, ptr, size);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* return the length in bytes. A trailing '\0' is added */
|
||
|
|
size_t bf_ftoa(char **pbuf, const bf_t *a2, int radix, limb_t prec,
|
||
|
|
bf_flags_t flags)
|
||
|
|
{
|
||
|
|
DynBuf s_s, *s = &s_s;
|
||
|
|
int radix_bits;
|
||
|
|
|
||
|
|
// bf_print_str("ftoa", a2);
|
||
|
|
// printf("radix=%d\n", radix);
|
||
|
|
dbuf_init2(s, a2->ctx, bf_dbuf_realloc);
|
||
|
|
if (a2->expn == BF_EXP_NAN) {
|
||
|
|
dbuf_putstr(s, "NaN");
|
||
|
|
} else {
|
||
|
|
if (a2->sign)
|
||
|
|
dbuf_putc(s, '-');
|
||
|
|
if (a2->expn == BF_EXP_INF) {
|
||
|
|
if (flags & BF_FTOA_JS_QUIRKS)
|
||
|
|
dbuf_putstr(s, "Infinity");
|
||
|
|
else
|
||
|
|
dbuf_putstr(s, "Inf");
|
||
|
|
} else {
|
||
|
|
int fmt;
|
||
|
|
slimb_t n_digits, n, i, n_max, n1;
|
||
|
|
bf_t a1_s, *a1;
|
||
|
|
bf_t a_s, *a = &a_s;
|
||
|
|
|
||
|
|
/* make a positive number */
|
||
|
|
a->tab = a2->tab;
|
||
|
|
a->len = a2->len;
|
||
|
|
a->expn = a2->expn;
|
||
|
|
a->sign = 0;
|
||
|
|
|
||
|
|
if ((radix & (radix - 1)) != 0)
|
||
|
|
radix_bits = 0;
|
||
|
|
else
|
||
|
|
radix_bits = ceil_log2(radix);
|
||
|
|
|
||
|
|
fmt = flags & BF_FTOA_FORMAT_MASK;
|
||
|
|
a1 = &a1_s;
|
||
|
|
bf_init(a2->ctx, a1);
|
||
|
|
if (fmt == BF_FTOA_FORMAT_FRAC) {
|
||
|
|
size_t pos, start;
|
||
|
|
/* one more digit for the rounding */
|
||
|
|
n = 1 + bf_mul_log2_radix(bf_max(a->expn, 0), radix, TRUE, TRUE);
|
||
|
|
n_digits = n + prec;
|
||
|
|
n1 = n;
|
||
|
|
bf_convert_to_radix(a1, &n1, a, radix, n_digits,
|
||
|
|
flags & BF_RND_MASK, TRUE);
|
||
|
|
start = s->size;
|
||
|
|
output_digits(s, a1, radix, n_digits, n);
|
||
|
|
/* remove leading zeros because we allocated one more digit */
|
||
|
|
pos = start;
|
||
|
|
while ((pos + 1) < s->size && s->buf[pos] == '0' &&
|
||
|
|
s->buf[pos + 1] != '.')
|
||
|
|
pos++;
|
||
|
|
if (pos > start) {
|
||
|
|
memmove(s->buf + start, s->buf + pos, s->size - pos);
|
||
|
|
s->size -= (pos - start);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
if (fmt == BF_FTOA_FORMAT_FIXED) {
|
||
|
|
n_digits = prec;
|
||
|
|
n_max = n_digits;
|
||
|
|
} else {
|
||
|
|
slimb_t n_digits_max, n_digits_min;
|
||
|
|
|
||
|
|
if (prec == BF_PREC_INF) {
|
||
|
|
assert(radix_bits != 0);
|
||
|
|
/* XXX: could use the exact number of bits */
|
||
|
|
prec = a->len * LIMB_BITS;
|
||
|
|
}
|
||
|
|
n_digits = 1 + bf_mul_log2_radix(prec, radix, TRUE, TRUE);
|
||
|
|
/* max number of digits for non exponential
|
||
|
|
notation. The rational is to have the same rule
|
||
|
|
as JS i.e. n_max = 21 for 64 bit float in base 10. */
|
||
|
|
n_max = n_digits + 4;
|
||
|
|
if (fmt == BF_FTOA_FORMAT_FREE_MIN) {
|
||
|
|
bf_t b_s, *b = &b_s;
|
||
|
|
|
||
|
|
/* find the minimum number of digits by
|
||
|
|
dichotomy. */
|
||
|
|
n_digits_max = n_digits;
|
||
|
|
n_digits_min = 1;
|
||
|
|
bf_init(a2->ctx, b);
|
||
|
|
while (n_digits_min < n_digits_max) {
|
||
|
|
n_digits = (n_digits_min + n_digits_max) / 2;
|
||
|
|
bf_convert_to_radix(a1, &n, a, radix, n_digits,
|
||
|
|
flags & BF_RND_MASK, FALSE);
|
||
|
|
/* convert back to a number and compare */
|
||
|
|
bf_mul_pow_radix(b, a1, radix, n - n_digits,
|
||
|
|
prec,
|
||
|
|
(flags & ~BF_RND_MASK) |
|
||
|
|
BF_RNDN);
|
||
|
|
if (bf_cmpu(b, a) == 0) {
|
||
|
|
n_digits_max = n_digits;
|
||
|
|
} else {
|
||
|
|
n_digits_min = n_digits + 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
bf_delete(b);
|
||
|
|
n_digits = n_digits_max;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
bf_convert_to_radix(a1, &n, a, radix, n_digits,
|
||
|
|
flags & BF_RND_MASK, FALSE);
|
||
|
|
if (a1->expn == BF_EXP_ZERO &&
|
||
|
|
fmt != BF_FTOA_FORMAT_FIXED &&
|
||
|
|
!(flags & BF_FTOA_FORCE_EXP)) {
|
||
|
|
/* just output zero */
|
||
|
|
dbuf_putstr(s, "0");
|
||
|
|
} else {
|
||
|
|
if (flags & BF_FTOA_ADD_PREFIX) {
|
||
|
|
if (radix == 16)
|
||
|
|
dbuf_putstr(s, "0x");
|
||
|
|
else if (radix == 8)
|
||
|
|
dbuf_putstr(s, "0o");
|
||
|
|
else if (radix == 2)
|
||
|
|
dbuf_putstr(s, "0b");
|
||
|
|
}
|
||
|
|
if (a1->expn == BF_EXP_ZERO)
|
||
|
|
n = 1;
|
||
|
|
if ((flags & BF_FTOA_FORCE_EXP) ||
|
||
|
|
n <= -6 || n > n_max) {
|
||
|
|
const char *fmt;
|
||
|
|
/* exponential notation */
|
||
|
|
output_digits(s, a1, radix, n_digits, 1);
|
||
|
|
if (radix_bits != 0 && radix <= 16) {
|
||
|
|
if (flags & BF_FTOA_JS_QUIRKS)
|
||
|
|
fmt = "p%+" PRId_LIMB;
|
||
|
|
else
|
||
|
|
fmt = "p%" PRId_LIMB;
|
||
|
|
dbuf_printf(s, fmt, (n - 1) * radix_bits);
|
||
|
|
} else {
|
||
|
|
if (flags & BF_FTOA_JS_QUIRKS)
|
||
|
|
fmt = "%c%+" PRId_LIMB;
|
||
|
|
else
|
||
|
|
fmt = "%c%" PRId_LIMB;
|
||
|
|
dbuf_printf(s, fmt,
|
||
|
|
radix <= 10 ? 'e' : '@', n - 1);
|
||
|
|
}
|
||
|
|
} else if (n <= 0) {
|
||
|
|
/* 0.x */
|
||
|
|
dbuf_putstr(s, "0.");
|
||
|
|
for(i = 0; i < -n; i++) {
|
||
|
|
dbuf_putc(s, '0');
|
||
|
|
}
|
||
|
|
output_digits(s, a1, radix, n_digits, n_digits);
|
||
|
|
} else {
|
||
|
|
if (n_digits <= n) {
|
||
|
|
/* no dot */
|
||
|
|
output_digits(s, a1, radix, n_digits, n_digits);
|
||
|
|
for(i = 0; i < (n - n_digits); i++)
|
||
|
|
dbuf_putc(s, '0');
|
||
|
|
} else {
|
||
|
|
output_digits(s, a1, radix, n_digits, n);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
bf_delete(a1);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
dbuf_putc(s, '\0');
|
||
|
|
*pbuf = (char *)s->buf;
|
||
|
|
return s->size - 1;
|
||
|
|
}
|
||
|
|
|
||
|
|
/***************************************************************/
|
||
|
|
/* transcendental functions */
|
||
|
|
|
||
|
|
/* Note: the algorithm is from MPFR */
|
||
|
|
static void bf_const_log2_rec(bf_t *T, bf_t *P, bf_t *Q, limb_t n1,
|
||
|
|
limb_t n2, BOOL need_P)
|
||
|
|
{
|
||
|
|
bf_context_t *s = T->ctx;
|
||
|
|
if ((n2 - n1) == 1) {
|
||
|
|
if (n1 == 0) {
|
||
|
|
bf_set_ui(P, 3);
|
||
|
|
} else {
|
||
|
|
bf_set_ui(P, n1);
|
||
|
|
P->sign = 1;
|
||
|
|
}
|
||
|
|
bf_set_ui(Q, 2 * n1 + 1);
|
||
|
|
Q->expn += 2;
|
||
|
|
bf_set(T, P);
|
||
|
|
} else {
|
||
|
|
limb_t m;
|
||
|
|
bf_t T1_s, *T1 = &T1_s;
|
||
|
|
bf_t P1_s, *P1 = &P1_s;
|
||
|
|
bf_t Q1_s, *Q1 = &Q1_s;
|
||
|
|
|
||
|
|
m = n1 + ((n2 - n1) >> 1);
|
||
|
|
bf_const_log2_rec(T, P, Q, n1, m, TRUE);
|
||
|
|
bf_init(s, T1);
|
||
|
|
bf_init(s, P1);
|
||
|
|
bf_init(s, Q1);
|
||
|
|
bf_const_log2_rec(T1, P1, Q1, m, n2, need_P);
|
||
|
|
bf_mul(T, T, Q1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_mul(T1, T1, P, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_add(T, T, T1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
if (need_P)
|
||
|
|
bf_mul(P, P, P1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_mul(Q, Q, Q1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_delete(T1);
|
||
|
|
bf_delete(P1);
|
||
|
|
bf_delete(Q1);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* compute log(2) with faithful rounding at precision 'prec' */
|
||
|
|
static void bf_const_log2_internal(bf_t *T, limb_t prec)
|
||
|
|
{
|
||
|
|
limb_t w, N;
|
||
|
|
bf_t P_s, *P = &P_s;
|
||
|
|
bf_t Q_s, *Q = &Q_s;
|
||
|
|
|
||
|
|
w = prec + 15;
|
||
|
|
N = w / 3 + 1;
|
||
|
|
bf_init(T->ctx, P);
|
||
|
|
bf_init(T->ctx, Q);
|
||
|
|
bf_const_log2_rec(T, P, Q, 0, N, FALSE);
|
||
|
|
bf_div(T, T, Q, prec, BF_RNDN);
|
||
|
|
bf_delete(P);
|
||
|
|
bf_delete(Q);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* PI constant */
|
||
|
|
|
||
|
|
#define CHUD_A 13591409
|
||
|
|
#define CHUD_B 545140134
|
||
|
|
#define CHUD_C 640320
|
||
|
|
#define CHUD_BITS_PER_TERM 47
|
||
|
|
|
||
|
|
static void chud_bs(bf_t *P, bf_t *Q, bf_t *G, int64_t a, int64_t b, int need_g,
|
||
|
|
limb_t prec)
|
||
|
|
{
|
||
|
|
bf_context_t *s = P->ctx;
|
||
|
|
int64_t c;
|
||
|
|
|
||
|
|
if (a == (b - 1)) {
|
||
|
|
bf_t T0, T1;
|
||
|
|
|
||
|
|
bf_init(s, &T0);
|
||
|
|
bf_init(s, &T1);
|
||
|
|
bf_set_ui(G, 2 * b - 1);
|
||
|
|
bf_mul_ui(G, G, 6 * b - 1, prec, BF_RNDN);
|
||
|
|
bf_mul_ui(G, G, 6 * b - 5, prec, BF_RNDN);
|
||
|
|
bf_set_ui(&T0, CHUD_B);
|
||
|
|
bf_mul_ui(&T0, &T0, b, prec, BF_RNDN);
|
||
|
|
bf_set_ui(&T1, CHUD_A);
|
||
|
|
bf_add(&T0, &T0, &T1, prec, BF_RNDN);
|
||
|
|
bf_mul(P, G, &T0, prec, BF_RNDN);
|
||
|
|
P->sign = b & 1;
|
||
|
|
|
||
|
|
bf_set_ui(Q, b);
|
||
|
|
bf_mul_ui(Q, Q, b, prec, BF_RNDN);
|
||
|
|
bf_mul_ui(Q, Q, b, prec, BF_RNDN);
|
||
|
|
bf_mul_ui(Q, Q, (uint64_t)CHUD_C * CHUD_C * CHUD_C / 24, prec, BF_RNDN);
|
||
|
|
bf_delete(&T0);
|
||
|
|
bf_delete(&T1);
|
||
|
|
} else {
|
||
|
|
bf_t P2, Q2, G2;
|
||
|
|
|
||
|
|
bf_init(s, &P2);
|
||
|
|
bf_init(s, &Q2);
|
||
|
|
bf_init(s, &G2);
|
||
|
|
|
||
|
|
c = (a + b) / 2;
|
||
|
|
chud_bs(P, Q, G, a, c, 1, prec);
|
||
|
|
chud_bs(&P2, &Q2, &G2, c, b, need_g, prec);
|
||
|
|
|
||
|
|
/* Q = Q1 * Q2 */
|
||
|
|
/* G = G1 * G2 */
|
||
|
|
/* P = P1 * Q2 + P2 * G1 */
|
||
|
|
bf_mul(&P2, &P2, G, prec, BF_RNDN);
|
||
|
|
if (!need_g)
|
||
|
|
bf_set_ui(G, 0);
|
||
|
|
bf_mul(P, P, &Q2, prec, BF_RNDN);
|
||
|
|
bf_add(P, P, &P2, prec, BF_RNDN);
|
||
|
|
bf_delete(&P2);
|
||
|
|
|
||
|
|
bf_mul(Q, Q, &Q2, prec, BF_RNDN);
|
||
|
|
bf_delete(&Q2);
|
||
|
|
if (need_g)
|
||
|
|
bf_mul(G, G, &G2, prec, BF_RNDN);
|
||
|
|
bf_delete(&G2);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* compute Pi with faithful rounding at precision 'prec' using the
|
||
|
|
Chudnovsky formula */
|
||
|
|
static void bf_const_pi_internal(bf_t *Q, limb_t prec)
|
||
|
|
{
|
||
|
|
bf_context_t *s = Q->ctx;
|
||
|
|
int64_t n, prec1;
|
||
|
|
bf_t P, G;
|
||
|
|
|
||
|
|
/* number of serie terms */
|
||
|
|
n = prec / CHUD_BITS_PER_TERM + 1;
|
||
|
|
/* XXX: precision analysis */
|
||
|
|
prec1 = prec + 32;
|
||
|
|
|
||
|
|
bf_init(s, &P);
|
||
|
|
bf_init(s, &G);
|
||
|
|
|
||
|
|
chud_bs(&P, Q, &G, 0, n, 0, BF_PREC_INF);
|
||
|
|
|
||
|
|
bf_mul_ui(&G, Q, CHUD_A, prec1, BF_RNDN);
|
||
|
|
bf_add(&P, &G, &P, prec1, BF_RNDN);
|
||
|
|
bf_div(Q, Q, &P, prec1, BF_RNDF);
|
||
|
|
|
||
|
|
bf_set_ui(&P, CHUD_C / 64);
|
||
|
|
bf_rsqrt(&G, &P, prec1);
|
||
|
|
bf_mul_ui(&G, &G, (uint64_t)CHUD_C * CHUD_C / (8 * 12), prec1, BF_RNDF);
|
||
|
|
bf_mul(Q, Q, &G, prec, BF_RNDN);
|
||
|
|
bf_delete(&P);
|
||
|
|
bf_delete(&G);
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_const_get(bf_t *T, limb_t prec, bf_flags_t flags,
|
||
|
|
BFConstCache *c,
|
||
|
|
void (*func)(bf_t *res, limb_t prec))
|
||
|
|
{
|
||
|
|
limb_t ziv_extra_bits, prec1;
|
||
|
|
|
||
|
|
ziv_extra_bits = 32;
|
||
|
|
for(;;) {
|
||
|
|
prec1 = prec + ziv_extra_bits;
|
||
|
|
if (c->prec < prec1) {
|
||
|
|
if (c->val.len == 0)
|
||
|
|
bf_init(T->ctx, &c->val);
|
||
|
|
func(&c->val, prec1);
|
||
|
|
c->prec = prec1;
|
||
|
|
} else {
|
||
|
|
prec1 = c->prec;
|
||
|
|
}
|
||
|
|
bf_set(T, &c->val);
|
||
|
|
if (!bf_can_round(T, prec, flags & BF_RND_MASK, prec1)) {
|
||
|
|
/* and more precision and retry */
|
||
|
|
ziv_extra_bits = ziv_extra_bits + (ziv_extra_bits / 2);
|
||
|
|
} else {
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return bf_round(T, prec, flags);
|
||
|
|
}
|
||
|
|
|
||
|
|
static void bf_const_free(BFConstCache *c)
|
||
|
|
{
|
||
|
|
bf_delete(&c->val);
|
||
|
|
memset(c, 0, sizeof(*c));
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_const_log2(bf_t *T, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_context_t *s = T->ctx;
|
||
|
|
return bf_const_get(T, prec, flags, &s->log2_cache, bf_const_log2_internal);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_const_pi(bf_t *T, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_context_t *s = T->ctx;
|
||
|
|
return bf_const_get(T, prec, flags, &s->pi_cache, bf_const_pi_internal);
|
||
|
|
}
|
||
|
|
|
||
|
|
void bf_clear_cache(bf_context_t *s)
|
||
|
|
{
|
||
|
|
#ifdef USE_FFT_MUL
|
||
|
|
fft_clear_cache(s);
|
||
|
|
#endif
|
||
|
|
bf_const_free(&s->log2_cache);
|
||
|
|
bf_const_free(&s->pi_cache);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* ZivFunc should compute the result 'r' with faithful rounding at
|
||
|
|
precision 'prec'. For efficiency purposes, the final bf_round()
|
||
|
|
does not need to be done in the function. */
|
||
|
|
typedef int ZivFunc(bf_t *r, const bf_t *a, limb_t prec, void *opaque);
|
||
|
|
|
||
|
|
static int bf_ziv_rounding(bf_t *r, const bf_t *a,
|
||
|
|
limb_t prec, bf_flags_t flags,
|
||
|
|
ZivFunc *f, void *opaque)
|
||
|
|
{
|
||
|
|
int rnd_mode, ret;
|
||
|
|
slimb_t prec1, ziv_extra_bits;
|
||
|
|
|
||
|
|
rnd_mode = flags & BF_RND_MASK;
|
||
|
|
if (rnd_mode == BF_RNDF) {
|
||
|
|
/* no need to iterate */
|
||
|
|
f(r, a, prec, opaque);
|
||
|
|
ret = 0;
|
||
|
|
} else {
|
||
|
|
ziv_extra_bits = 32;
|
||
|
|
for(;;) {
|
||
|
|
prec1 = prec + ziv_extra_bits;
|
||
|
|
ret = f(r, a, prec1, opaque);
|
||
|
|
if (ret & (BF_ST_OVERFLOW | BF_ST_UNDERFLOW)) {
|
||
|
|
/* should never happen because it indicates the
|
||
|
|
rounding cannot be done correctly, but we do not
|
||
|
|
catch all the cases */
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
/* if the result is exact, we can stop */
|
||
|
|
if (!(ret & BF_ST_INEXACT)) {
|
||
|
|
ret = 0;
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if (bf_can_round(r, prec, rnd_mode, prec1)) {
|
||
|
|
ret = BF_ST_INEXACT;
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
ziv_extra_bits = ziv_extra_bits * 2;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return bf_round(r, prec, flags) | ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Compute the exponential using faithful rounding at precision 'prec'.
|
||
|
|
Note: the algorithm is from MPFR */
|
||
|
|
static int bf_exp_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
slimb_t n, K, l, i, prec1;
|
||
|
|
|
||
|
|
assert(r != a);
|
||
|
|
|
||
|
|
/* argument reduction:
|
||
|
|
T = a - n*log(2) with 0 <= T < log(2) and n integer.
|
||
|
|
*/
|
||
|
|
bf_init(s, T);
|
||
|
|
if (a->expn <= -1) {
|
||
|
|
/* 0 <= abs(a) <= 0.5 */
|
||
|
|
if (a->sign)
|
||
|
|
n = -1;
|
||
|
|
else
|
||
|
|
n = 0;
|
||
|
|
} else {
|
||
|
|
bf_const_log2(T, LIMB_BITS, BF_RNDZ);
|
||
|
|
bf_div(T, a, T, LIMB_BITS, BF_RNDD);
|
||
|
|
bf_get_limb(&n, T, 0);
|
||
|
|
}
|
||
|
|
|
||
|
|
K = bf_isqrt((prec + 1) / 2);
|
||
|
|
l = (prec - 1) / K + 1;
|
||
|
|
/* XXX: precision analysis ? */
|
||
|
|
prec1 = prec + (K + 2 * l + 18) + K + 8;
|
||
|
|
if (a->expn > 0)
|
||
|
|
prec1 += a->expn;
|
||
|
|
// printf("n=%ld K=%ld prec1=%ld\n", n, K, prec1);
|
||
|
|
|
||
|
|
bf_const_log2(T, prec1, BF_RNDF);
|
||
|
|
bf_mul_si(T, T, n, prec1, BF_RNDN);
|
||
|
|
bf_sub(T, a, T, prec1, BF_RNDN);
|
||
|
|
|
||
|
|
/* reduce the range of T */
|
||
|
|
bf_mul_2exp(T, -K, BF_PREC_INF, BF_RNDZ);
|
||
|
|
|
||
|
|
/* Taylor expansion around zero :
|
||
|
|
1 + x + x^2/2 + ... + x^n/n!
|
||
|
|
= (1 + x * (1 + x/2 * (1 + ... (x/n))))
|
||
|
|
*/
|
||
|
|
{
|
||
|
|
bf_t U_s, *U = &U_s;
|
||
|
|
|
||
|
|
bf_init(s, U);
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
for(i = l ; i >= 1; i--) {
|
||
|
|
bf_set_ui(U, i);
|
||
|
|
bf_div(U, T, U, prec1, BF_RNDN);
|
||
|
|
bf_mul(r, r, U, prec1, BF_RNDN);
|
||
|
|
bf_add_si(r, r, 1, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
bf_delete(U);
|
||
|
|
}
|
||
|
|
bf_delete(T);
|
||
|
|
|
||
|
|
/* undo the range reduction */
|
||
|
|
for(i = 0; i < K; i++) {
|
||
|
|
bf_mul(r, r, r, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* undo the argument reduction */
|
||
|
|
bf_mul_2exp(r, n, BF_PREC_INF, BF_RNDZ);
|
||
|
|
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_exp(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
assert(r != a);
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
if (a->sign)
|
||
|
|
bf_set_zero(r, 0);
|
||
|
|
else
|
||
|
|
bf_set_inf(r, 0);
|
||
|
|
} else {
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* crude overflow and underflow tests */
|
||
|
|
if (a->expn > 0) {
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
bf_t log2_s, *log2 = &log2_s;
|
||
|
|
slimb_t e_min, e_max;
|
||
|
|
e_max = (limb_t)1 << (bf_get_exp_bits(flags) - 1);
|
||
|
|
e_min = -e_max + 3;
|
||
|
|
if (flags & BF_FLAG_SUBNORMAL)
|
||
|
|
e_min -= (prec - 1);
|
||
|
|
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_init(s, log2);
|
||
|
|
bf_const_log2(log2, LIMB_BITS, BF_RNDU);
|
||
|
|
bf_mul_ui(T, log2, e_max, LIMB_BITS, BF_RNDU);
|
||
|
|
/* a > e_max * log(2) implies exp(a) > e_max */
|
||
|
|
if (bf_cmp_lt(T, a) > 0) {
|
||
|
|
/* overflow */
|
||
|
|
bf_delete(T);
|
||
|
|
bf_delete(log2);
|
||
|
|
return bf_set_overflow(r, 0, prec, flags);
|
||
|
|
}
|
||
|
|
/* a < e_min * log(2) implies exp(a) < e_min */
|
||
|
|
bf_mul_si(T, log2, e_min, LIMB_BITS, BF_RNDD);
|
||
|
|
if (bf_cmp_lt(a, T)) {
|
||
|
|
int rnd_mode = flags & BF_RND_MASK;
|
||
|
|
|
||
|
|
/* underflow */
|
||
|
|
bf_delete(T);
|
||
|
|
bf_delete(log2);
|
||
|
|
if (rnd_mode == BF_RNDU) {
|
||
|
|
/* set the smallest value */
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
r->expn = e_min;
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, 0);
|
||
|
|
}
|
||
|
|
return BF_ST_UNDERFLOW | BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
bf_delete(log2);
|
||
|
|
bf_delete(T);
|
||
|
|
}
|
||
|
|
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_exp_internal, NULL);
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_log_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
bf_t U_s, *U = &U_s;
|
||
|
|
bf_t V_s, *V = &V_s;
|
||
|
|
slimb_t n, prec1, l, i, K;
|
||
|
|
|
||
|
|
assert(r != a);
|
||
|
|
|
||
|
|
bf_init(s, T);
|
||
|
|
/* argument reduction 1 */
|
||
|
|
/* T=a*2^n with 2/3 <= T <= 4/3 */
|
||
|
|
{
|
||
|
|
bf_t U_s, *U = &U_s;
|
||
|
|
bf_set(T, a);
|
||
|
|
n = T->expn;
|
||
|
|
T->expn = 0;
|
||
|
|
/* U= ~ 2/3 */
|
||
|
|
bf_init(s, U);
|
||
|
|
bf_set_ui(U, 0xaaaaaaaa);
|
||
|
|
U->expn = 0;
|
||
|
|
if (bf_cmp_lt(T, U)) {
|
||
|
|
T->expn++;
|
||
|
|
n--;
|
||
|
|
}
|
||
|
|
bf_delete(U);
|
||
|
|
}
|
||
|
|
// printf("n=%ld\n", n);
|
||
|
|
// bf_print_str("T", T);
|
||
|
|
|
||
|
|
/* XXX: precision analysis */
|
||
|
|
/* number of iterations for argument reduction 2 */
|
||
|
|
K = bf_isqrt((prec + 1) / 2);
|
||
|
|
/* order of Taylor expansion */
|
||
|
|
l = prec / (2 * K) + 1;
|
||
|
|
/* precision of the intermediate computations */
|
||
|
|
prec1 = prec + K + 2 * l + 32;
|
||
|
|
|
||
|
|
bf_init(s, U);
|
||
|
|
bf_init(s, V);
|
||
|
|
|
||
|
|
/* Note: cancellation occurs here, so we use more precision (XXX:
|
||
|
|
reduce the precision by computing the exact cancellation) */
|
||
|
|
bf_add_si(T, T, -1, BF_PREC_INF, BF_RNDN);
|
||
|
|
|
||
|
|
/* argument reduction 2 */
|
||
|
|
for(i = 0; i < K; i++) {
|
||
|
|
/* T = T / (1 + sqrt(1 + T)) */
|
||
|
|
bf_add_si(U, T, 1, prec1, BF_RNDN);
|
||
|
|
bf_sqrt(V, U, prec1, BF_RNDF);
|
||
|
|
bf_add_si(U, V, 1, prec1, BF_RNDN);
|
||
|
|
bf_div(T, T, U, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
|
||
|
|
{
|
||
|
|
bf_t Y_s, *Y = &Y_s;
|
||
|
|
bf_t Y2_s, *Y2 = &Y2_s;
|
||
|
|
bf_init(s, Y);
|
||
|
|
bf_init(s, Y2);
|
||
|
|
|
||
|
|
/* compute ln(1+x) = ln((1+y)/(1-y)) with y=x/(2+x)
|
||
|
|
= y + y^3/3 + ... + y^(2*l + 1) / (2*l+1)
|
||
|
|
with Y=Y^2
|
||
|
|
= y*(1+Y/3+Y^2/5+...) = y*(1+Y*(1/3+Y*(1/5 + ...)))
|
||
|
|
*/
|
||
|
|
bf_add_si(Y, T, 2, prec1, BF_RNDN);
|
||
|
|
bf_div(Y, T, Y, prec1, BF_RNDN);
|
||
|
|
|
||
|
|
bf_mul(Y2, Y, Y, prec1, BF_RNDN);
|
||
|
|
bf_set_ui(r, 0);
|
||
|
|
for(i = l; i >= 1; i--) {
|
||
|
|
bf_set_ui(U, 1);
|
||
|
|
bf_set_ui(V, 2 * i + 1);
|
||
|
|
bf_div(U, U, V, prec1, BF_RNDN);
|
||
|
|
bf_add(r, r, U, prec1, BF_RNDN);
|
||
|
|
bf_mul(r, r, Y2, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
bf_add_si(r, r, 1, prec1, BF_RNDN);
|
||
|
|
bf_mul(r, r, Y, prec1, BF_RNDN);
|
||
|
|
bf_delete(Y);
|
||
|
|
bf_delete(Y2);
|
||
|
|
}
|
||
|
|
bf_delete(V);
|
||
|
|
bf_delete(U);
|
||
|
|
|
||
|
|
/* multiplication by 2 for the Taylor expansion and undo the
|
||
|
|
argument reduction 2*/
|
||
|
|
bf_mul_2exp(r, K + 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
|
||
|
|
/* undo the argument reduction 1 */
|
||
|
|
bf_const_log2(T, prec1, BF_RNDF);
|
||
|
|
bf_mul_si(T, T, n, prec1, BF_RNDN);
|
||
|
|
bf_add(r, r, T, prec1, BF_RNDN);
|
||
|
|
|
||
|
|
bf_delete(T);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_log(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
|
||
|
|
assert(r != a);
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
if (a->sign) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, 0);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, 1);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if (a->sign) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
}
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_set_ui(T, 1);
|
||
|
|
if (bf_cmp_eq(a, T)) {
|
||
|
|
bf_set_zero(r, 0);
|
||
|
|
bf_delete(T);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
bf_delete(T);
|
||
|
|
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_log_internal, NULL);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* x and y finite and x > 0 */
|
||
|
|
/* XXX: overflow/underflow handling */
|
||
|
|
static int bf_pow_generic(bf_t *r, const bf_t *x, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
const bf_t *y = opaque;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
limb_t prec1;
|
||
|
|
|
||
|
|
bf_init(s, T);
|
||
|
|
/* XXX: proof for the added precision */
|
||
|
|
prec1 = prec + 32;
|
||
|
|
bf_log(T, x, prec1, BF_RNDF);
|
||
|
|
bf_mul(T, T, y, prec1, BF_RNDF);
|
||
|
|
bf_exp(r, T, prec1, BF_RNDF);
|
||
|
|
bf_delete(T);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* x and y finite, x > 0, y integer and y fits on one limb */
|
||
|
|
/* XXX: overflow/underflow handling */
|
||
|
|
static int bf_pow_int(bf_t *r, const bf_t *x, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
const bf_t *y = opaque;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
limb_t prec1;
|
||
|
|
int ret;
|
||
|
|
slimb_t y1;
|
||
|
|
|
||
|
|
bf_get_limb(&y1, y, 0);
|
||
|
|
if (y1 < 0)
|
||
|
|
y1 = -y1;
|
||
|
|
/* XXX: proof for the added precision */
|
||
|
|
prec1 = prec + ceil_log2(y1) * 2 + 8;
|
||
|
|
ret = bf_pow_ui(r, x, y1 < 0 ? -y1 : y1, prec1, BF_RNDN);
|
||
|
|
if (y->sign) {
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_set_ui(T, 1);
|
||
|
|
ret |= bf_div(r, T, r, prec1, BF_RNDN);
|
||
|
|
bf_delete(T);
|
||
|
|
}
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* x must be a finite non zero float. Return TRUE if there is a
|
||
|
|
floating point number r such as x=r^(2^n) and return this floating
|
||
|
|
point number 'r'. Otherwise return FALSE and r is undefined. */
|
||
|
|
static BOOL check_exact_power2n(bf_t *r, const bf_t *x, slimb_t n)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
slimb_t e, i, er;
|
||
|
|
limb_t v;
|
||
|
|
|
||
|
|
/* x = m*2^e with m odd integer */
|
||
|
|
e = bf_get_exp_min(x);
|
||
|
|
/* fast check on the exponent */
|
||
|
|
if (n > (LIMB_BITS - 1)) {
|
||
|
|
if (e != 0)
|
||
|
|
return FALSE;
|
||
|
|
er = 0;
|
||
|
|
} else {
|
||
|
|
if ((e & (((limb_t)1 << n) - 1)) != 0)
|
||
|
|
return FALSE;
|
||
|
|
er = e >> n;
|
||
|
|
}
|
||
|
|
/* every perfect odd square = 1 modulo 8 */
|
||
|
|
v = get_bits(x->tab, x->len, x->len * LIMB_BITS - x->expn + e);
|
||
|
|
if ((v & 7) != 1)
|
||
|
|
return FALSE;
|
||
|
|
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_set(T, x);
|
||
|
|
T->expn -= e;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
if (i != 0)
|
||
|
|
bf_set(T, r);
|
||
|
|
if (bf_sqrtrem(r, NULL, T) != 0)
|
||
|
|
return FALSE;
|
||
|
|
}
|
||
|
|
r->expn += er;
|
||
|
|
return TRUE;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* prec = BF_PREC_INF is accepted for x and y integers and y >= 0 */
|
||
|
|
int bf_pow(bf_t *r, const bf_t *x, const bf_t *y, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
bf_t ytmp_s;
|
||
|
|
BOOL y_is_int, y_is_odd;
|
||
|
|
int r_sign, ret, rnd_mode;
|
||
|
|
slimb_t y_emin;
|
||
|
|
|
||
|
|
if (x->len == 0 || y->len == 0) {
|
||
|
|
if (y->expn == BF_EXP_ZERO) {
|
||
|
|
/* pow(x, 0) = 1 */
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
} else if (x->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else {
|
||
|
|
int cmp_x_abs_1;
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
cmp_x_abs_1 = bf_cmpu(x, r);
|
||
|
|
if (cmp_x_abs_1 == 0 && (flags & BF_POW_JS_QUICKS) &&
|
||
|
|
(y->expn >= BF_EXP_INF)) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (cmp_x_abs_1 == 0 &&
|
||
|
|
(!x->sign || y->expn != BF_EXP_NAN)) {
|
||
|
|
/* pow(1, y) = 1 even if y = NaN */
|
||
|
|
/* pow(-1, +/-inf) = 1 */
|
||
|
|
} else if (y->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
} else if (y->expn == BF_EXP_INF) {
|
||
|
|
if (y->sign == (cmp_x_abs_1 > 0)) {
|
||
|
|
bf_set_zero(r, 0);
|
||
|
|
} else {
|
||
|
|
bf_set_inf(r, 0);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
y_emin = bf_get_exp_min(y);
|
||
|
|
y_is_odd = (y_emin == 0);
|
||
|
|
if (y->sign == (x->expn == BF_EXP_ZERO)) {
|
||
|
|
bf_set_inf(r, y_is_odd & x->sign);
|
||
|
|
if (y->sign) {
|
||
|
|
/* pow(0, y) with y < 0 */
|
||
|
|
return BF_ST_DIVIDE_ZERO;
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, y_is_odd & x->sign);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_set(T, x);
|
||
|
|
y_emin = bf_get_exp_min(y);
|
||
|
|
y_is_int = (y_emin >= 0);
|
||
|
|
rnd_mode = flags & BF_RND_MASK;
|
||
|
|
if (x->sign) {
|
||
|
|
if (!y_is_int) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
bf_delete(T);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
}
|
||
|
|
y_is_odd = (y_emin == 0);
|
||
|
|
r_sign = y_is_odd;
|
||
|
|
/* change the directed rounding mode if the sign of the result
|
||
|
|
is changed */
|
||
|
|
if (r_sign && (rnd_mode == BF_RNDD || rnd_mode == BF_RNDU))
|
||
|
|
flags ^= 1;
|
||
|
|
bf_neg(T);
|
||
|
|
} else {
|
||
|
|
r_sign = 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
if (bf_cmp_eq(T, r)) {
|
||
|
|
/* abs(x) = 1: nothing more to do */
|
||
|
|
ret = 0;
|
||
|
|
} else if (y_is_int) {
|
||
|
|
slimb_t T_bits, e;
|
||
|
|
int_pow:
|
||
|
|
T_bits = T->expn - bf_get_exp_min(T);
|
||
|
|
if (T_bits == 1) {
|
||
|
|
/* pow(2^b, y) = 2^(b*y) */
|
||
|
|
bf_mul_si(T, y, T->expn - 1, LIMB_BITS, BF_RNDZ);
|
||
|
|
bf_get_limb(&e, T, 0);
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
ret = bf_mul_2exp(r, e, prec, flags);
|
||
|
|
} else if (prec == BF_PREC_INF) {
|
||
|
|
slimb_t y1;
|
||
|
|
/* specific case for infinite precision (integer case) */
|
||
|
|
bf_get_limb(&y1, y, 0);
|
||
|
|
assert(!y->sign);
|
||
|
|
/* x must be an integer, so abs(x) >= 2 */
|
||
|
|
if (y1 >= ((slimb_t)1 << BF_EXP_BITS_MAX)) {
|
||
|
|
bf_delete(T);
|
||
|
|
return bf_set_overflow(r, 0, BF_PREC_INF, flags);
|
||
|
|
}
|
||
|
|
ret = bf_pow_ui(r, T, y1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
} else {
|
||
|
|
if (y->expn <= 31) {
|
||
|
|
/* small enough power: use exponentiation in all cases */
|
||
|
|
} else if (y->sign) {
|
||
|
|
/* cannot be exact */
|
||
|
|
goto general_case;
|
||
|
|
} else {
|
||
|
|
if (rnd_mode == BF_RNDF)
|
||
|
|
goto general_case; /* no need to track exact results */
|
||
|
|
/* see if the result has a chance to be exact:
|
||
|
|
if x=a*2^b (a odd), x^y=a^y*2^(b*y)
|
||
|
|
x^y needs a precision of at least floor_log2(a)*y bits
|
||
|
|
*/
|
||
|
|
bf_mul_si(r, y, T_bits - 1, LIMB_BITS, BF_RNDZ);
|
||
|
|
bf_get_limb(&e, r, 0);
|
||
|
|
if (prec < e)
|
||
|
|
goto general_case;
|
||
|
|
}
|
||
|
|
ret = bf_ziv_rounding(r, T, prec, flags, bf_pow_int, (void *)y);
|
||
|
|
}
|
||
|
|
} else {
|
||
|
|
if (rnd_mode != BF_RNDF) {
|
||
|
|
bf_t *y1;
|
||
|
|
if (y_emin < 0 && check_exact_power2n(r, T, -y_emin)) {
|
||
|
|
/* the problem is reduced to a power to an integer */
|
||
|
|
#if 0
|
||
|
|
printf("\nn=%ld\n", -y_emin);
|
||
|
|
bf_print_str("T", T);
|
||
|
|
bf_print_str("r", r);
|
||
|
|
#endif
|
||
|
|
bf_set(T, r);
|
||
|
|
y1 = &ytmp_s;
|
||
|
|
y1->tab = y->tab;
|
||
|
|
y1->len = y->len;
|
||
|
|
y1->sign = y->sign;
|
||
|
|
y1->expn = y->expn - y_emin;
|
||
|
|
y = y1;
|
||
|
|
goto int_pow;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
general_case:
|
||
|
|
ret = bf_ziv_rounding(r, T, prec, flags, bf_pow_generic, (void *)y);
|
||
|
|
}
|
||
|
|
bf_delete(T);
|
||
|
|
r->sign = r_sign;
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* compute sqrt(-2*x-x^2) to get |sin(x)| from cos(x) - 1. */
|
||
|
|
static void bf_sqrt_sin(bf_t *r, const bf_t *x, limb_t prec1)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_set(T, x);
|
||
|
|
bf_mul(r, T, T, prec1, BF_RNDN);
|
||
|
|
bf_mul_2exp(T, 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_add(T, T, r, prec1, BF_RNDN);
|
||
|
|
bf_neg(T);
|
||
|
|
bf_sqrt(r, T, prec1, BF_RNDF);
|
||
|
|
bf_delete(T);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_sincos(bf_t *s, bf_t *c, const bf_t *a, limb_t prec)
|
||
|
|
{
|
||
|
|
bf_context_t *s1 = a->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
bf_t U_s, *U = &U_s;
|
||
|
|
bf_t r_s, *r = &r_s;
|
||
|
|
slimb_t K, prec1, i, l, mod, prec2;
|
||
|
|
int is_neg;
|
||
|
|
|
||
|
|
assert(c != a && s != a);
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
if (c)
|
||
|
|
bf_set_nan(c);
|
||
|
|
if (s)
|
||
|
|
bf_set_nan(s);
|
||
|
|
return 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
if (c)
|
||
|
|
bf_set_nan(c);
|
||
|
|
if (s)
|
||
|
|
bf_set_nan(s);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
if (c)
|
||
|
|
bf_set_ui(c, 1);
|
||
|
|
if (s)
|
||
|
|
bf_set_zero(s, a->sign);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
bf_init(s1, T);
|
||
|
|
bf_init(s1, U);
|
||
|
|
bf_init(s1, r);
|
||
|
|
|
||
|
|
/* XXX: precision analysis */
|
||
|
|
K = bf_isqrt(prec / 2);
|
||
|
|
l = prec / (2 * K) + 1;
|
||
|
|
prec1 = prec + 2 * K + l + 8;
|
||
|
|
|
||
|
|
/* after the modulo reduction, -pi/4 <= T <= pi/4 */
|
||
|
|
if (a->expn <= -1) {
|
||
|
|
/* abs(a) <= 0.25: no modulo reduction needed */
|
||
|
|
bf_set(T, a);
|
||
|
|
mod = 0;
|
||
|
|
} else {
|
||
|
|
slimb_t cancel;
|
||
|
|
cancel = 0;
|
||
|
|
for(;;) {
|
||
|
|
prec2 = prec1 + cancel;
|
||
|
|
bf_const_pi(U, prec2, BF_RNDF);
|
||
|
|
bf_mul_2exp(U, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_remquo(&mod, T, a, U, prec2, BF_RNDN);
|
||
|
|
// printf("T.expn=%ld prec2=%ld\n", T->expn, prec2);
|
||
|
|
if (mod == 0 || (T->expn != BF_EXP_ZERO &&
|
||
|
|
(T->expn + prec2) >= (prec1 - 1)))
|
||
|
|
break;
|
||
|
|
/* increase the number of bits until the precision is good enough */
|
||
|
|
cancel = bf_max(-T->expn, (cancel + 1) * 3 / 2);
|
||
|
|
}
|
||
|
|
mod &= 3;
|
||
|
|
}
|
||
|
|
|
||
|
|
is_neg = T->sign;
|
||
|
|
|
||
|
|
/* compute cosm1(x) = cos(x) - 1 */
|
||
|
|
bf_mul(T, T, T, prec1, BF_RNDN);
|
||
|
|
bf_mul_2exp(T, -2 * K, BF_PREC_INF, BF_RNDZ);
|
||
|
|
|
||
|
|
/* Taylor expansion:
|
||
|
|
-x^2/2 + x^4/4! - x^6/6! + ...
|
||
|
|
*/
|
||
|
|
bf_set_ui(r, 1);
|
||
|
|
for(i = l ; i >= 1; i--) {
|
||
|
|
bf_set_ui(U, 2 * i - 1);
|
||
|
|
bf_mul_ui(U, U, 2 * i, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_div(U, T, U, prec1, BF_RNDN);
|
||
|
|
bf_mul(r, r, U, prec1, BF_RNDN);
|
||
|
|
bf_neg(r);
|
||
|
|
if (i != 1)
|
||
|
|
bf_add_si(r, r, 1, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
bf_delete(U);
|
||
|
|
|
||
|
|
/* undo argument reduction:
|
||
|
|
cosm1(2*x)= 2*(2*cosm1(x)+cosm1(x)^2)
|
||
|
|
*/
|
||
|
|
for(i = 0; i < K; i++) {
|
||
|
|
bf_mul(T, r, r, prec1, BF_RNDN);
|
||
|
|
bf_mul_2exp(r, 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
bf_add(r, r, T, prec1, BF_RNDN);
|
||
|
|
bf_mul_2exp(r, 1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
}
|
||
|
|
bf_delete(T);
|
||
|
|
|
||
|
|
if (c) {
|
||
|
|
if ((mod & 1) == 0) {
|
||
|
|
bf_add_si(c, r, 1, prec1, BF_RNDN);
|
||
|
|
} else {
|
||
|
|
bf_sqrt_sin(c, r, prec1);
|
||
|
|
c->sign = is_neg ^ 1;
|
||
|
|
}
|
||
|
|
c->sign ^= mod >> 1;
|
||
|
|
}
|
||
|
|
if (s) {
|
||
|
|
if ((mod & 1) == 0) {
|
||
|
|
bf_sqrt_sin(s, r, prec1);
|
||
|
|
s->sign = is_neg;
|
||
|
|
} else {
|
||
|
|
bf_add_si(s, r, 1, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
s->sign ^= mod >> 1;
|
||
|
|
}
|
||
|
|
bf_delete(r);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_cos_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
return bf_sincos(NULL, r, a, prec);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_cos(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_cos_internal, NULL);
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_sin_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
return bf_sincos(r, NULL, a, prec);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_sin(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_sin_internal, NULL);
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_tan_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
limb_t prec1;
|
||
|
|
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, a->sign);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* XXX: precision analysis */
|
||
|
|
prec1 = prec + 8;
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_sincos(r, T, a, prec1);
|
||
|
|
bf_div(r, r, T, prec1, BF_RNDF);
|
||
|
|
bf_delete(T);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_tan(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_tan_internal, NULL);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* if add_pi2 is true, add pi/2 to the result (used for acos(x) to
|
||
|
|
avoid cancellation) */
|
||
|
|
static int bf_atan_internal(bf_t *r, const bf_t *a, limb_t prec,
|
||
|
|
void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
BOOL add_pi2 = (BOOL)(intptr_t)opaque;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
bf_t U_s, *U = &U_s;
|
||
|
|
bf_t V_s, *V = &V_s;
|
||
|
|
bf_t X2_s, *X2 = &X2_s;
|
||
|
|
int cmp_1;
|
||
|
|
slimb_t prec1, i, K, l;
|
||
|
|
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return 0;
|
||
|
|
} else {
|
||
|
|
if (a->expn == BF_EXP_INF)
|
||
|
|
i = 1 - 2 * a->sign;
|
||
|
|
else
|
||
|
|
i = 0;
|
||
|
|
i += add_pi2;
|
||
|
|
/* return i*(pi/2) with -1 <= i <= 2 */
|
||
|
|
if (i == 0) {
|
||
|
|
bf_set_zero(r, add_pi2 ? 0 : a->sign);
|
||
|
|
return 0;
|
||
|
|
} else {
|
||
|
|
/* PI or PI/2 */
|
||
|
|
bf_const_pi(r, prec, BF_RNDF);
|
||
|
|
if (i != 2)
|
||
|
|
bf_mul_2exp(r, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
r->sign = (i < 0);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_set_ui(T, 1);
|
||
|
|
cmp_1 = bf_cmpu(a, T);
|
||
|
|
if (cmp_1 == 0 && !add_pi2) {
|
||
|
|
/* short cut: abs(a) == 1 -> +/-pi/4 */
|
||
|
|
bf_const_pi(r, prec, BF_RNDF);
|
||
|
|
bf_mul_2exp(r, -2, BF_PREC_INF, BF_RNDZ);
|
||
|
|
r->sign = a->sign;
|
||
|
|
bf_delete(T);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* XXX: precision analysis */
|
||
|
|
K = bf_isqrt((prec + 1) / 2);
|
||
|
|
l = prec / (2 * K) + 1;
|
||
|
|
prec1 = prec + K + 2 * l + 32;
|
||
|
|
// printf("prec=%ld K=%ld l=%ld prec1=%ld\n", prec, K, l, prec1);
|
||
|
|
|
||
|
|
if (cmp_1 > 0) {
|
||
|
|
bf_set_ui(T, 1);
|
||
|
|
bf_div(T, T, a, prec1, BF_RNDN);
|
||
|
|
} else {
|
||
|
|
bf_set(T, a);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* abs(T) <= 1 */
|
||
|
|
|
||
|
|
/* argument reduction */
|
||
|
|
|
||
|
|
bf_init(s, U);
|
||
|
|
bf_init(s, V);
|
||
|
|
bf_init(s, X2);
|
||
|
|
for(i = 0; i < K; i++) {
|
||
|
|
/* T = T / (1 + sqrt(1 + T^2)) */
|
||
|
|
bf_mul(U, T, T, prec1, BF_RNDN);
|
||
|
|
bf_add_si(U, U, 1, prec1, BF_RNDN);
|
||
|
|
bf_sqrt(V, U, prec1, BF_RNDN);
|
||
|
|
bf_add_si(V, V, 1, prec1, BF_RNDN);
|
||
|
|
bf_div(T, T, V, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Taylor series:
|
||
|
|
x - x^3/3 + ... + (-1)^ l * y^(2*l + 1) / (2*l+1)
|
||
|
|
*/
|
||
|
|
bf_mul(X2, T, T, prec1, BF_RNDN);
|
||
|
|
bf_set_ui(r, 0);
|
||
|
|
for(i = l; i >= 1; i--) {
|
||
|
|
bf_set_si(U, 1);
|
||
|
|
bf_set_ui(V, 2 * i + 1);
|
||
|
|
bf_div(U, U, V, prec1, BF_RNDN);
|
||
|
|
bf_neg(r);
|
||
|
|
bf_add(r, r, U, prec1, BF_RNDN);
|
||
|
|
bf_mul(r, r, X2, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
bf_neg(r);
|
||
|
|
bf_add_si(r, r, 1, prec1, BF_RNDN);
|
||
|
|
bf_mul(r, r, T, prec1, BF_RNDN);
|
||
|
|
|
||
|
|
/* undo the argument reduction */
|
||
|
|
bf_mul_2exp(r, K, BF_PREC_INF, BF_RNDZ);
|
||
|
|
|
||
|
|
bf_delete(U);
|
||
|
|
bf_delete(V);
|
||
|
|
bf_delete(X2);
|
||
|
|
|
||
|
|
i = add_pi2;
|
||
|
|
if (cmp_1 > 0) {
|
||
|
|
/* undo the inversion : r = sign(a)*PI/2 - r */
|
||
|
|
bf_neg(r);
|
||
|
|
i += 1 - 2 * a->sign;
|
||
|
|
}
|
||
|
|
/* add i*(pi/2) with -1 <= i <= 2 */
|
||
|
|
if (i != 0) {
|
||
|
|
bf_const_pi(T, prec1, BF_RNDF);
|
||
|
|
if (i != 2)
|
||
|
|
bf_mul_2exp(T, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
T->sign = (i < 0);
|
||
|
|
bf_add(r, T, r, prec1, BF_RNDN);
|
||
|
|
}
|
||
|
|
|
||
|
|
bf_delete(T);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_atan(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_atan_internal, (void *)FALSE);
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_atan2_internal(bf_t *r, const bf_t *y, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
const bf_t *x = opaque;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
limb_t prec1;
|
||
|
|
int ret;
|
||
|
|
|
||
|
|
if (y->expn == BF_EXP_NAN || x->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* compute atan(y/x) assumming inf/inf = 1 and 0/0 = 0 */
|
||
|
|
bf_init(s, T);
|
||
|
|
prec1 = prec + 32;
|
||
|
|
if (y->expn == BF_EXP_INF && x->expn == BF_EXP_INF) {
|
||
|
|
bf_set_ui(T, 1);
|
||
|
|
T->sign = y->sign ^ x->sign;
|
||
|
|
} else if (y->expn == BF_EXP_ZERO && x->expn == BF_EXP_ZERO) {
|
||
|
|
bf_set_zero(T, y->sign ^ x->sign);
|
||
|
|
} else {
|
||
|
|
bf_div(T, y, x, prec1, BF_RNDF);
|
||
|
|
}
|
||
|
|
ret = bf_atan(r, T, prec1, BF_RNDF);
|
||
|
|
|
||
|
|
if (x->sign) {
|
||
|
|
/* if x < 0 (it includes -0), return sign(y)*pi + atan(y/x) */
|
||
|
|
bf_const_pi(T, prec1, BF_RNDF);
|
||
|
|
T->sign = y->sign;
|
||
|
|
bf_add(r, r, T, prec1, BF_RNDN);
|
||
|
|
ret |= BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
bf_delete(T);
|
||
|
|
return ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_atan2(bf_t *r, const bf_t *y, const bf_t *x,
|
||
|
|
limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_ziv_rounding(r, y, prec, flags, bf_atan2_internal, (void *)x);
|
||
|
|
}
|
||
|
|
|
||
|
|
static int bf_asin_internal(bf_t *r, const bf_t *a, limb_t prec, void *opaque)
|
||
|
|
{
|
||
|
|
bf_context_t *s = r->ctx;
|
||
|
|
BOOL is_acos = (BOOL)(intptr_t)opaque;
|
||
|
|
bf_t T_s, *T = &T_s;
|
||
|
|
limb_t prec1, prec2;
|
||
|
|
int res;
|
||
|
|
|
||
|
|
if (a->len == 0) {
|
||
|
|
if (a->expn == BF_EXP_NAN) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return 0;
|
||
|
|
} else if (a->expn == BF_EXP_INF) {
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else {
|
||
|
|
if (is_acos) {
|
||
|
|
bf_const_pi(r, prec, BF_RNDF);
|
||
|
|
bf_mul_2exp(r, -1, BF_PREC_INF, BF_RNDZ);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
} else {
|
||
|
|
bf_set_zero(r, a->sign);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
bf_init(s, T);
|
||
|
|
bf_set_ui(T, 1);
|
||
|
|
res = bf_cmpu(a, T);
|
||
|
|
if (res > 0) {
|
||
|
|
bf_delete(T);
|
||
|
|
bf_set_nan(r);
|
||
|
|
return BF_ST_INVALID_OP;
|
||
|
|
} else if (res == 0 && a->sign == 0 && is_acos) {
|
||
|
|
bf_set_zero(r, 0);
|
||
|
|
bf_delete(T);
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* asin(x) = atan(x/sqrt(1-x^2))
|
||
|
|
acos(x) = pi/2 - asin(x) */
|
||
|
|
prec1 = prec + 8;
|
||
|
|
/* increase the precision in x^2 to compensate the cancellation in
|
||
|
|
(1-x^2) if x is close to 1 */
|
||
|
|
/* XXX: use less precision when possible */
|
||
|
|
if (a->expn >= 0)
|
||
|
|
prec2 = BF_PREC_INF;
|
||
|
|
else
|
||
|
|
prec2 = prec1;
|
||
|
|
bf_mul(T, a, a, prec2, BF_RNDN);
|
||
|
|
bf_neg(T);
|
||
|
|
bf_add_si(T, T, 1, prec2, BF_RNDN);
|
||
|
|
|
||
|
|
bf_sqrt(r, T, prec1, BF_RNDN);
|
||
|
|
bf_div(T, a, r, prec1, BF_RNDN);
|
||
|
|
if (is_acos)
|
||
|
|
bf_neg(T);
|
||
|
|
bf_atan_internal(r, T, prec1, (void *)(intptr_t)is_acos);
|
||
|
|
bf_delete(T);
|
||
|
|
return BF_ST_INEXACT;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_asin(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_asin_internal, (void *)FALSE);
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_acos(bf_t *r, const bf_t *a, limb_t prec, bf_flags_t flags)
|
||
|
|
{
|
||
|
|
return bf_ziv_rounding(r, a, prec, flags, bf_asin_internal, (void *)TRUE);
|
||
|
|
}
|
||
|
|
|
||
|
|
#ifdef USE_FFT_MUL
|
||
|
|
/***************************************************************/
|
||
|
|
/* Integer multiplication with FFT */
|
||
|
|
|
||
|
|
/* or LIMB_BITS at bit position 'pos' in tab */
|
||
|
|
static inline void put_bits(limb_t *tab, limb_t len, slimb_t pos, limb_t val)
|
||
|
|
{
|
||
|
|
limb_t i;
|
||
|
|
int p;
|
||
|
|
|
||
|
|
i = pos >> LIMB_LOG2_BITS;
|
||
|
|
p = pos & (LIMB_BITS - 1);
|
||
|
|
if (i < len)
|
||
|
|
tab[i] |= val << p;
|
||
|
|
if (p != 0) {
|
||
|
|
i++;
|
||
|
|
if (i < len) {
|
||
|
|
tab[i] |= val >> (LIMB_BITS - p);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
|
||
|
|
typedef double NTTLimb;
|
||
|
|
|
||
|
|
/* we must have: modulo >= 1 << NTT_MOD_LOG2_MIN */
|
||
|
|
#define NTT_MOD_LOG2_MIN 50
|
||
|
|
#define NTT_MOD_LOG2_MAX 51
|
||
|
|
#define NB_MODS 5
|
||
|
|
#define NTT_PROOT_2EXP 39
|
||
|
|
static const int ntt_int_bits[NB_MODS] = { 254, 203, 152, 101, 50, };
|
||
|
|
|
||
|
|
static const limb_t ntt_mods[NB_MODS] = { 0x00073a8000000001, 0x0007858000000001, 0x0007a38000000001, 0x0007a68000000001, 0x0007fd8000000001,
|
||
|
|
};
|
||
|
|
|
||
|
|
static const limb_t ntt_proot[2][NB_MODS] = {
|
||
|
|
{ 0x00056198d44332c8, 0x0002eb5d640aad39, 0x00047e31eaa35fd0, 0x0005271ac118a150, 0x00075e0ce8442bd5, },
|
||
|
|
{ 0x000461169761bcc5, 0x0002dac3cb2da688, 0x0004abc97751e3bf, 0x000656778fc8c485, 0x0000dc6469c269fa, },
|
||
|
|
};
|
||
|
|
|
||
|
|
static const limb_t ntt_mods_cr[NB_MODS * (NB_MODS - 1) / 2] = {
|
||
|
|
0x00020e4da740da8e, 0x0004c3dc09c09c1d, 0x000063bd097b4271, 0x000799d8f18f18fd,
|
||
|
|
0x0005384222222264, 0x000572b07c1f07fe, 0x00035cd08888889a,
|
||
|
|
0x00066015555557e3, 0x000725960b60b623,
|
||
|
|
0x0002fc1fa1d6ce12,
|
||
|
|
};
|
||
|
|
|
||
|
|
#else
|
||
|
|
|
||
|
|
typedef limb_t NTTLimb;
|
||
|
|
|
||
|
|
#if LIMB_BITS == 64
|
||
|
|
|
||
|
|
#define NTT_MOD_LOG2_MIN 61
|
||
|
|
#define NTT_MOD_LOG2_MAX 62
|
||
|
|
#define NB_MODS 5
|
||
|
|
#define NTT_PROOT_2EXP 51
|
||
|
|
static const int ntt_int_bits[NB_MODS] = { 307, 246, 185, 123, 61, };
|
||
|
|
|
||
|
|
static const limb_t ntt_mods[NB_MODS] = { 0x28d8000000000001, 0x2a88000000000001, 0x2ed8000000000001, 0x3508000000000001, 0x3aa8000000000001,
|
||
|
|
};
|
||
|
|
|
||
|
|
static const limb_t ntt_proot[2][NB_MODS] = {
|
||
|
|
{ 0x1b8ea61034a2bea7, 0x21a9762de58206fb, 0x02ca782f0756a8ea, 0x278384537a3e50a1, 0x106e13fee74ce0ab, },
|
||
|
|
{ 0x233513af133e13b8, 0x1d13140d1c6f75f1, 0x12cde57f97e3eeda, 0x0d6149e23cbe654f, 0x36cd204f522a1379, },
|
||
|
|
};
|
||
|
|
|
||
|
|
static const limb_t ntt_mods_cr[NB_MODS * (NB_MODS - 1) / 2] = {
|
||
|
|
0x08a9ed097b425eea, 0x18a44aaaaaaaaab3, 0x2493f57f57f57f5d, 0x126b8d0649a7f8d4,
|
||
|
|
0x09d80ed7303b5ccc, 0x25b8bcf3cf3cf3d5, 0x2ce6ce63398ce638,
|
||
|
|
0x0e31fad40a57eb59, 0x02a3529fd4a7f52f,
|
||
|
|
0x3a5493e93e93e94a,
|
||
|
|
};
|
||
|
|
|
||
|
|
#elif LIMB_BITS == 32
|
||
|
|
|
||
|
|
/* we must have: modulo >= 1 << NTT_MOD_LOG2_MIN */
|
||
|
|
#define NTT_MOD_LOG2_MIN 29
|
||
|
|
#define NTT_MOD_LOG2_MAX 30
|
||
|
|
#define NB_MODS 5
|
||
|
|
#define NTT_PROOT_2EXP 20
|
||
|
|
static const int ntt_int_bits[NB_MODS] = { 148, 119, 89, 59, 29, };
|
||
|
|
|
||
|
|
static const limb_t ntt_mods[NB_MODS] = { 0x0000000032b00001, 0x0000000033700001, 0x0000000036d00001, 0x0000000037300001, 0x000000003e500001,
|
||
|
|
};
|
||
|
|
|
||
|
|
static const limb_t ntt_proot[2][NB_MODS] = {
|
||
|
|
{ 0x0000000032525f31, 0x0000000005eb3b37, 0x00000000246eda9f, 0x0000000035f25901, 0x00000000022f5768, },
|
||
|
|
{ 0x00000000051eba1a, 0x00000000107be10e, 0x000000001cd574e0, 0x00000000053806e6, 0x000000002cd6bf98, },
|
||
|
|
};
|
||
|
|
|
||
|
|
static const limb_t ntt_mods_cr[NB_MODS * (NB_MODS - 1) / 2] = {
|
||
|
|
0x000000000449559a, 0x000000001eba6ca9, 0x000000002ec18e46, 0x000000000860160b,
|
||
|
|
0x000000000d321307, 0x000000000bf51120, 0x000000000f662938,
|
||
|
|
0x000000000932ab3e, 0x000000002f40eef8,
|
||
|
|
0x000000002e760905,
|
||
|
|
};
|
||
|
|
|
||
|
|
#endif /* LIMB_BITS */
|
||
|
|
|
||
|
|
#endif /* !AVX2 */
|
||
|
|
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
#define NTT_TRIG_K_MAX 18
|
||
|
|
#else
|
||
|
|
#define NTT_TRIG_K_MAX 19
|
||
|
|
#endif
|
||
|
|
|
||
|
|
typedef struct BFNTTState {
|
||
|
|
bf_context_t *ctx;
|
||
|
|
|
||
|
|
/* used for mul_mod_fast() */
|
||
|
|
limb_t ntt_mods_div[NB_MODS];
|
||
|
|
|
||
|
|
limb_t ntt_proot_pow[NB_MODS][2][NTT_PROOT_2EXP + 1];
|
||
|
|
limb_t ntt_proot_pow_inv[NB_MODS][2][NTT_PROOT_2EXP + 1];
|
||
|
|
NTTLimb *ntt_trig[NB_MODS][2][NTT_TRIG_K_MAX + 1];
|
||
|
|
/* 1/2^n mod m */
|
||
|
|
limb_t ntt_len_inv[NB_MODS][NTT_PROOT_2EXP + 1][2];
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
__m256d ntt_mods_cr_vec[NB_MODS * (NB_MODS - 1) / 2];
|
||
|
|
__m256d ntt_mods_vec[NB_MODS];
|
||
|
|
__m256d ntt_mods_inv_vec[NB_MODS];
|
||
|
|
#else
|
||
|
|
limb_t ntt_mods_cr_inv[NB_MODS * (NB_MODS - 1) / 2];
|
||
|
|
#endif
|
||
|
|
} BFNTTState;
|
||
|
|
|
||
|
|
static NTTLimb *get_trig(BFNTTState *s, int k, int inverse, int m_idx);
|
||
|
|
|
||
|
|
/* add modulo with up to (LIMB_BITS-1) bit modulo */
|
||
|
|
static inline limb_t add_mod(limb_t a, limb_t b, limb_t m)
|
||
|
|
{
|
||
|
|
limb_t r;
|
||
|
|
r = a + b;
|
||
|
|
if (r >= m)
|
||
|
|
r -= m;
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* sub modulo with up to LIMB_BITS bit modulo */
|
||
|
|
static inline limb_t sub_mod(limb_t a, limb_t b, limb_t m)
|
||
|
|
{
|
||
|
|
limb_t r;
|
||
|
|
r = a - b;
|
||
|
|
if (r > a)
|
||
|
|
r += m;
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* return (r0+r1*B) mod m
|
||
|
|
precondition: 0 <= r0+r1*B < 2^(64+NTT_MOD_LOG2_MIN)
|
||
|
|
*/
|
||
|
|
static inline limb_t mod_fast(dlimb_t r,
|
||
|
|
limb_t m, limb_t m_inv)
|
||
|
|
{
|
||
|
|
limb_t a1, q, t0, r1, r0;
|
||
|
|
|
||
|
|
a1 = r >> NTT_MOD_LOG2_MIN;
|
||
|
|
|
||
|
|
q = ((dlimb_t)a1 * m_inv) >> LIMB_BITS;
|
||
|
|
r = r - (dlimb_t)q * m - m * 2;
|
||
|
|
r1 = r >> LIMB_BITS;
|
||
|
|
t0 = (slimb_t)r1 >> 1;
|
||
|
|
r += m & t0;
|
||
|
|
r0 = r;
|
||
|
|
r1 = r >> LIMB_BITS;
|
||
|
|
r0 += m & r1;
|
||
|
|
return r0;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* faster version using precomputed modulo inverse.
|
||
|
|
precondition: 0 <= a * b < 2^(64+NTT_MOD_LOG2_MIN) */
|
||
|
|
static inline limb_t mul_mod_fast(limb_t a, limb_t b,
|
||
|
|
limb_t m, limb_t m_inv)
|
||
|
|
{
|
||
|
|
dlimb_t r;
|
||
|
|
r = (dlimb_t)a * (dlimb_t)b;
|
||
|
|
return mod_fast(r, m, m_inv);
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline limb_t init_mul_mod_fast(limb_t m)
|
||
|
|
{
|
||
|
|
dlimb_t t;
|
||
|
|
assert(m < (limb_t)1 << NTT_MOD_LOG2_MAX);
|
||
|
|
assert(m >= (limb_t)1 << NTT_MOD_LOG2_MIN);
|
||
|
|
t = (dlimb_t)1 << (LIMB_BITS + NTT_MOD_LOG2_MIN);
|
||
|
|
return t / m;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Faster version used when the multiplier is constant. 0 <= a < 2^64,
|
||
|
|
0 <= b < m. */
|
||
|
|
static inline limb_t mul_mod_fast2(limb_t a, limb_t b,
|
||
|
|
limb_t m, limb_t b_inv)
|
||
|
|
{
|
||
|
|
limb_t r, q;
|
||
|
|
|
||
|
|
q = ((dlimb_t)a * (dlimb_t)b_inv) >> LIMB_BITS;
|
||
|
|
r = a * b - q * m;
|
||
|
|
if (r >= m)
|
||
|
|
r -= m;
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Faster version used when the multiplier is constant. 0 <= a < 2^64,
|
||
|
|
0 <= b < m. Let r = a * b mod m. The return value is 'r' or 'r +
|
||
|
|
m'. */
|
||
|
|
static inline limb_t mul_mod_fast3(limb_t a, limb_t b,
|
||
|
|
limb_t m, limb_t b_inv)
|
||
|
|
{
|
||
|
|
limb_t r, q;
|
||
|
|
|
||
|
|
q = ((dlimb_t)a * (dlimb_t)b_inv) >> LIMB_BITS;
|
||
|
|
r = a * b - q * m;
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline limb_t init_mul_mod_fast2(limb_t b, limb_t m)
|
||
|
|
{
|
||
|
|
return ((dlimb_t)b << LIMB_BITS) / m;
|
||
|
|
}
|
||
|
|
|
||
|
|
#ifdef __AVX2__
|
||
|
|
|
||
|
|
static inline limb_t ntt_limb_to_int(NTTLimb a, limb_t m)
|
||
|
|
{
|
||
|
|
slimb_t v;
|
||
|
|
v = a;
|
||
|
|
if (v < 0)
|
||
|
|
v += m;
|
||
|
|
if (v >= m)
|
||
|
|
v -= m;
|
||
|
|
return v;
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline NTTLimb int_to_ntt_limb(limb_t a, limb_t m)
|
||
|
|
{
|
||
|
|
return (slimb_t)a;
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline NTTLimb int_to_ntt_limb2(limb_t a, limb_t m)
|
||
|
|
{
|
||
|
|
if (a >= (m / 2))
|
||
|
|
a -= m;
|
||
|
|
return (slimb_t)a;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* return r + m if r < 0 otherwise r. */
|
||
|
|
static inline __m256d ntt_mod1(__m256d r, __m256d m)
|
||
|
|
{
|
||
|
|
return _mm256_blendv_pd(r, r + m, r);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* input: abs(r) < 2 * m. Output: abs(r) < m */
|
||
|
|
static inline __m256d ntt_mod(__m256d r, __m256d mf, __m256d m2f)
|
||
|
|
{
|
||
|
|
return _mm256_blendv_pd(r, r + m2f, r) - mf;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* input: abs(a*b) < 2 * m^2, output: abs(r) < m */
|
||
|
|
static inline __m256d ntt_mul_mod(__m256d a, __m256d b, __m256d mf,
|
||
|
|
__m256d m_inv)
|
||
|
|
{
|
||
|
|
__m256d r, q, ab1, ab0, qm0, qm1;
|
||
|
|
ab1 = a * b;
|
||
|
|
q = _mm256_round_pd(ab1 * m_inv, 0); /* round to nearest */
|
||
|
|
qm1 = q * mf;
|
||
|
|
qm0 = _mm256_fmsub_pd(q, mf, qm1); /* low part */
|
||
|
|
ab0 = _mm256_fmsub_pd(a, b, ab1); /* low part */
|
||
|
|
r = (ab1 - qm1) + (ab0 - qm0);
|
||
|
|
return r;
|
||
|
|
}
|
||
|
|
|
||
|
|
static void *bf_aligned_malloc(bf_context_t *s, size_t size, size_t align)
|
||
|
|
{
|
||
|
|
void *ptr;
|
||
|
|
void **ptr1;
|
||
|
|
ptr = bf_malloc(s, size + sizeof(void *) + align - 1);
|
||
|
|
if (!ptr)
|
||
|
|
return NULL;
|
||
|
|
ptr1 = (void **)(((uintptr_t)ptr + sizeof(void *) + align - 1) &
|
||
|
|
~(align - 1));
|
||
|
|
ptr1[-1] = ptr;
|
||
|
|
return ptr1;
|
||
|
|
}
|
||
|
|
|
||
|
|
static void bf_aligned_free(bf_context_t *s, void *ptr)
|
||
|
|
{
|
||
|
|
if (!ptr)
|
||
|
|
return;
|
||
|
|
bf_free(s, ((void **)ptr)[-1]);
|
||
|
|
}
|
||
|
|
|
||
|
|
static void *ntt_malloc(BFNTTState *s, size_t size)
|
||
|
|
{
|
||
|
|
return bf_aligned_malloc(s->ctx, size, 64);
|
||
|
|
}
|
||
|
|
|
||
|
|
static void ntt_free(BFNTTState *s, void *ptr)
|
||
|
|
{
|
||
|
|
bf_aligned_free(s->ctx, ptr);
|
||
|
|
}
|
||
|
|
|
||
|
|
static no_inline void ntt_fft(BFNTTState *s,
|
||
|
|
NTTLimb *out_buf, NTTLimb *in_buf,
|
||
|
|
NTTLimb *tmp_buf, int fft_len_log2,
|
||
|
|
int inverse, int m_idx)
|
||
|
|
{
|
||
|
|
limb_t nb_blocks, fft_per_block, p, k, n, stride_in, i, j;
|
||
|
|
NTTLimb *tab_in, *tab_out, *tmp, *trig;
|
||
|
|
__m256d m_inv, mf, m2f, c, a0, a1, b0, b1;
|
||
|
|
limb_t m;
|
||
|
|
int l;
|
||
|
|
|
||
|
|
m = ntt_mods[m_idx];
|
||
|
|
|
||
|
|
m_inv = _mm256_set1_pd(1.0 / (double)m);
|
||
|
|
mf = _mm256_set1_pd(m);
|
||
|
|
m2f = _mm256_set1_pd(m * 2);
|
||
|
|
|
||
|
|
n = (limb_t)1 << fft_len_log2;
|
||
|
|
assert(n >= 8);
|
||
|
|
stride_in = n / 2;
|
||
|
|
|
||
|
|
tab_in = in_buf;
|
||
|
|
tab_out = tmp_buf;
|
||
|
|
trig = get_trig(s, fft_len_log2, inverse, m_idx);
|
||
|
|
p = 0;
|
||
|
|
for(k = 0; k < stride_in; k += 4) {
|
||
|
|
a0 = _mm256_load_pd(&tab_in[k]);
|
||
|
|
a1 = _mm256_load_pd(&tab_in[k + stride_in]);
|
||
|
|
c = _mm256_load_pd(trig);
|
||
|
|
trig += 4;
|
||
|
|
b0 = ntt_mod(a0 + a1, mf, m2f);
|
||
|
|
b1 = ntt_mul_mod(a0 - a1, c, mf, m_inv);
|
||
|
|
a0 = _mm256_permute2f128_pd(b0, b1, 0x20);
|
||
|
|
a1 = _mm256_permute2f128_pd(b0, b1, 0x31);
|
||
|
|
a0 = _mm256_permute4x64_pd(a0, 0xd8);
|
||
|
|
a1 = _mm256_permute4x64_pd(a1, 0xd8);
|
||
|
|
_mm256_store_pd(&tab_out[p], a0);
|
||
|
|
_mm256_store_pd(&tab_out[p + 4], a1);
|
||
|
|
p += 2 * 4;
|
||
|
|
}
|
||
|
|
tmp = tab_in;
|
||
|
|
tab_in = tab_out;
|
||
|
|
tab_out = tmp;
|
||
|
|
|
||
|
|
trig = get_trig(s, fft_len_log2 - 1, inverse, m_idx);
|
||
|
|
p = 0;
|
||
|
|
for(k = 0; k < stride_in; k += 4) {
|
||
|
|
a0 = _mm256_load_pd(&tab_in[k]);
|
||
|
|
a1 = _mm256_load_pd(&tab_in[k + stride_in]);
|
||
|
|
c = _mm256_setr_pd(trig[0], trig[0], trig[1], trig[1]);
|
||
|
|
trig += 2;
|
||
|
|
b0 = ntt_mod(a0 + a1, mf, m2f);
|
||
|
|
b1 = ntt_mul_mod(a0 - a1, c, mf, m_inv);
|
||
|
|
a0 = _mm256_permute2f128_pd(b0, b1, 0x20);
|
||
|
|
a1 = _mm256_permute2f128_pd(b0, b1, 0x31);
|
||
|
|
_mm256_store_pd(&tab_out[p], a0);
|
||
|
|
_mm256_store_pd(&tab_out[p + 4], a1);
|
||
|
|
p += 2 * 4;
|
||
|
|
}
|
||
|
|
tmp = tab_in;
|
||
|
|
tab_in = tab_out;
|
||
|
|
tab_out = tmp;
|
||
|
|
|
||
|
|
nb_blocks = n / 4;
|
||
|
|
fft_per_block = 4;
|
||
|
|
|
||
|
|
l = fft_len_log2 - 2;
|
||
|
|
while (nb_blocks != 2) {
|
||
|
|
nb_blocks >>= 1;
|
||
|
|
p = 0;
|
||
|
|
k = 0;
|
||
|
|
trig = get_trig(s, l, inverse, m_idx);
|
||
|
|
for(i = 0; i < nb_blocks; i++) {
|
||
|
|
c = _mm256_set1_pd(trig[0]);
|
||
|
|
trig++;
|
||
|
|
for(j = 0; j < fft_per_block; j += 4) {
|
||
|
|
a0 = _mm256_load_pd(&tab_in[k + j]);
|
||
|
|
a1 = _mm256_load_pd(&tab_in[k + j + stride_in]);
|
||
|
|
b0 = ntt_mod(a0 + a1, mf, m2f);
|
||
|
|
b1 = ntt_mul_mod(a0 - a1, c, mf, m_inv);
|
||
|
|
_mm256_store_pd(&tab_out[p + j], b0);
|
||
|
|
_mm256_store_pd(&tab_out[p + j + fft_per_block], b1);
|
||
|
|
}
|
||
|
|
k += fft_per_block;
|
||
|
|
p += 2 * fft_per_block;
|
||
|
|
}
|
||
|
|
fft_per_block <<= 1;
|
||
|
|
l--;
|
||
|
|
tmp = tab_in;
|
||
|
|
tab_in = tab_out;
|
||
|
|
tab_out = tmp;
|
||
|
|
}
|
||
|
|
|
||
|
|
tab_out = out_buf;
|
||
|
|
for(k = 0; k < stride_in; k += 4) {
|
||
|
|
a0 = _mm256_load_pd(&tab_in[k]);
|
||
|
|
a1 = _mm256_load_pd(&tab_in[k + stride_in]);
|
||
|
|
b0 = ntt_mod(a0 + a1, mf, m2f);
|
||
|
|
b1 = ntt_mod(a0 - a1, mf, m2f);
|
||
|
|
_mm256_store_pd(&tab_out[k], b0);
|
||
|
|
_mm256_store_pd(&tab_out[k + stride_in], b1);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static void ntt_vec_mul(BFNTTState *s,
|
||
|
|
NTTLimb *tab1, NTTLimb *tab2, limb_t fft_len_log2,
|
||
|
|
int k_tot, int m_idx)
|
||
|
|
{
|
||
|
|
limb_t i, c_inv, n, m;
|
||
|
|
__m256d m_inv, mf, a, b, c;
|
||
|
|
|
||
|
|
m = ntt_mods[m_idx];
|
||
|
|
c_inv = s->ntt_len_inv[m_idx][k_tot][0];
|
||
|
|
m_inv = _mm256_set1_pd(1.0 / (double)m);
|
||
|
|
mf = _mm256_set1_pd(m);
|
||
|
|
c = _mm256_set1_pd(int_to_ntt_limb(c_inv, m));
|
||
|
|
n = (limb_t)1 << fft_len_log2;
|
||
|
|
for(i = 0; i < n; i += 4) {
|
||
|
|
a = _mm256_load_pd(&tab1[i]);
|
||
|
|
b = _mm256_load_pd(&tab2[i]);
|
||
|
|
a = ntt_mul_mod(a, b, mf, m_inv);
|
||
|
|
a = ntt_mul_mod(a, c, mf, m_inv);
|
||
|
|
_mm256_store_pd(&tab1[i], a);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static no_inline void mul_trig(NTTLimb *buf,
|
||
|
|
limb_t n, limb_t c1, limb_t m, limb_t m_inv1)
|
||
|
|
{
|
||
|
|
limb_t i, c2, c3, c4;
|
||
|
|
__m256d c, c_mul, a0, mf, m_inv;
|
||
|
|
assert(n >= 2);
|
||
|
|
|
||
|
|
mf = _mm256_set1_pd(m);
|
||
|
|
m_inv = _mm256_set1_pd(1.0 / (double)m);
|
||
|
|
|
||
|
|
c2 = mul_mod_fast(c1, c1, m, m_inv1);
|
||
|
|
c3 = mul_mod_fast(c2, c1, m, m_inv1);
|
||
|
|
c4 = mul_mod_fast(c2, c2, m, m_inv1);
|
||
|
|
c = _mm256_setr_pd(1, int_to_ntt_limb(c1, m),
|
||
|
|
int_to_ntt_limb(c2, m), int_to_ntt_limb(c3, m));
|
||
|
|
c_mul = _mm256_set1_pd(int_to_ntt_limb(c4, m));
|
||
|
|
for(i = 0; i < n; i += 4) {
|
||
|
|
a0 = _mm256_load_pd(&buf[i]);
|
||
|
|
a0 = ntt_mul_mod(a0, c, mf, m_inv);
|
||
|
|
_mm256_store_pd(&buf[i], a0);
|
||
|
|
c = ntt_mul_mod(c, c_mul, mf, m_inv);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
#else
|
||
|
|
|
||
|
|
static void *ntt_malloc(BFNTTState *s, size_t size)
|
||
|
|
{
|
||
|
|
return bf_malloc(s->ctx, size);
|
||
|
|
}
|
||
|
|
|
||
|
|
static void ntt_free(BFNTTState *s, void *ptr)
|
||
|
|
{
|
||
|
|
bf_free(s->ctx, ptr);
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline limb_t ntt_limb_to_int(NTTLimb a, limb_t m)
|
||
|
|
{
|
||
|
|
if (a >= m)
|
||
|
|
a -= m;
|
||
|
|
return a;
|
||
|
|
}
|
||
|
|
|
||
|
|
static inline NTTLimb int_to_ntt_limb(slimb_t a, limb_t m)
|
||
|
|
{
|
||
|
|
return a;
|
||
|
|
}
|
||
|
|
|
||
|
|
static no_inline void ntt_fft(BFNTTState *s, NTTLimb *out_buf, NTTLimb *in_buf,
|
||
|
|
NTTLimb *tmp_buf, int fft_len_log2,
|
||
|
|
int inverse, int m_idx)
|
||
|
|
{
|
||
|
|
limb_t nb_blocks, fft_per_block, p, k, n, stride_in, i, j, m, m2;
|
||
|
|
NTTLimb *tab_in, *tab_out, *tmp, a0, a1, b0, b1, c, *trig, c_inv;
|
||
|
|
int l;
|
||
|
|
|
||
|
|
m = ntt_mods[m_idx];
|
||
|
|
m2 = 2 * m;
|
||
|
|
n = (limb_t)1 << fft_len_log2;
|
||
|
|
nb_blocks = n;
|
||
|
|
fft_per_block = 1;
|
||
|
|
stride_in = n / 2;
|
||
|
|
tab_in = in_buf;
|
||
|
|
tab_out = tmp_buf;
|
||
|
|
l = fft_len_log2;
|
||
|
|
while (nb_blocks != 2) {
|
||
|
|
nb_blocks >>= 1;
|
||
|
|
p = 0;
|
||
|
|
k = 0;
|
||
|
|
trig = get_trig(s, l, inverse, m_idx);
|
||
|
|
for(i = 0; i < nb_blocks; i++) {
|
||
|
|
c = trig[0];
|
||
|
|
c_inv = trig[1];
|
||
|
|
trig += 2;
|
||
|
|
for(j = 0; j < fft_per_block; j++) {
|
||
|
|
a0 = tab_in[k + j];
|
||
|
|
a1 = tab_in[k + j + stride_in];
|
||
|
|
b0 = add_mod(a0, a1, m2);
|
||
|
|
b1 = a0 - a1 + m2;
|
||
|
|
b1 = mul_mod_fast3(b1, c, m, c_inv);
|
||
|
|
tab_out[p + j] = b0;
|
||
|
|
tab_out[p + j + fft_per_block] = b1;
|
||
|
|
}
|
||
|
|
k += fft_per_block;
|
||
|
|
p += 2 * fft_per_block;
|
||
|
|
}
|
||
|
|
fft_per_block <<= 1;
|
||
|
|
l--;
|
||
|
|
tmp = tab_in;
|
||
|
|
tab_in = tab_out;
|
||
|
|
tab_out = tmp;
|
||
|
|
}
|
||
|
|
/* no twiddle in last step */
|
||
|
|
tab_out = out_buf;
|
||
|
|
for(k = 0; k < stride_in; k++) {
|
||
|
|
a0 = tab_in[k];
|
||
|
|
a1 = tab_in[k + stride_in];
|
||
|
|
b0 = add_mod(a0, a1, m2);
|
||
|
|
b1 = sub_mod(a0, a1, m2);
|
||
|
|
tab_out[k] = b0;
|
||
|
|
tab_out[k + stride_in] = b1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static void ntt_vec_mul(BFNTTState *s,
|
||
|
|
NTTLimb *tab1, NTTLimb *tab2, int fft_len_log2,
|
||
|
|
int k_tot, int m_idx)
|
||
|
|
{
|
||
|
|
limb_t i, norm, norm_inv, a, n, m, m_inv;
|
||
|
|
|
||
|
|
m = ntt_mods[m_idx];
|
||
|
|
m_inv = s->ntt_mods_div[m_idx];
|
||
|
|
norm = s->ntt_len_inv[m_idx][k_tot][0];
|
||
|
|
norm_inv = s->ntt_len_inv[m_idx][k_tot][1];
|
||
|
|
n = (limb_t)1 << fft_len_log2;
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
a = tab1[i];
|
||
|
|
/* need to reduce the range so that the product is <
|
||
|
|
2^(LIMB_BITS+NTT_MOD_LOG2_MIN) */
|
||
|
|
if (a >= m)
|
||
|
|
a -= m;
|
||
|
|
a = mul_mod_fast(a, tab2[i], m, m_inv);
|
||
|
|
a = mul_mod_fast3(a, norm, m, norm_inv);
|
||
|
|
tab1[i] = a;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static no_inline void mul_trig(NTTLimb *buf,
|
||
|
|
limb_t n, limb_t c_mul, limb_t m, limb_t m_inv)
|
||
|
|
{
|
||
|
|
limb_t i, c0, c_mul_inv;
|
||
|
|
|
||
|
|
c0 = 1;
|
||
|
|
c_mul_inv = init_mul_mod_fast2(c_mul, m);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
buf[i] = mul_mod_fast(buf[i], c0, m, m_inv);
|
||
|
|
c0 = mul_mod_fast2(c0, c_mul, m, c_mul_inv);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
#endif /* !AVX2 */
|
||
|
|
|
||
|
|
static no_inline NTTLimb *get_trig(BFNTTState *s,
|
||
|
|
int k, int inverse1, int m_idx1)
|
||
|
|
{
|
||
|
|
NTTLimb *tab;
|
||
|
|
limb_t i, n2, c, c_mul, m, c_mul_inv;
|
||
|
|
int m_idx, inverse;
|
||
|
|
|
||
|
|
if (k > NTT_TRIG_K_MAX)
|
||
|
|
return NULL;
|
||
|
|
|
||
|
|
for(;;) {
|
||
|
|
tab = s->ntt_trig[m_idx1][inverse1][k];
|
||
|
|
if (tab)
|
||
|
|
return tab;
|
||
|
|
n2 = (limb_t)1 << (k - 1);
|
||
|
|
for(m_idx = 0; m_idx < NB_MODS; m_idx++) {
|
||
|
|
m = ntt_mods[m_idx];
|
||
|
|
for(inverse = 0; inverse < 2; inverse++) {
|
||
|
|
#ifdef __AVX2__
|
||
|
|
tab = ntt_malloc(s, sizeof(NTTLimb) * n2);
|
||
|
|
#else
|
||
|
|
tab = ntt_malloc(s, sizeof(NTTLimb) * n2 * 2);
|
||
|
|
#endif
|
||
|
|
c = 1;
|
||
|
|
c_mul = s->ntt_proot_pow[m_idx][inverse][k];
|
||
|
|
c_mul_inv = s->ntt_proot_pow_inv[m_idx][inverse][k];
|
||
|
|
for(i = 0; i < n2; i++) {
|
||
|
|
#ifdef __AVX2__
|
||
|
|
tab[i] = int_to_ntt_limb2(c, m);
|
||
|
|
#else
|
||
|
|
tab[2 * i] = int_to_ntt_limb(c, m);
|
||
|
|
tab[2 * i + 1] = init_mul_mod_fast2(c, m);
|
||
|
|
#endif
|
||
|
|
c = mul_mod_fast2(c, c_mul, m, c_mul_inv);
|
||
|
|
}
|
||
|
|
s->ntt_trig[m_idx][inverse][k] = tab;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
void fft_clear_cache(bf_context_t *s1)
|
||
|
|
{
|
||
|
|
int m_idx, inverse, k;
|
||
|
|
BFNTTState *s = s1->ntt_state;
|
||
|
|
if (s) {
|
||
|
|
for(m_idx = 0; m_idx < NB_MODS; m_idx++) {
|
||
|
|
for(inverse = 0; inverse < 2; inverse++) {
|
||
|
|
for(k = 0; k < NTT_TRIG_K_MAX + 1; k++) {
|
||
|
|
if (s->ntt_trig[m_idx][inverse][k]) {
|
||
|
|
ntt_free(s, s->ntt_trig[m_idx][inverse][k]);
|
||
|
|
s->ntt_trig[m_idx][inverse][k] = NULL;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
bf_aligned_free(s1, s);
|
||
|
|
#else
|
||
|
|
bf_free(s1, s);
|
||
|
|
#endif
|
||
|
|
s1->ntt_state = NULL;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
#define STRIP_LEN 16
|
||
|
|
|
||
|
|
/* dst = buf1, src = buf2 */
|
||
|
|
static void ntt_fft_partial(BFNTTState *s, NTTLimb *buf1,
|
||
|
|
int k1, int k2, limb_t n1, limb_t n2, int inverse,
|
||
|
|
limb_t m_idx)
|
||
|
|
{
|
||
|
|
limb_t i, j, c_mul, c0, m, m_inv, strip_len, l;
|
||
|
|
NTTLimb *buf2, *buf3;
|
||
|
|
|
||
|
|
buf3 = ntt_malloc(s, sizeof(NTTLimb) * n1);
|
||
|
|
if (k2 == 0) {
|
||
|
|
ntt_fft(s, buf1, buf1, buf3, k1, inverse, m_idx);
|
||
|
|
} else {
|
||
|
|
strip_len = STRIP_LEN;
|
||
|
|
buf2 = ntt_malloc(s, sizeof(NTTLimb) * n1 * strip_len);
|
||
|
|
|
||
|
|
m = ntt_mods[m_idx];
|
||
|
|
m_inv = s->ntt_mods_div[m_idx];
|
||
|
|
c0 = s->ntt_proot_pow[m_idx][inverse][k1 + k2];
|
||
|
|
c_mul = 1;
|
||
|
|
assert((n2 % strip_len) == 0);
|
||
|
|
for(j = 0; j < n2; j += strip_len) {
|
||
|
|
for(i = 0; i < n1; i++) {
|
||
|
|
for(l = 0; l < strip_len; l++) {
|
||
|
|
buf2[i + l * n1] = buf1[i * n2 + (j + l)];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
for(l = 0; l < strip_len; l++) {
|
||
|
|
if (inverse)
|
||
|
|
mul_trig(buf2 + l * n1, n1, c_mul, m, m_inv);
|
||
|
|
ntt_fft(s, buf2 + l * n1, buf2 + l * n1, buf3, k1, inverse, m_idx);
|
||
|
|
if (!inverse)
|
||
|
|
mul_trig(buf2 + l * n1, n1, c_mul, m, m_inv);
|
||
|
|
c_mul = mul_mod_fast(c_mul, c0, m, m_inv);
|
||
|
|
}
|
||
|
|
|
||
|
|
for(i = 0; i < n1; i++) {
|
||
|
|
for(l = 0; l < strip_len; l++) {
|
||
|
|
buf1[i * n2 + (j + l)] = buf2[i + l *n1];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
ntt_free(s, buf2);
|
||
|
|
}
|
||
|
|
ntt_free(s, buf3);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
/* dst = buf1, src = buf2, tmp = buf3 */
|
||
|
|
static void ntt_conv(BFNTTState *s, NTTLimb *buf1, NTTLimb *buf2,
|
||
|
|
int k, int k_tot, limb_t m_idx)
|
||
|
|
{
|
||
|
|
limb_t n1, n2, i;
|
||
|
|
int k1, k2;
|
||
|
|
|
||
|
|
if (k <= NTT_TRIG_K_MAX) {
|
||
|
|
k1 = k;
|
||
|
|
} else {
|
||
|
|
/* recursive split of the FFT */
|
||
|
|
k1 = bf_min(k / 2, NTT_TRIG_K_MAX);
|
||
|
|
}
|
||
|
|
k2 = k - k1;
|
||
|
|
n1 = (limb_t)1 << k1;
|
||
|
|
n2 = (limb_t)1 << k2;
|
||
|
|
|
||
|
|
ntt_fft_partial(s, buf1, k1, k2, n1, n2, 0, m_idx);
|
||
|
|
ntt_fft_partial(s, buf2, k1, k2, n1, n2, 0, m_idx);
|
||
|
|
if (k2 == 0) {
|
||
|
|
ntt_vec_mul(s, buf1, buf2, k, k_tot, m_idx);
|
||
|
|
} else {
|
||
|
|
for(i = 0; i < n1; i++) {
|
||
|
|
ntt_conv(s, buf1 + i * n2, buf2 + i * n2, k2, k_tot, m_idx);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
ntt_fft_partial(s, buf1, k1, k2, n1, n2, 1, m_idx);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
static no_inline void limb_to_ntt(BFNTTState *s,
|
||
|
|
NTTLimb *tabr, limb_t fft_len,
|
||
|
|
const limb_t *taba, limb_t a_len, int dpl,
|
||
|
|
int first_m_idx, int nb_mods)
|
||
|
|
{
|
||
|
|
slimb_t i, n;
|
||
|
|
dlimb_t a, b;
|
||
|
|
int j, shift;
|
||
|
|
limb_t base_mask1, a0, a1, a2, r, m, m_inv;
|
||
|
|
|
||
|
|
#if 0
|
||
|
|
for(i = 0; i < a_len; i++) {
|
||
|
|
printf("%" PRId64 ": " FMT_LIMB "\n",
|
||
|
|
(int64_t)i, taba[i]);
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
memset(tabr, 0, sizeof(NTTLimb) * fft_len * nb_mods);
|
||
|
|
shift = dpl & (LIMB_BITS - 1);
|
||
|
|
if (shift == 0)
|
||
|
|
base_mask1 = -1;
|
||
|
|
else
|
||
|
|
base_mask1 = ((limb_t)1 << shift) - 1;
|
||
|
|
n = bf_min(fft_len, (a_len * LIMB_BITS + dpl - 1) / dpl);
|
||
|
|
for(i = 0; i < n; i++) {
|
||
|
|
a0 = get_bits(taba, a_len, i * dpl);
|
||
|
|
if (dpl <= LIMB_BITS) {
|
||
|
|
a0 &= base_mask1;
|
||
|
|
a = a0;
|
||
|
|
} else {
|
||
|
|
a1 = get_bits(taba, a_len, i * dpl + LIMB_BITS);
|
||
|
|
if (dpl <= (LIMB_BITS + NTT_MOD_LOG2_MIN)) {
|
||
|
|
a = a0 | ((dlimb_t)(a1 & base_mask1) << LIMB_BITS);
|
||
|
|
} else {
|
||
|
|
if (dpl > 2 * LIMB_BITS) {
|
||
|
|
a2 = get_bits(taba, a_len, i * dpl + LIMB_BITS * 2) &
|
||
|
|
base_mask1;
|
||
|
|
} else {
|
||
|
|
a1 &= base_mask1;
|
||
|
|
a2 = 0;
|
||
|
|
}
|
||
|
|
// printf("a=0x%016lx%016lx%016lx\n", a2, a1, a0);
|
||
|
|
a = (a0 >> (LIMB_BITS - NTT_MOD_LOG2_MAX + NTT_MOD_LOG2_MIN)) |
|
||
|
|
((dlimb_t)a1 << (NTT_MOD_LOG2_MAX - NTT_MOD_LOG2_MIN)) |
|
||
|
|
((dlimb_t)a2 << (LIMB_BITS + NTT_MOD_LOG2_MAX - NTT_MOD_LOG2_MIN));
|
||
|
|
a0 &= ((limb_t)1 << (LIMB_BITS - NTT_MOD_LOG2_MAX + NTT_MOD_LOG2_MIN)) - 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
for(j = 0; j < nb_mods; j++) {
|
||
|
|
m = ntt_mods[first_m_idx + j];
|
||
|
|
m_inv = s->ntt_mods_div[first_m_idx + j];
|
||
|
|
r = mod_fast(a, m, m_inv);
|
||
|
|
if (dpl > (LIMB_BITS + NTT_MOD_LOG2_MIN)) {
|
||
|
|
b = ((dlimb_t)r << (LIMB_BITS - NTT_MOD_LOG2_MAX + NTT_MOD_LOG2_MIN)) | a0;
|
||
|
|
r = mod_fast(b, m, m_inv);
|
||
|
|
}
|
||
|
|
tabr[i + j * fft_len] = int_to_ntt_limb(r, m);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
|
||
|
|
#define VEC_LEN 4
|
||
|
|
|
||
|
|
typedef union {
|
||
|
|
__m256d v;
|
||
|
|
double d[4];
|
||
|
|
} VecUnion;
|
||
|
|
|
||
|
|
static no_inline void ntt_to_limb(BFNTTState *s, limb_t *tabr, limb_t r_len,
|
||
|
|
const NTTLimb *buf, int fft_len_log2, int dpl,
|
||
|
|
int nb_mods)
|
||
|
|
{
|
||
|
|
const limb_t *mods = ntt_mods + NB_MODS - nb_mods;
|
||
|
|
const __m256d *mods_cr_vec, *mf, *m_inv;
|
||
|
|
VecUnion y[NB_MODS];
|
||
|
|
limb_t u[NB_MODS], carry[NB_MODS], fft_len, base_mask1, r;
|
||
|
|
slimb_t i, len, pos;
|
||
|
|
int j, k, l, shift, n_limb1, p;
|
||
|
|
dlimb_t t;
|
||
|
|
|
||
|
|
j = NB_MODS * (NB_MODS - 1) / 2 - nb_mods * (nb_mods - 1) / 2;
|
||
|
|
mods_cr_vec = s->ntt_mods_cr_vec + j;
|
||
|
|
mf = s->ntt_mods_vec + NB_MODS - nb_mods;
|
||
|
|
m_inv = s->ntt_mods_inv_vec + NB_MODS - nb_mods;
|
||
|
|
|
||
|
|
shift = dpl & (LIMB_BITS - 1);
|
||
|
|
if (shift == 0)
|
||
|
|
base_mask1 = -1;
|
||
|
|
else
|
||
|
|
base_mask1 = ((limb_t)1 << shift) - 1;
|
||
|
|
n_limb1 = ((unsigned)dpl - 1) / LIMB_BITS;
|
||
|
|
for(j = 0; j < NB_MODS; j++)
|
||
|
|
carry[j] = 0;
|
||
|
|
for(j = 0; j < NB_MODS; j++)
|
||
|
|
u[j] = 0; /* avoid warnings */
|
||
|
|
memset(tabr, 0, sizeof(limb_t) * r_len);
|
||
|
|
fft_len = (limb_t)1 << fft_len_log2;
|
||
|
|
len = bf_min(fft_len, (r_len * LIMB_BITS + dpl - 1) / dpl);
|
||
|
|
len = (len + VEC_LEN - 1) & ~(VEC_LEN - 1);
|
||
|
|
i = 0;
|
||
|
|
while (i < len) {
|
||
|
|
for(j = 0; j < nb_mods; j++)
|
||
|
|
y[j].v = *(__m256d *)&buf[i + fft_len * j];
|
||
|
|
|
||
|
|
/* Chinese remainder to get mixed radix representation */
|
||
|
|
l = 0;
|
||
|
|
for(j = 0; j < nb_mods - 1; j++) {
|
||
|
|
y[j].v = ntt_mod1(y[j].v, mf[j]);
|
||
|
|
for(k = j + 1; k < nb_mods; k++) {
|
||
|
|
y[k].v = ntt_mul_mod(y[k].v - y[j].v,
|
||
|
|
mods_cr_vec[l], mf[k], m_inv[k]);
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
y[j].v = ntt_mod1(y[j].v, mf[j]);
|
||
|
|
|
||
|
|
for(p = 0; p < VEC_LEN; p++) {
|
||
|
|
/* back to normal representation */
|
||
|
|
u[0] = (int64_t)y[nb_mods - 1].d[p];
|
||
|
|
l = 1;
|
||
|
|
for(j = nb_mods - 2; j >= 1; j--) {
|
||
|
|
r = (int64_t)y[j].d[p];
|
||
|
|
for(k = 0; k < l; k++) {
|
||
|
|
t = (dlimb_t)u[k] * mods[j] + r;
|
||
|
|
r = t >> LIMB_BITS;
|
||
|
|
u[k] = t;
|
||
|
|
}
|
||
|
|
u[l] = r;
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
/* XXX: for nb_mods = 5, l should be 4 */
|
||
|
|
|
||
|
|
/* last step adds the carry */
|
||
|
|
r = (int64_t)y[0].d[p];
|
||
|
|
for(k = 0; k < l; k++) {
|
||
|
|
t = (dlimb_t)u[k] * mods[j] + r + carry[k];
|
||
|
|
r = t >> LIMB_BITS;
|
||
|
|
u[k] = t;
|
||
|
|
}
|
||
|
|
u[l] = r + carry[l];
|
||
|
|
|
||
|
|
#if 0
|
||
|
|
printf("%" PRId64 ": ", i);
|
||
|
|
for(j = nb_mods - 1; j >= 0; j--) {
|
||
|
|
printf(" %019" PRIu64, u[j]);
|
||
|
|
}
|
||
|
|
printf("\n");
|
||
|
|
#endif
|
||
|
|
|
||
|
|
/* write the digits */
|
||
|
|
pos = i * dpl;
|
||
|
|
for(j = 0; j < n_limb1; j++) {
|
||
|
|
put_bits(tabr, r_len, pos, u[j]);
|
||
|
|
pos += LIMB_BITS;
|
||
|
|
}
|
||
|
|
put_bits(tabr, r_len, pos, u[n_limb1] & base_mask1);
|
||
|
|
/* shift by dpl digits and set the carry */
|
||
|
|
if (shift == 0) {
|
||
|
|
for(j = n_limb1 + 1; j < nb_mods; j++)
|
||
|
|
carry[j - (n_limb1 + 1)] = u[j];
|
||
|
|
} else {
|
||
|
|
for(j = n_limb1; j < nb_mods - 1; j++) {
|
||
|
|
carry[j - n_limb1] = (u[j] >> shift) |
|
||
|
|
(u[j + 1] << (LIMB_BITS - shift));
|
||
|
|
}
|
||
|
|
carry[nb_mods - 1 - n_limb1] = u[nb_mods - 1] >> shift;
|
||
|
|
}
|
||
|
|
i++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
#else
|
||
|
|
static no_inline void ntt_to_limb(BFNTTState *s, limb_t *tabr, limb_t r_len,
|
||
|
|
const NTTLimb *buf, int fft_len_log2, int dpl,
|
||
|
|
int nb_mods)
|
||
|
|
{
|
||
|
|
const limb_t *mods = ntt_mods + NB_MODS - nb_mods;
|
||
|
|
const limb_t *mods_cr, *mods_cr_inv;
|
||
|
|
limb_t y[NB_MODS], u[NB_MODS], carry[NB_MODS], fft_len, base_mask1, r;
|
||
|
|
slimb_t i, len, pos;
|
||
|
|
int j, k, l, shift, n_limb1;
|
||
|
|
dlimb_t t;
|
||
|
|
|
||
|
|
j = NB_MODS * (NB_MODS - 1) / 2 - nb_mods * (nb_mods - 1) / 2;
|
||
|
|
mods_cr = ntt_mods_cr + j;
|
||
|
|
mods_cr_inv = s->ntt_mods_cr_inv + j;
|
||
|
|
|
||
|
|
shift = dpl & (LIMB_BITS - 1);
|
||
|
|
if (shift == 0)
|
||
|
|
base_mask1 = -1;
|
||
|
|
else
|
||
|
|
base_mask1 = ((limb_t)1 << shift) - 1;
|
||
|
|
n_limb1 = ((unsigned)dpl - 1) / LIMB_BITS;
|
||
|
|
for(j = 0; j < NB_MODS; j++)
|
||
|
|
carry[j] = 0;
|
||
|
|
for(j = 0; j < NB_MODS; j++)
|
||
|
|
u[j] = 0; /* avoid warnings */
|
||
|
|
memset(tabr, 0, sizeof(limb_t) * r_len);
|
||
|
|
fft_len = (limb_t)1 << fft_len_log2;
|
||
|
|
len = bf_min(fft_len, (r_len * LIMB_BITS + dpl - 1) / dpl);
|
||
|
|
for(i = 0; i < len; i++) {
|
||
|
|
for(j = 0; j < nb_mods; j++) {
|
||
|
|
y[j] = ntt_limb_to_int(buf[i + fft_len * j], mods[j]);
|
||
|
|
}
|
||
|
|
|
||
|
|
/* Chinese remainder to get mixed radix representation */
|
||
|
|
l = 0;
|
||
|
|
for(j = 0; j < nb_mods - 1; j++) {
|
||
|
|
for(k = j + 1; k < nb_mods; k++) {
|
||
|
|
limb_t m;
|
||
|
|
m = mods[k];
|
||
|
|
/* Note: there is no overflow in the sub_mod() because
|
||
|
|
the modulos are sorted by increasing order */
|
||
|
|
y[k] = mul_mod_fast2(y[k] - y[j] + m,
|
||
|
|
mods_cr[l], m, mods_cr_inv[l]);
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* back to normal representation */
|
||
|
|
u[0] = y[nb_mods - 1];
|
||
|
|
l = 1;
|
||
|
|
for(j = nb_mods - 2; j >= 1; j--) {
|
||
|
|
r = y[j];
|
||
|
|
for(k = 0; k < l; k++) {
|
||
|
|
t = (dlimb_t)u[k] * mods[j] + r;
|
||
|
|
r = t >> LIMB_BITS;
|
||
|
|
u[k] = t;
|
||
|
|
}
|
||
|
|
u[l] = r;
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* last step adds the carry */
|
||
|
|
r = y[0];
|
||
|
|
for(k = 0; k < l; k++) {
|
||
|
|
t = (dlimb_t)u[k] * mods[j] + r + carry[k];
|
||
|
|
r = t >> LIMB_BITS;
|
||
|
|
u[k] = t;
|
||
|
|
}
|
||
|
|
u[l] = r + carry[l];
|
||
|
|
|
||
|
|
#if 0
|
||
|
|
printf("%" PRId64 ": ", (int64_t)i);
|
||
|
|
for(j = nb_mods - 1; j >= 0; j--) {
|
||
|
|
printf(" " FMT_LIMB, u[j]);
|
||
|
|
}
|
||
|
|
printf("\n");
|
||
|
|
#endif
|
||
|
|
|
||
|
|
/* write the digits */
|
||
|
|
pos = i * dpl;
|
||
|
|
for(j = 0; j < n_limb1; j++) {
|
||
|
|
put_bits(tabr, r_len, pos, u[j]);
|
||
|
|
pos += LIMB_BITS;
|
||
|
|
}
|
||
|
|
put_bits(tabr, r_len, pos, u[n_limb1] & base_mask1);
|
||
|
|
/* shift by dpl digits and set the carry */
|
||
|
|
if (shift == 0) {
|
||
|
|
for(j = n_limb1 + 1; j < nb_mods; j++)
|
||
|
|
carry[j - (n_limb1 + 1)] = u[j];
|
||
|
|
} else {
|
||
|
|
for(j = n_limb1; j < nb_mods - 1; j++) {
|
||
|
|
carry[j - n_limb1] = (u[j] >> shift) |
|
||
|
|
(u[j + 1] << (LIMB_BITS - shift));
|
||
|
|
}
|
||
|
|
carry[nb_mods - 1 - n_limb1] = u[nb_mods - 1] >> shift;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
|
||
|
|
static int ntt_static_init(bf_context_t *s1)
|
||
|
|
{
|
||
|
|
BFNTTState *s;
|
||
|
|
int inverse, i, j, k, l;
|
||
|
|
limb_t c, c_inv, c_inv2, m, m_inv;
|
||
|
|
|
||
|
|
if (s1->ntt_state)
|
||
|
|
return 0;
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
s = bf_aligned_malloc(s1, sizeof(*s), 64);
|
||
|
|
#else
|
||
|
|
s = bf_malloc(s1, sizeof(*s));
|
||
|
|
#endif
|
||
|
|
if (!s)
|
||
|
|
return -1;
|
||
|
|
memset(s, 0, sizeof(*s));
|
||
|
|
s1->ntt_state = s;
|
||
|
|
s->ctx = s1;
|
||
|
|
|
||
|
|
for(j = 0; j < NB_MODS; j++) {
|
||
|
|
m = ntt_mods[j];
|
||
|
|
m_inv = init_mul_mod_fast(m);
|
||
|
|
s->ntt_mods_div[j] = m_inv;
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
s->ntt_mods_vec[j] = _mm256_set1_pd(m);
|
||
|
|
s->ntt_mods_inv_vec[j] = _mm256_set1_pd(1.0 / (double)m);
|
||
|
|
#endif
|
||
|
|
c_inv2 = (m + 1) / 2; /* 1/2 */
|
||
|
|
c_inv = 1;
|
||
|
|
for(i = 0; i <= NTT_PROOT_2EXP; i++) {
|
||
|
|
s->ntt_len_inv[j][i][0] = c_inv;
|
||
|
|
s->ntt_len_inv[j][i][1] = init_mul_mod_fast2(c_inv, m);
|
||
|
|
c_inv = mul_mod_fast(c_inv, c_inv2, m, m_inv);
|
||
|
|
}
|
||
|
|
|
||
|
|
for(inverse = 0; inverse < 2; inverse++) {
|
||
|
|
c = ntt_proot[inverse][j];
|
||
|
|
for(i = 0; i < NTT_PROOT_2EXP; i++) {
|
||
|
|
s->ntt_proot_pow[j][inverse][NTT_PROOT_2EXP - i] = c;
|
||
|
|
s->ntt_proot_pow_inv[j][inverse][NTT_PROOT_2EXP - i] =
|
||
|
|
init_mul_mod_fast2(c, m);
|
||
|
|
c = mul_mod_fast(c, c, m, m_inv);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
l = 0;
|
||
|
|
for(j = 0; j < NB_MODS - 1; j++) {
|
||
|
|
for(k = j + 1; k < NB_MODS; k++) {
|
||
|
|
#if defined(__AVX2__)
|
||
|
|
s->ntt_mods_cr_vec[l] = _mm256_set1_pd(int_to_ntt_limb2(ntt_mods_cr[l],
|
||
|
|
ntt_mods[k]));
|
||
|
|
#else
|
||
|
|
s->ntt_mods_cr_inv[l] = init_mul_mod_fast2(ntt_mods_cr[l],
|
||
|
|
ntt_mods[k]);
|
||
|
|
#endif
|
||
|
|
l++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
int bf_get_fft_size(int *pdpl, int *pnb_mods, limb_t len)
|
||
|
|
{
|
||
|
|
int dpl, fft_len_log2, n_bits, nb_mods, dpl_found, fft_len_log2_found;
|
||
|
|
int int_bits, nb_mods_found;
|
||
|
|
limb_t cost, min_cost;
|
||
|
|
|
||
|
|
min_cost = -1;
|
||
|
|
dpl_found = 0;
|
||
|
|
nb_mods_found = 4;
|
||
|
|
fft_len_log2_found = 0;
|
||
|
|
for(nb_mods = 3; nb_mods <= NB_MODS; nb_mods++) {
|
||
|
|
int_bits = ntt_int_bits[NB_MODS - nb_mods];
|
||
|
|
dpl = bf_min((int_bits - 4) / 2,
|
||
|
|
2 * LIMB_BITS + 2 * NTT_MOD_LOG2_MIN - NTT_MOD_LOG2_MAX);
|
||
|
|
for(;;) {
|
||
|
|
fft_len_log2 = ceil_log2((len * LIMB_BITS + dpl - 1) / dpl);
|
||
|
|
if (fft_len_log2 > NTT_PROOT_2EXP)
|
||
|
|
goto next;
|
||
|
|
n_bits = fft_len_log2 + 2 * dpl;
|
||
|
|
if (n_bits <= int_bits) {
|
||
|
|
cost = ((limb_t)(fft_len_log2 + 1) << fft_len_log2) * nb_mods;
|
||
|
|
// printf("n=%d dpl=%d: cost=%" PRId64 "\n", nb_mods, dpl, (int64_t)cost);
|
||
|
|
if (cost < min_cost) {
|
||
|
|
min_cost = cost;
|
||
|
|
dpl_found = dpl;
|
||
|
|
nb_mods_found = nb_mods;
|
||
|
|
fft_len_log2_found = fft_len_log2;
|
||
|
|
}
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
dpl--;
|
||
|
|
if (dpl == 0)
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
next: ;
|
||
|
|
}
|
||
|
|
if (!dpl_found)
|
||
|
|
abort();
|
||
|
|
/* limit dpl if possible to reduce fixed cost of limb/NTT conversion */
|
||
|
|
if (dpl_found > (LIMB_BITS + NTT_MOD_LOG2_MIN) &&
|
||
|
|
((limb_t)(LIMB_BITS + NTT_MOD_LOG2_MIN) << fft_len_log2_found) >=
|
||
|
|
len * LIMB_BITS) {
|
||
|
|
dpl_found = LIMB_BITS + NTT_MOD_LOG2_MIN;
|
||
|
|
}
|
||
|
|
*pnb_mods = nb_mods_found;
|
||
|
|
*pdpl = dpl_found;
|
||
|
|
return fft_len_log2_found;
|
||
|
|
}
|
||
|
|
|
||
|
|
static no_inline void fft_mul(bf_t *res, limb_t *a_tab, limb_t a_len,
|
||
|
|
limb_t *b_tab, limb_t b_len, int mul_flags)
|
||
|
|
{
|
||
|
|
bf_context_t *s1 = res->ctx;
|
||
|
|
BFNTTState *s;
|
||
|
|
int dpl, fft_len_log2, j, nb_mods, reduced_mem;
|
||
|
|
slimb_t len, fft_len;
|
||
|
|
NTTLimb *buf1, *buf2, *ptr;
|
||
|
|
#if defined(USE_MUL_CHECK)
|
||
|
|
limb_t ha, hb, hr, h_ref;
|
||
|
|
#endif
|
||
|
|
|
||
|
|
ntt_static_init(s1);
|
||
|
|
s = s1->ntt_state;
|
||
|
|
|
||
|
|
/* find the optimal number of digits per limb (dpl) */
|
||
|
|
len = a_len + b_len;
|
||
|
|
fft_len_log2 = bf_get_fft_size(&dpl, &nb_mods, len);
|
||
|
|
fft_len = (uint64_t)1 << fft_len_log2;
|
||
|
|
// printf("len=%" PRId64 " fft_len_log2=%d dpl=%d\n", len, fft_len_log2, dpl);
|
||
|
|
#if defined(USE_MUL_CHECK)
|
||
|
|
ha = mp_mod1(a_tab, a_len, BF_CHKSUM_MOD, 0);
|
||
|
|
hb = mp_mod1(b_tab, b_len, BF_CHKSUM_MOD, 0);
|
||
|
|
#endif
|
||
|
|
if ((mul_flags & (FFT_MUL_R_OVERLAP_A | FFT_MUL_R_OVERLAP_B)) == 0) {
|
||
|
|
bf_resize(res, 0);
|
||
|
|
} else if (mul_flags & FFT_MUL_R_OVERLAP_B) {
|
||
|
|
limb_t *tmp_tab, tmp_len;
|
||
|
|
/* it is better to free 'b' first */
|
||
|
|
tmp_tab = a_tab;
|
||
|
|
a_tab = b_tab;
|
||
|
|
b_tab = tmp_tab;
|
||
|
|
tmp_len = a_len;
|
||
|
|
a_len = b_len;
|
||
|
|
b_len = tmp_len;
|
||
|
|
}
|
||
|
|
buf1 = ntt_malloc(s, sizeof(NTTLimb) * fft_len * nb_mods);
|
||
|
|
limb_to_ntt(s, buf1, fft_len, a_tab, a_len, dpl,
|
||
|
|
NB_MODS - nb_mods, nb_mods);
|
||
|
|
if ((mul_flags & (FFT_MUL_R_OVERLAP_A | FFT_MUL_R_OVERLAP_B)) ==
|
||
|
|
FFT_MUL_R_OVERLAP_A) {
|
||
|
|
bf_resize(res, 0);
|
||
|
|
}
|
||
|
|
reduced_mem = (fft_len_log2 >= 14);
|
||
|
|
if (!reduced_mem) {
|
||
|
|
buf2 = ntt_malloc(s, sizeof(NTTLimb) * fft_len * nb_mods);
|
||
|
|
limb_to_ntt(s, buf2, fft_len, b_tab, b_len, dpl,
|
||
|
|
NB_MODS - nb_mods, nb_mods);
|
||
|
|
bf_resize(res, 0); /* in case res == b */
|
||
|
|
} else {
|
||
|
|
buf2 = ntt_malloc(s, sizeof(NTTLimb) * fft_len);
|
||
|
|
}
|
||
|
|
for(j = 0; j < nb_mods; j++) {
|
||
|
|
if (reduced_mem) {
|
||
|
|
limb_to_ntt(s, buf2, fft_len, b_tab, b_len, dpl,
|
||
|
|
NB_MODS - nb_mods + j, 1);
|
||
|
|
ptr = buf2;
|
||
|
|
} else {
|
||
|
|
ptr = buf2 + fft_len * j;
|
||
|
|
}
|
||
|
|
ntt_conv(s, buf1 + fft_len * j, ptr,
|
||
|
|
fft_len_log2, fft_len_log2, j + NB_MODS - nb_mods);
|
||
|
|
}
|
||
|
|
bf_resize(res, 0); /* in case res == b and reduced mem */
|
||
|
|
ntt_free(s, buf2);
|
||
|
|
bf_resize(res, len);
|
||
|
|
ntt_to_limb(s, res->tab, len, buf1, fft_len_log2, dpl, nb_mods);
|
||
|
|
ntt_free(s, buf1);
|
||
|
|
#if defined(USE_MUL_CHECK)
|
||
|
|
hr = mp_mod1(res->tab, len, BF_CHKSUM_MOD, 0);
|
||
|
|
h_ref = mul_mod(ha, hb, BF_CHKSUM_MOD);
|
||
|
|
if (hr != h_ref) {
|
||
|
|
printf("ntt_mul_error: len=%" PRId_LIMB " fft_len_log2=%d dpl=%d nb_mods=%d\n",
|
||
|
|
len, fft_len_log2, dpl, nb_mods);
|
||
|
|
// printf("ha=0x" FMT_LIMB" hb=0x" FMT_LIMB " hr=0x" FMT_LIMB " expected=0x" FMT_LIMB "\n", ha, hb, hr, h_ref);
|
||
|
|
exit(1);
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
}
|
||
|
|
|
||
|
|
#else /* USE_FFT_MUL */
|
||
|
|
|
||
|
|
int bf_get_fft_size(int *pdpl, int *pnb_mods, limb_t len)
|
||
|
|
{
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
#endif /* !USE_FFT_MUL */
|