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dust3d/thirdparty/quickjs/quickjs-2019-07-09-dust3d/libbf.c

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/*
* Tiny arbitrary precision floating point library
*
* Copyright (c) 2017-2018 Fabrice Bellard
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
* THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
#include <stdlib.h>
#include <stdio.h>
#include <inttypes.h>
#include <math.h>
#include <string.h>
#include <assert.h>
#ifdef __AVX2__
#include <immintrin.h>
#endif
#include "cutils.h"
#include "libbf.h"
/* enable it to check the multiplication result */
//#define USE_MUL_CHECK
/* enable it to use FFT/NTT multiplication */
#define USE_FFT_MUL
//#define inline __attribute__((always_inline))
#define unused __attribute__((unused))
#ifdef __AVX2__
#define FFT_MUL_THRESHOLD 100 /* in limbs of the smallest factor */
#else
#define FFT_MUL_THRESHOLD 100 /* in limbs of the smallest factor */
#endif
/* XXX: adjust */
#define BASECASE_DIV_THRESHOLD_B 300
#define BASECASE_DIV_THRESHOLD_Q 300
#if LIMB_BITS == 64
#define FMT_LIMB1 "%" PRIx64
#define FMT_LIMB "%016" PRIx64
#define PRId_LIMB PRId64
#define PRIu_LIMB PRIu64
#else
#define FMT_LIMB1 "%x"
#define FMT_LIMB "%08x"
#define PRId_LIMB "d"
#define PRIu_LIMB "u"
#endif
typedef int bf_op2_func_t(bf_t *r, const bf_t *a, const bf_t *b, limb_t prec,
bf_flags_t flags);
#ifdef USE_FFT_MUL
#define FFT_MUL_R_OVERLAP_A (1 << 0)
#define FFT_MUL_R_OVERLAP_B (1 << 1)
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);
static void fft_clear_cache(bf_context_t *s);
#endif
/* could leading zeros */
static inline int clz(limb_t a)
{
if (a == 0) {
return LIMB_BITS;
} else {
#if LIMB_BITS == 64
return clz64(a);
#else
return clz32(a);
#endif
}
}
static inline int ctz(limb_t a)
{
if (a == 0) {
return LIMB_BITS;
} else {
#if LIMB_BITS == 64
return ctz64(a);
#else
return ctz32(a);
#endif
}
}
static inline int ceil_log2(limb_t a)
{
if (a <= 1)
return 0;
else
return LIMB_BITS - clz(a - 1);
}
#if 0
static inline slimb_t ceil_div(slimb_t a, limb_t b)
{
if (a >= 0)
return (a + b - 1) / b;
else
return a / (slimb_t)b;
}
static inline slimb_t floor_div(slimb_t a, limb_t b)
{
if (a >= 0) {
return a / b;
} else {
return (a - b + 1) / (slimb_t)b;
}
}
#endif
#define malloc(s) malloc_is_forbidden(s)
#define free(p) free_is_forbidden(p)
#define realloc(p, s) realloc_is_forbidden(p, s)
void bf_context_init(bf_context_t *s, bf_realloc_func_t *realloc_func,
void *realloc_opaque)
{
memset(s, 0, sizeof(*s));
s->realloc_func = realloc_func;
s->realloc_opaque = realloc_opaque;
}
void bf_context_end(bf_context_t *s)
{
bf_clear_cache(s);
}
/* 'size' must be > 0 */
static void *bf_malloc(bf_context_t *s, size_t size)
{
return bf_realloc(s, NULL, size);
}
static void bf_free(bf_context_t *s, void *ptr)
{
bf_realloc(s, ptr, 0);
}
void bf_init(bf_context_t *s, bf_t *r)
{
r->ctx = s;
r->sign = 0;
r->expn = BF_EXP_ZERO;
r->len = 0;
r->tab = NULL;
}
void bf_resize(bf_t *r, limb_t len)
{
if (len != r->len) {
r->tab = bf_realloc(r->ctx, r->tab, len * sizeof(limb_t));
r->len = len;
}
}
void bf_set_ui(bf_t *r, uint64_t a)
{
r->sign = 0;
if (a == 0) {
r->expn = BF_EXP_ZERO;
bf_resize(r, 0);
}
#if LIMB_BITS == 32
else if (a <= 0xffffffff)
#else
else
#endif
{
int shift;
bf_resize(r, 1);
shift = clz(a);
r->tab[0] = a << shift;
r->expn = LIMB_BITS - shift;
}
#if LIMB_BITS == 32
else {
uint32_t a1, a0;
int shift;
bf_resize(r, 2);
a0 = a;
a1 = a >> 32;
shift = clz(a1);
r->tab[0] = a0 << shift;
r->tab[1] = (a1 << shift) | (a0 >> (LIMB_BITS - shift));
r->expn = 2 * LIMB_BITS - shift;
}
#endif
}
void bf_set_si(bf_t *r, int64_t a)
{
if (a < 0) {
bf_set_ui(r, -a);
r->sign = 1;
} else {
bf_set_ui(r, a);
}
}
void bf_set_nan(bf_t *r)
{
bf_resize(r, 0);
r->expn = BF_EXP_NAN;
r->sign = 0;
}
void bf_set_zero(bf_t *r, int is_neg)
{
bf_resize(r, 0);
r->expn = BF_EXP_ZERO;
r->sign = is_neg;
}
void bf_set_inf(bf_t *r, int is_neg)
{
bf_resize(r, 0);
r->expn = BF_EXP_INF;
r->sign = is_neg;
}
void bf_set(bf_t *r, const bf_t *a)
{
if (r == a)
return;
r->sign = a->sign;
r->expn = a->expn;
bf_resize(r, a->len);
memcpy(r->tab, a->tab, a->len * sizeof(limb_t));
}
/* equivalent to bf_set(r, a); bf_delete(a) */
void bf_move(bf_t *r, bf_t *a)
{
bf_context_t *s = r->ctx;
if (r == a)
return;
bf_free(s, r->tab);
*r = *a;
}
static limb_t get_limbz(const bf_t *a, limb_t idx)
{
if (idx >= a->len)
return 0;
else
return a->tab[idx];
}
/* get LIMB_BITS at bit position 'pos' in tab */
static inline limb_t get_bits(const limb_t *tab, limb_t len, slimb_t pos)
{
limb_t i, a0, a1;
int p;
i = pos >> LIMB_LOG2_BITS;
p = pos & (LIMB_BITS - 1);
if (i < len)
a0 = tab[i];
else
a0 = 0;
if (p == 0) {
return a0;
} else {
i++;
if (i < len)
a1 = tab[i];
else
a1 = 0;
return (a0 >> p) | (a1 << (LIMB_BITS - p));
}
}
static inline limb_t get_bit(const limb_t *tab, limb_t len, slimb_t pos)
{
slimb_t i;
i = pos >> LIMB_LOG2_BITS;
if (i < 0 || i >= len)
return 0;
return (tab[i] >> (pos & (LIMB_BITS - 1))) & 1;
}
static inline limb_t limb_mask(int start, int last)
{
limb_t v;
int n;
n = last - start + 1;
if (n == LIMB_BITS)
v = -1;
else
v = (((limb_t)1 << n) - 1) << start;
return v;
}
/* return != 0 if one bit between 0 and bit_pos inclusive is not zero. */
static inline limb_t scan_bit_nz(const bf_t *r, slimb_t bit_pos)
{
slimb_t pos;
limb_t v;
pos = bit_pos >> LIMB_LOG2_BITS;
if (pos < 0)
return 0;
v = r->tab[pos] & limb_mask(0, bit_pos & (LIMB_BITS - 1));
if (v != 0)
return 1;
pos--;
while (pos >= 0) {
if (r->tab[pos] != 0)
return 1;
pos--;
}
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,
slimb_t prec, int rnd_mode)
{
int add_one, inexact;
limb_t bit1, bit0;
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));
}
/* get the bit at 'prec' */
bit1 = get_bit(r->tab, l, l * LIMB_BITS - 1 - prec);
inexact = (bit1 | bit0) != 0;
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 */
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