glibc 2.29 compatibility

This commit is contained in:
Amos Bird 2019-09-28 22:36:56 +08:00
parent fd877a62e0
commit dbc352fdf9
33 changed files with 2294 additions and 159 deletions

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@ -68,8 +68,8 @@ if (SANITIZE)
endif ()
elseif (SANITIZE STREQUAL "undefined")
set (CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} ${SAN_FLAGS} -fsanitize=undefined -fno-sanitize-recover=all")
set (CMAKE_C_FLAGS "${CMAKE_C_FLAGS} ${SAN_FLAGS} -fsanitize=undefined -fno-sanitize-recover=all")
set (CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} ${SAN_FLAGS} -fsanitize=undefined -fno-sanitize-recover=all -fno-sanitize=float-divide-by-zero")
set (CMAKE_C_FLAGS "${CMAKE_C_FLAGS} ${SAN_FLAGS} -fsanitize=undefined -fno-sanitize-recover=all -fno-sanitize=float-divide-by-zero")
if (CMAKE_CXX_COMPILER_ID STREQUAL "GNU")
set (CMAKE_EXE_LINKER_FLAGS "${CMAKE_EXE_LINKER_FLAGS} -fsanitize=undefined")
endif()

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@ -174,17 +174,42 @@ obstacle to adoption, that text has been removed.
double preciseExp10(double x)
{
static const double p10[] = {
1e-15, 1e-14, 1e-13, 1e-12, 1e-11, 1e-10,
1e-9, 1e-8, 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1,
1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9,
1e10, 1e11, 1e12, 1e13, 1e14, 1e15
};
static const double p10[]
= {1e-15, 1e-14, 1e-13, 1e-12, 1e-11, 1e-10, 1e-9, 1e-8, 1e-7, 1e-6, 1e-5, 1e-4, 1e-3, 1e-2, 1e-1, 1,
1e+1, 1e+2, 1e+3, 1e+4, 1e+5, 1e+6, 1e+7, 1e+8, 1e+9, 1e+10, 1e+11, 1e+12, 1e+13, 1e+14, 1e+15, 1e+16,
1e+17, 1e+18, 1e+19, 1e+20, 1e+21, 1e+22, 1e+23, 1e+24, 1e+25, 1e+26, 1e+27, 1e+28, 1e+29, 1e+30, 1e+31, 1e+32,
1e+33, 1e+34, 1e+35, 1e+36, 1e+37, 1e+38, 1e+39, 1e+40, 1e+41, 1e+42, 1e+43, 1e+44, 1e+45, 1e+46, 1e+47, 1e+48,
1e+49, 1e+50, 1e+51, 1e+52, 1e+53, 1e+54, 1e+55, 1e+56, 1e+57, 1e+58, 1e+59, 1e+60, 1e+61, 1e+62, 1e+63, 1e+64,
1e+65, 1e+66, 1e+67, 1e+68, 1e+69, 1e+70, 1e+71, 1e+72, 1e+73, 1e+74, 1e+75, 1e+76, 1e+77, 1e+78, 1e+79, 1e+80,
1e+81, 1e+82, 1e+83, 1e+84, 1e+85, 1e+86, 1e+87, 1e+88, 1e+89, 1e+90, 1e+91, 1e+92, 1e+93, 1e+94, 1e+95, 1e+96,
1e+97, 1e+98, 1e+99, 1e+100, 1e+101, 1e+102, 1e+103, 1e+104, 1e+105, 1e+106, 1e+107, 1e+108, 1e+109, 1e+110, 1e+111, 1e+112,
1e+113, 1e+114, 1e+115, 1e+116, 1e+117, 1e+118, 1e+119, 1e+120, 1e+121, 1e+122, 1e+123, 1e+124, 1e+125, 1e+126, 1e+127, 1e+128,
1e+129, 1e+130, 1e+131, 1e+132, 1e+133, 1e+134, 1e+135, 1e+136, 1e+137, 1e+138, 1e+139, 1e+140, 1e+141, 1e+142, 1e+143, 1e+144,
1e+145, 1e+146, 1e+147, 1e+148, 1e+149, 1e+150, 1e+151, 1e+152, 1e+153, 1e+154, 1e+155, 1e+156, 1e+157, 1e+158, 1e+159, 1e+160,
1e+161, 1e+162, 1e+163, 1e+164, 1e+165, 1e+166, 1e+167, 1e+168, 1e+169, 1e+170, 1e+171, 1e+172, 1e+173, 1e+174, 1e+175, 1e+176,
1e+177, 1e+178, 1e+179, 1e+180, 1e+181, 1e+182, 1e+183, 1e+184, 1e+185, 1e+186, 1e+187, 1e+188, 1e+189, 1e+190, 1e+191, 1e+192,
1e+193, 1e+194, 1e+195, 1e+196, 1e+197, 1e+198, 1e+199, 1e+200, 1e+201, 1e+202, 1e+203, 1e+204, 1e+205, 1e+206, 1e+207, 1e+208,
1e+209, 1e+210, 1e+211, 1e+212, 1e+213, 1e+214, 1e+215, 1e+216, 1e+217, 1e+218, 1e+219, 1e+220, 1e+221, 1e+222, 1e+223, 1e+224,
1e+225, 1e+226, 1e+227, 1e+228, 1e+229, 1e+230, 1e+231, 1e+232, 1e+233, 1e+234, 1e+235, 1e+236, 1e+237, 1e+238, 1e+239, 1e+240,
1e+241, 1e+242, 1e+243, 1e+244, 1e+245, 1e+246, 1e+247, 1e+248, 1e+249, 1e+250, 1e+251, 1e+252, 1e+253, 1e+254, 1e+255, 1e+256,
1e+257, 1e+258, 1e+259, 1e+260, 1e+261, 1e+262, 1e+263, 1e+264, 1e+265, 1e+266, 1e+267, 1e+268, 1e+269, 1e+270, 1e+271, 1e+272,
1e+273, 1e+274, 1e+275, 1e+276, 1e+277, 1e+278, 1e+279, 1e+280, 1e+281, 1e+282, 1e+283, 1e+284, 1e+285, 1e+286, 1e+287, 1e+288,
1e+289, 1e+290, 1e+291, 1e+292, 1e+293, 1e+294, 1e+295, 1e+296, 1e+297, 1e+298, 1e+299, 1e+300, 1e+301, 1e+302, 1e+303, 1e+304,
1e+305, 1e+306, 1e+307, 1e+308};
double n, y = modf(x, &n);
union {double f; uint64_t i;} u = {n};
/* fabs(n) < 16 without raising invalid on nan */
if ((u.i>>52 & 0x7ff) < 0x3ff+4) {
if (!y) return p10[(int)n+15];
if (n > 308)
return x > 0 ? INFINITY : -INFINITY;
if (!y)
return p10[(int)n + 15];
union
{
double f;
uint64_t i;
} u = {n};
if ((u.i >> 52 & 0x7ff) < 0x3ff + 4)
{
y = exp2(3.32192809488736234787031942948939 * y);
return y * p10[(int)n + 15];
}

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@ -4,3 +4,5 @@ git://git.musl-libc.org/musl
c10bc61508dc52b8315084e628f36a6c3c2dabb1
NOTE: Files was edited.
NOTE: Math related files are pulled from commit 6ad514e4e278f0c3b18eb2db1d45638c9af1c07f.

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@ -0,0 +1,6 @@
#include "libm.h"
double __math_divzero(uint32_t sign)
{
return fp_barrier(sign ? -1.0 : 1.0) / 0.0;
}

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@ -1,15 +1,4 @@
#include <stdint.h>
/* fp_barrier returns its input, but limits code transformations
as if it had a side-effect (e.g. observable io) and returned
an arbitrary value. */
static inline float fp_barrierf(float x)
{
volatile float y = x;
return y;
}
#include "libm.h"
float __math_divzerof(uint32_t sign)
{

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@ -0,0 +1,6 @@
#include "libm.h"
double __math_invalid(double x)
{
return (x - x) / (x - x);
}

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@ -1,3 +1,5 @@
#include "libm.h"
float __math_invalidf(float x)
{
return (x - x) / (x - x);

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@ -0,0 +1,6 @@
#include "libm.h"
double __math_oflow(uint32_t sign)
{
return __math_xflow(sign, 0x1p769);
}

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@ -0,0 +1,6 @@
#include "libm.h"
float __math_oflowf(uint32_t sign)
{
return __math_xflowf(sign, 0x1p97f);
}

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@ -0,0 +1,6 @@
#include "libm.h"
double __math_uflow(uint32_t sign)
{
return __math_xflow(sign, 0x1p-767);
}

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@ -0,0 +1,6 @@
#include "libm.h"
float __math_uflowf(uint32_t sign)
{
return __math_xflowf(sign, 0x1p-95f);
}

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@ -0,0 +1,6 @@
#include "libm.h"
double __math_xflow(uint32_t sign, double y)
{
return eval_as_double(fp_barrier(sign ? -y : y) * y);
}

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@ -0,0 +1,6 @@
#include "libm.h"
float __math_xflowf(uint32_t sign, float y)
{
return eval_as_float(fp_barrierf(sign ? -y : y) * y);
}

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@ -0,0 +1,134 @@
/*
* Double-precision e^x function.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "exp_data.h"
#define N (1 << EXP_TABLE_BITS)
#define InvLn2N __exp_data.invln2N
#define NegLn2hiN __exp_data.negln2hiN
#define NegLn2loN __exp_data.negln2loN
#define Shift __exp_data.shift
#define T __exp_data.tab
#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
/* Handle cases that may overflow or underflow when computing the result that
is scale*(1+TMP) without intermediate rounding. The bit representation of
scale is in SBITS, however it has a computed exponent that may have
overflown into the sign bit so that needs to be adjusted before using it as
a double. (int32_t)KI is the k used in the argument reduction and exponent
adjustment of scale, positive k here means the result may overflow and
negative k means the result may underflow. */
static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
{
double_t scale, y;
if ((ki & 0x80000000) == 0) {
/* k > 0, the exponent of scale might have overflowed by <= 460. */
sbits -= 1009ull << 52;
scale = asdouble(sbits);
y = 0x1p1009 * (scale + scale * tmp);
return eval_as_double(y);
}
/* k < 0, need special care in the subnormal range. */
sbits += 1022ull << 52;
scale = asdouble(sbits);
y = scale + scale * tmp;
if (y < 1.0) {
/* Round y to the right precision before scaling it into the subnormal
range to avoid double rounding that can cause 0.5+E/2 ulp error where
E is the worst-case ulp error outside the subnormal range. So this
is only useful if the goal is better than 1 ulp worst-case error. */
double_t hi, lo;
lo = scale - y + scale * tmp;
hi = 1.0 + y;
lo = 1.0 - hi + y + lo;
y = eval_as_double(hi + lo) - 1.0;
/* Avoid -0.0 with downward rounding. */
if (WANT_ROUNDING && y == 0.0)
y = 0.0;
/* The underflow exception needs to be signaled explicitly. */
fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
}
y = 0x1p-1022 * y;
return eval_as_double(y);
}
/* Top 12 bits of a double (sign and exponent bits). */
static inline uint32_t top12(double x)
{
return asuint64(x) >> 52;
}
double exp(double x)
{
uint32_t abstop;
uint64_t ki, idx, top, sbits;
double_t kd, z, r, r2, scale, tail, tmp;
abstop = top12(x) & 0x7ff;
if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
if (abstop - top12(0x1p-54) >= 0x80000000)
/* Avoid spurious underflow for tiny x. */
/* Note: 0 is common input. */
return WANT_ROUNDING ? 1.0 + x : 1.0;
if (abstop >= top12(1024.0)) {
if (asuint64(x) == asuint64(-INFINITY))
return 0.0;
if (abstop >= top12(INFINITY))
return 1.0 + x;
if (asuint64(x) >> 63)
return __math_uflow(0);
else
return __math_oflow(0);
}
/* Large x is special cased below. */
abstop = 0;
}
/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
z = InvLn2N * x;
#if TOINT_INTRINSICS
kd = roundtoint(z);
ki = converttoint(z);
#elif EXP_USE_TOINT_NARROW
/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
kd = eval_as_double(z + Shift);
ki = asuint64(kd) >> 16;
kd = (double_t)(int32_t)ki;
#else
/* z - kd is in [-1, 1] in non-nearest rounding modes. */
kd = eval_as_double(z + Shift);
ki = asuint64(kd);
kd -= Shift;
#endif
r = x + kd * NegLn2hiN + kd * NegLn2loN;
/* 2^(k/N) ~= scale * (1 + tail). */
idx = 2 * (ki % N);
top = ki << (52 - EXP_TABLE_BITS);
tail = asdouble(T[idx]);
/* This is only a valid scale when -1023*N < k < 1024*N. */
sbits = T[idx + 1] + top;
/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r;
/* Without fma the worst case error is 0.25/N ulp larger. */
/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
if (predict_false(abstop == 0))
return specialcase(tmp, sbits, ki);
scale = asdouble(sbits);
/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
is no spurious underflow here even without fma. */
return eval_as_double(scale + scale * tmp);
}

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@ -0,0 +1,121 @@
/*
* Double-precision 2^x function.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "exp_data.h"
#define N (1 << EXP_TABLE_BITS)
#define Shift __exp_data.exp2_shift
#define T __exp_data.tab
#define C1 __exp_data.exp2_poly[0]
#define C2 __exp_data.exp2_poly[1]
#define C3 __exp_data.exp2_poly[2]
#define C4 __exp_data.exp2_poly[3]
#define C5 __exp_data.exp2_poly[4]
/* Handle cases that may overflow or underflow when computing the result that
is scale*(1+TMP) without intermediate rounding. The bit representation of
scale is in SBITS, however it has a computed exponent that may have
overflown into the sign bit so that needs to be adjusted before using it as
a double. (int32_t)KI is the k used in the argument reduction and exponent
adjustment of scale, positive k here means the result may overflow and
negative k means the result may underflow. */
static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
{
double_t scale, y;
if ((ki & 0x80000000) == 0) {
/* k > 0, the exponent of scale might have overflowed by 1. */
sbits -= 1ull << 52;
scale = asdouble(sbits);
y = 2 * (scale + scale * tmp);
return eval_as_double(y);
}
/* k < 0, need special care in the subnormal range. */
sbits += 1022ull << 52;
scale = asdouble(sbits);
y = scale + scale * tmp;
if (y < 1.0) {
/* Round y to the right precision before scaling it into the subnormal
range to avoid double rounding that can cause 0.5+E/2 ulp error where
E is the worst-case ulp error outside the subnormal range. So this
is only useful if the goal is better than 1 ulp worst-case error. */
double_t hi, lo;
lo = scale - y + scale * tmp;
hi = 1.0 + y;
lo = 1.0 - hi + y + lo;
y = eval_as_double(hi + lo) - 1.0;
/* Avoid -0.0 with downward rounding. */
if (WANT_ROUNDING && y == 0.0)
y = 0.0;
/* The underflow exception needs to be signaled explicitly. */
fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
}
y = 0x1p-1022 * y;
return eval_as_double(y);
}
/* Top 12 bits of a double (sign and exponent bits). */
static inline uint32_t top12(double x)
{
return asuint64(x) >> 52;
}
double exp2(double x)
{
uint32_t abstop;
uint64_t ki, idx, top, sbits;
double_t kd, r, r2, scale, tail, tmp;
abstop = top12(x) & 0x7ff;
if (predict_false(abstop - top12(0x1p-54) >= top12(512.0) - top12(0x1p-54))) {
if (abstop - top12(0x1p-54) >= 0x80000000)
/* Avoid spurious underflow for tiny x. */
/* Note: 0 is common input. */
return WANT_ROUNDING ? 1.0 + x : 1.0;
if (abstop >= top12(1024.0)) {
if (asuint64(x) == asuint64(-INFINITY))
return 0.0;
if (abstop >= top12(INFINITY))
return 1.0 + x;
if (!(asuint64(x) >> 63))
return __math_oflow(0);
else if (asuint64(x) >= asuint64(-1075.0))
return __math_uflow(0);
}
if (2 * asuint64(x) > 2 * asuint64(928.0))
/* Large x is special cased below. */
abstop = 0;
}
/* exp2(x) = 2^(k/N) * 2^r, with 2^r in [2^(-1/2N),2^(1/2N)]. */
/* x = k/N + r, with int k and r in [-1/2N, 1/2N]. */
kd = eval_as_double(x + Shift);
ki = asuint64(kd); /* k. */
kd -= Shift; /* k/N for int k. */
r = x - kd;
/* 2^(k/N) ~= scale * (1 + tail). */
idx = 2 * (ki % N);
top = ki << (52 - EXP_TABLE_BITS);
tail = asdouble(T[idx]);
/* This is only a valid scale when -1023*N < k < 1024*N. */
sbits = T[idx + 1] + top;
/* exp2(x) = 2^(k/N) * 2^r ~= scale + scale * (tail + 2^r - 1). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r;
/* Without fma the worst case error is 0.5/N ulp larger. */
/* Worst case error is less than 0.5+0.86/N+(abs poly error * 2^53) ulp. */
tmp = tail + r * C1 + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
if (predict_false(abstop == 0))
return specialcase(tmp, sbits, ki);
scale = asdouble(sbits);
/* Note: tmp == 0 or |tmp| > 2^-65 and scale > 2^-928, so there
is no spurious underflow here even without fma. */
return eval_as_double(scale + scale * tmp);
}

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@ -1,127 +1,69 @@
/* origin: FreeBSD /usr/src/lib/msun/src/s_exp2f.c */
/*-
* Copyright (c) 2005 David Schultz <das@FreeBSD.ORG>
* All rights reserved.
/*
* Single-precision 2^x function.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
* SUCH DAMAGE.
* Copyright (c) 2017-2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
#define TBLSIZE 16
static const float
redux = 0x1.8p23f / TBLSIZE,
P1 = 0x1.62e430p-1f,
P2 = 0x1.ebfbe0p-3f,
P3 = 0x1.c6b348p-5f,
P4 = 0x1.3b2c9cp-7f;
static const double exp2ft[TBLSIZE] = {
0x1.6a09e667f3bcdp-1,
0x1.7a11473eb0187p-1,
0x1.8ace5422aa0dbp-1,
0x1.9c49182a3f090p-1,
0x1.ae89f995ad3adp-1,
0x1.c199bdd85529cp-1,
0x1.d5818dcfba487p-1,
0x1.ea4afa2a490dap-1,
0x1.0000000000000p+0,
0x1.0b5586cf9890fp+0,
0x1.172b83c7d517bp+0,
0x1.2387a6e756238p+0,
0x1.306fe0a31b715p+0,
0x1.3dea64c123422p+0,
0x1.4bfdad5362a27p+0,
0x1.5ab07dd485429p+0,
};
#include "libm.h"
#include "exp2f_data.h"
/*
* exp2f(x): compute the base 2 exponential of x
*
* Accuracy: Peak error < 0.501 ulp; location of peak: -0.030110927.
*
* Method: (equally-spaced tables)
*
* Reduce x:
* x = k + y, for integer k and |y| <= 1/2.
* Thus we have exp2f(x) = 2**k * exp2(y).
*
* Reduce y:
* y = i/TBLSIZE + z for integer i near y * TBLSIZE.
* Thus we have exp2(y) = exp2(i/TBLSIZE) * exp2(z),
* with |z| <= 2**-(TBLSIZE+1).
*
* We compute exp2(i/TBLSIZE) via table lookup and exp2(z) via a
* degree-4 minimax polynomial with maximum error under 1.4 * 2**-33.
* Using double precision for everything except the reduction makes
* roundoff error insignificant and simplifies the scaling step.
*
* This method is due to Tang, but I do not use his suggested parameters:
*
* Tang, P. Table-driven Implementation of the Exponential Function
* in IEEE Floating-Point Arithmetic. TOMS 15(2), 144-157 (1989).
EXP2F_TABLE_BITS = 5
EXP2F_POLY_ORDER = 3
ULP error: 0.502 (nearest rounding.)
Relative error: 1.69 * 2^-34 in [-1/64, 1/64] (before rounding.)
Wrong count: 168353 (all nearest rounding wrong results with fma.)
Non-nearest ULP error: 1 (rounded ULP error)
*/
#define N (1 << EXP2F_TABLE_BITS)
#define T __exp2f_data.tab
#define C __exp2f_data.poly
#define SHIFT __exp2f_data.shift_scaled
static inline uint32_t top12(float x)
{
return asuint(x) >> 20;
}
float exp2f(float x)
{
double_t t, r, z;
union {float f; uint32_t i;} u = {x};
union {double f; uint64_t i;} uk;
uint32_t ix, i0, k;
uint32_t abstop;
uint64_t ki, t;
double_t kd, xd, z, r, r2, y, s;
/* Filter out exceptional cases. */
ix = u.i & 0x7fffffff;
if (ix > 0x42fc0000) { /* |x| > 126 */
if (ix > 0x7f800000) /* NaN */
return x;
if (u.i >= 0x43000000 && u.i < 0x80000000) { /* x >= 128 */
x *= 0x1p127f;
return x;
}
if (u.i >= 0x80000000) { /* x < -126 */
if (u.i >= 0xc3160000 || (u.i & 0x0000ffff))
{ volatile float tmp; tmp = (-0x1p-149f/x); (void)tmp; }
if (u.i >= 0xc3160000) /* x <= -150 */
return 0;
}
} else if (ix <= 0x33000000) { /* |x| <= 0x1p-25 */
return 1.0f + x;
xd = (double_t)x;
abstop = top12(x) & 0x7ff;
if (predict_false(abstop >= top12(128.0f))) {
/* |x| >= 128 or x is nan. */
if (asuint(x) == asuint(-INFINITY))
return 0.0f;
if (abstop >= top12(INFINITY))
return x + x;
if (x > 0.0f)
return __math_oflowf(0);
if (x <= -150.0f)
return __math_uflowf(0);
}
/* Reduce x, computing z, i0, and k. */
u.f = x + redux;
i0 = u.i;
i0 += TBLSIZE / 2;
k = i0 / TBLSIZE;
uk.i = (uint64_t)(0x3ff + k)<<52;
i0 &= TBLSIZE - 1;
u.f -= redux;
z = x - u.f;
/* Compute r = exp2(y) = exp2ft[i0] * p(z). */
r = exp2ft[i0];
t = r * z;
r = r + t * (P1 + z * P2) + t * (z * z) * (P3 + z * P4);
/* x = k/N + r with r in [-1/(2N), 1/(2N)] and int k. */
kd = eval_as_double(xd + SHIFT);
ki = asuint64(kd);
kd -= SHIFT; /* k/N for int k. */
r = xd - kd;
/* Scale by 2**k */
return r * uk.f;
/* exp2(x) = 2^(k/N) * 2^r ~= s * (C0*r^3 + C1*r^2 + C2*r + 1) */
t = T[ki % N];
t += ki << (52 - EXP2F_TABLE_BITS);
s = asdouble(t);
z = C[0] * r + C[1];
r2 = r * r;
y = C[2] * r + 1;
y = z * r2 + y;
y = y * s;
return eval_as_float(y);
}

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/*
* Shared data between expf, exp2f and powf.
*
* Copyright (c) 2017-2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include "exp2f_data.h"
#define N (1 << EXP2F_TABLE_BITS)
const struct exp2f_data __exp2f_data = {
/* tab[i] = uint(2^(i/N)) - (i << 52-BITS)
used for computing 2^(k/N) for an int |k| < 150 N as
double(tab[k%N] + (k << 52-BITS)) */
.tab = {
0x3ff0000000000000, 0x3fefd9b0d3158574, 0x3fefb5586cf9890f, 0x3fef9301d0125b51,
0x3fef72b83c7d517b, 0x3fef54873168b9aa, 0x3fef387a6e756238, 0x3fef1e9df51fdee1,
0x3fef06fe0a31b715, 0x3feef1a7373aa9cb, 0x3feedea64c123422, 0x3feece086061892d,
0x3feebfdad5362a27, 0x3feeb42b569d4f82, 0x3feeab07dd485429, 0x3feea47eb03a5585,
0x3feea09e667f3bcd, 0x3fee9f75e8ec5f74, 0x3feea11473eb0187, 0x3feea589994cce13,
0x3feeace5422aa0db, 0x3feeb737b0cdc5e5, 0x3feec49182a3f090, 0x3feed503b23e255d,
0x3feee89f995ad3ad, 0x3feeff76f2fb5e47, 0x3fef199bdd85529c, 0x3fef3720dcef9069,
0x3fef5818dcfba487, 0x3fef7c97337b9b5f, 0x3fefa4afa2a490da, 0x3fefd0765b6e4540,
},
.shift_scaled = 0x1.8p+52 / N,
.poly = {
0x1.c6af84b912394p-5, 0x1.ebfce50fac4f3p-3, 0x1.62e42ff0c52d6p-1,
},
.shift = 0x1.8p+52,
.invln2_scaled = 0x1.71547652b82fep+0 * N,
.poly_scaled = {
0x1.c6af84b912394p-5/N/N/N, 0x1.ebfce50fac4f3p-3/N/N, 0x1.62e42ff0c52d6p-1/N,
},
};

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/*
* Copyright (c) 2017-2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#ifndef _EXP2F_DATA_H
#define _EXP2F_DATA_H
#include "musl_features.h"
#include <stdint.h>
/* Shared between expf, exp2f and powf. */
#define EXP2F_TABLE_BITS 5
#define EXP2F_POLY_ORDER 3
extern hidden const struct exp2f_data {
uint64_t tab[1 << EXP2F_TABLE_BITS];
double shift_scaled;
double poly[EXP2F_POLY_ORDER];
double shift;
double invln2_scaled;
double poly_scaled[EXP2F_POLY_ORDER];
} __exp2f_data;
#endif

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/*
* Shared data between exp, exp2 and pow.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include "exp_data.h"
#define N (1 << EXP_TABLE_BITS)
const struct exp_data __exp_data = {
// N/ln2
.invln2N = 0x1.71547652b82fep0 * N,
// -ln2/N
.negln2hiN = -0x1.62e42fefa0000p-8,
.negln2loN = -0x1.cf79abc9e3b3ap-47,
// Used for rounding when !TOINT_INTRINSICS
#if EXP_USE_TOINT_NARROW
.shift = 0x1800000000.8p0,
#else
.shift = 0x1.8p52,
#endif
// exp polynomial coefficients.
.poly = {
// abs error: 1.555*2^-66
// ulp error: 0.509 (0.511 without fma)
// if |x| < ln2/256+eps
// abs error if |x| < ln2/256+0x1p-15: 1.09*2^-65
// abs error if |x| < ln2/128: 1.7145*2^-56
0x1.ffffffffffdbdp-2,
0x1.555555555543cp-3,
0x1.55555cf172b91p-5,
0x1.1111167a4d017p-7,
},
.exp2_shift = 0x1.8p52 / N,
// exp2 polynomial coefficients.
.exp2_poly = {
// abs error: 1.2195*2^-65
// ulp error: 0.507 (0.511 without fma)
// if |x| < 1/256
// abs error if |x| < 1/128: 1.9941*2^-56
0x1.62e42fefa39efp-1,
0x1.ebfbdff82c424p-3,
0x1.c6b08d70cf4b5p-5,
0x1.3b2abd24650ccp-7,
0x1.5d7e09b4e3a84p-10,
},
// 2^(k/N) ~= H[k]*(1 + T[k]) for int k in [0,N)
// tab[2*k] = asuint64(T[k])
// tab[2*k+1] = asuint64(H[k]) - (k << 52)/N
.tab = {
0x0, 0x3ff0000000000000,
0x3c9b3b4f1a88bf6e, 0x3feff63da9fb3335,
0xbc7160139cd8dc5d, 0x3fefec9a3e778061,
0xbc905e7a108766d1, 0x3fefe315e86e7f85,
0x3c8cd2523567f613, 0x3fefd9b0d3158574,
0xbc8bce8023f98efa, 0x3fefd06b29ddf6de,
0x3c60f74e61e6c861, 0x3fefc74518759bc8,
0x3c90a3e45b33d399, 0x3fefbe3ecac6f383,
0x3c979aa65d837b6d, 0x3fefb5586cf9890f,
0x3c8eb51a92fdeffc, 0x3fefac922b7247f7,
0x3c3ebe3d702f9cd1, 0x3fefa3ec32d3d1a2,
0xbc6a033489906e0b, 0x3fef9b66affed31b,
0xbc9556522a2fbd0e, 0x3fef9301d0125b51,
0xbc5080ef8c4eea55, 0x3fef8abdc06c31cc,
0xbc91c923b9d5f416, 0x3fef829aaea92de0,
0x3c80d3e3e95c55af, 0x3fef7a98c8a58e51,
0xbc801b15eaa59348, 0x3fef72b83c7d517b,
0xbc8f1ff055de323d, 0x3fef6af9388c8dea,
0x3c8b898c3f1353bf, 0x3fef635beb6fcb75,
0xbc96d99c7611eb26, 0x3fef5be084045cd4,
0x3c9aecf73e3a2f60, 0x3fef54873168b9aa,
0xbc8fe782cb86389d, 0x3fef4d5022fcd91d,
0x3c8a6f4144a6c38d, 0x3fef463b88628cd6,
0x3c807a05b0e4047d, 0x3fef3f49917ddc96,
0x3c968efde3a8a894, 0x3fef387a6e756238,
0x3c875e18f274487d, 0x3fef31ce4fb2a63f,
0x3c80472b981fe7f2, 0x3fef2b4565e27cdd,
0xbc96b87b3f71085e, 0x3fef24dfe1f56381,
0x3c82f7e16d09ab31, 0x3fef1e9df51fdee1,
0xbc3d219b1a6fbffa, 0x3fef187fd0dad990,
0x3c8b3782720c0ab4, 0x3fef1285a6e4030b,
0x3c6e149289cecb8f, 0x3fef0cafa93e2f56,
0x3c834d754db0abb6, 0x3fef06fe0a31b715,
0x3c864201e2ac744c, 0x3fef0170fc4cd831,
0x3c8fdd395dd3f84a, 0x3feefc08b26416ff,
0xbc86a3803b8e5b04, 0x3feef6c55f929ff1,
0xbc924aedcc4b5068, 0x3feef1a7373aa9cb,
0xbc9907f81b512d8e, 0x3feeecae6d05d866,
0xbc71d1e83e9436d2, 0x3feee7db34e59ff7,
0xbc991919b3ce1b15, 0x3feee32dc313a8e5,
0x3c859f48a72a4c6d, 0x3feedea64c123422,
0xbc9312607a28698a, 0x3feeda4504ac801c,
0xbc58a78f4817895b, 0x3feed60a21f72e2a,
0xbc7c2c9b67499a1b, 0x3feed1f5d950a897,
0x3c4363ed60c2ac11, 0x3feece086061892d,
0x3c9666093b0664ef, 0x3feeca41ed1d0057,
0x3c6ecce1daa10379, 0x3feec6a2b5c13cd0,
0x3c93ff8e3f0f1230, 0x3feec32af0d7d3de,
0x3c7690cebb7aafb0, 0x3feebfdad5362a27,
0x3c931dbdeb54e077, 0x3feebcb299fddd0d,
0xbc8f94340071a38e, 0x3feeb9b2769d2ca7,
0xbc87deccdc93a349, 0x3feeb6daa2cf6642,
0xbc78dec6bd0f385f, 0x3feeb42b569d4f82,
0xbc861246ec7b5cf6, 0x3feeb1a4ca5d920f,
0x3c93350518fdd78e, 0x3feeaf4736b527da,
0x3c7b98b72f8a9b05, 0x3feead12d497c7fd,
0x3c9063e1e21c5409, 0x3feeab07dd485429,
0x3c34c7855019c6ea, 0x3feea9268a5946b7,
0x3c9432e62b64c035, 0x3feea76f15ad2148,
0xbc8ce44a6199769f, 0x3feea5e1b976dc09,
0xbc8c33c53bef4da8, 0x3feea47eb03a5585,
0xbc845378892be9ae, 0x3feea34634ccc320,
0xbc93cedd78565858, 0x3feea23882552225,
0x3c5710aa807e1964, 0x3feea155d44ca973,
0xbc93b3efbf5e2228, 0x3feea09e667f3bcd,
0xbc6a12ad8734b982, 0x3feea012750bdabf,
0xbc6367efb86da9ee, 0x3fee9fb23c651a2f,
0xbc80dc3d54e08851, 0x3fee9f7df9519484,
0xbc781f647e5a3ecf, 0x3fee9f75e8ec5f74,
0xbc86ee4ac08b7db0, 0x3fee9f9a48a58174,
0xbc8619321e55e68a, 0x3fee9feb564267c9,
0x3c909ccb5e09d4d3, 0x3feea0694fde5d3f,
0xbc7b32dcb94da51d, 0x3feea11473eb0187,
0x3c94ecfd5467c06b, 0x3feea1ed0130c132,
0x3c65ebe1abd66c55, 0x3feea2f336cf4e62,
0xbc88a1c52fb3cf42, 0x3feea427543e1a12,
0xbc9369b6f13b3734, 0x3feea589994cce13,
0xbc805e843a19ff1e, 0x3feea71a4623c7ad,
0xbc94d450d872576e, 0x3feea8d99b4492ed,
0x3c90ad675b0e8a00, 0x3feeaac7d98a6699,
0x3c8db72fc1f0eab4, 0x3feeace5422aa0db,
0xbc65b6609cc5e7ff, 0x3feeaf3216b5448c,
0x3c7bf68359f35f44, 0x3feeb1ae99157736,
0xbc93091fa71e3d83, 0x3feeb45b0b91ffc6,
0xbc5da9b88b6c1e29, 0x3feeb737b0cdc5e5,
0xbc6c23f97c90b959, 0x3feeba44cbc8520f,
0xbc92434322f4f9aa, 0x3feebd829fde4e50,
0xbc85ca6cd7668e4b, 0x3feec0f170ca07ba,
0x3c71affc2b91ce27, 0x3feec49182a3f090,
0x3c6dd235e10a73bb, 0x3feec86319e32323,
0xbc87c50422622263, 0x3feecc667b5de565,
0x3c8b1c86e3e231d5, 0x3feed09bec4a2d33,
0xbc91bbd1d3bcbb15, 0x3feed503b23e255d,
0x3c90cc319cee31d2, 0x3feed99e1330b358,
0x3c8469846e735ab3, 0x3feede6b5579fdbf,
0xbc82dfcd978e9db4, 0x3feee36bbfd3f37a,
0x3c8c1a7792cb3387, 0x3feee89f995ad3ad,
0xbc907b8f4ad1d9fa, 0x3feeee07298db666,
0xbc55c3d956dcaeba, 0x3feef3a2b84f15fb,
0xbc90a40e3da6f640, 0x3feef9728de5593a,
0xbc68d6f438ad9334, 0x3feeff76f2fb5e47,
0xbc91eee26b588a35, 0x3fef05b030a1064a,
0x3c74ffd70a5fddcd, 0x3fef0c1e904bc1d2,
0xbc91bdfbfa9298ac, 0x3fef12c25bd71e09,
0x3c736eae30af0cb3, 0x3fef199bdd85529c,
0x3c8ee3325c9ffd94, 0x3fef20ab5fffd07a,
0x3c84e08fd10959ac, 0x3fef27f12e57d14b,
0x3c63cdaf384e1a67, 0x3fef2f6d9406e7b5,
0x3c676b2c6c921968, 0x3fef3720dcef9069,
0xbc808a1883ccb5d2, 0x3fef3f0b555dc3fa,
0xbc8fad5d3ffffa6f, 0x3fef472d4a07897c,
0xbc900dae3875a949, 0x3fef4f87080d89f2,
0x3c74a385a63d07a7, 0x3fef5818dcfba487,
0xbc82919e2040220f, 0x3fef60e316c98398,
0x3c8e5a50d5c192ac, 0x3fef69e603db3285,
0x3c843a59ac016b4b, 0x3fef7321f301b460,
0xbc82d52107b43e1f, 0x3fef7c97337b9b5f,
0xbc892ab93b470dc9, 0x3fef864614f5a129,
0x3c74b604603a88d3, 0x3fef902ee78b3ff6,
0x3c83c5ec519d7271, 0x3fef9a51fbc74c83,
0xbc8ff7128fd391f0, 0x3fefa4afa2a490da,
0xbc8dae98e223747d, 0x3fefaf482d8e67f1,
0x3c8ec3bc41aa2008, 0x3fefba1bee615a27,
0x3c842b94c3a9eb32, 0x3fefc52b376bba97,
0x3c8a64a931d185ee, 0x3fefd0765b6e4540,
0xbc8e37bae43be3ed, 0x3fefdbfdad9cbe14,
0x3c77893b4d91cd9d, 0x3fefe7c1819e90d8,
0x3c5305c14160cc89, 0x3feff3c22b8f71f1,
},
};

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/*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#ifndef _EXP_DATA_H
#define _EXP_DATA_H
#include "musl_features.h"
#include <stdint.h>
#define EXP_TABLE_BITS 7
#define EXP_POLY_ORDER 5
#define EXP_USE_TOINT_NARROW 0
#define EXP2_POLY_ORDER 5
extern hidden const struct exp_data {
double invln2N;
double shift;
double negln2hiN;
double negln2loN;
double poly[4]; /* Last four coefficients. */
double exp2_shift;
double exp2_poly[EXP2_POLY_ORDER];
uint64_t tab[2*(1 << EXP_TABLE_BITS)];
} __exp_data;
#endif

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#ifndef _LIBM_H
#define _LIBM_H
#include <stdint.h>
#include <float.h>
#include <math.h>
#include <endian.h>
#include "musl_features.h"
#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 && __BYTE_ORDER == __LITTLE_ENDIAN
union ldshape {
long double f;
struct {
uint64_t m;
uint16_t se;
} i;
};
#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 && __BYTE_ORDER == __BIG_ENDIAN
/* This is the m68k variant of 80-bit long double, and this definition only works
* on archs where the alignment requirement of uint64_t is <= 4. */
union ldshape {
long double f;
struct {
uint16_t se;
uint16_t pad;
uint64_t m;
} i;
};
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 && __BYTE_ORDER == __LITTLE_ENDIAN
union ldshape {
long double f;
struct {
uint64_t lo;
uint32_t mid;
uint16_t top;
uint16_t se;
} i;
struct {
uint64_t lo;
uint64_t hi;
} i2;
};
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 && __BYTE_ORDER == __BIG_ENDIAN
union ldshape {
long double f;
struct {
uint16_t se;
uint16_t top;
uint32_t mid;
uint64_t lo;
} i;
struct {
uint64_t hi;
uint64_t lo;
} i2;
};
#else
#error Unsupported long double representation
#endif
/* Support non-nearest rounding mode. */
#define WANT_ROUNDING 1
/* Support signaling NaNs. */
#define WANT_SNAN 0
#if WANT_SNAN
#error SNaN is unsupported
#else
#define issignalingf_inline(x) 0
#define issignaling_inline(x) 0
#endif
#ifndef TOINT_INTRINSICS
#define TOINT_INTRINSICS 0
#endif
#if TOINT_INTRINSICS
/* Round x to nearest int in all rounding modes, ties have to be rounded
consistently with converttoint so the results match. If the result
would be outside of [-2^31, 2^31-1] then the semantics is unspecified. */
static double_t roundtoint(double_t);
/* Convert x to nearest int in all rounding modes, ties have to be rounded
consistently with roundtoint. If the result is not representible in an
int32_t then the semantics is unspecified. */
static int32_t converttoint(double_t);
#endif
/* Helps static branch prediction so hot path can be better optimized. */
#ifdef __GNUC__
#define predict_true(x) __builtin_expect(!!(x), 1)
#define predict_false(x) __builtin_expect(x, 0)
#else
#define predict_true(x) (x)
#define predict_false(x) (x)
#endif
/* Evaluate an expression as the specified type. With standard excess
precision handling a type cast or assignment is enough (with
-ffloat-store an assignment is required, in old compilers argument
passing and return statement may not drop excess precision). */
static inline float eval_as_float(float x)
{
float y = x;
return y;
}
static inline double eval_as_double(double x)
{
double y = x;
return y;
}
/* fp_barrier returns its input, but limits code transformations
as if it had a side-effect (e.g. observable io) and returned
an arbitrary value. */
#ifndef fp_barrierf
#define fp_barrierf fp_barrierf
static inline float fp_barrierf(float x)
{
volatile float y = x;
return y;
}
#endif
#ifndef fp_barrier
#define fp_barrier fp_barrier
static inline double fp_barrier(double x)
{
volatile double y = x;
return y;
}
#endif
#ifndef fp_barrierl
#define fp_barrierl fp_barrierl
static inline long double fp_barrierl(long double x)
{
volatile long double y = x;
return y;
}
#endif
/* fp_force_eval ensures that the input value is computed when that's
otherwise unused. To prevent the constant folding of the input
expression, an additional fp_barrier may be needed or a compilation
mode that does so (e.g. -frounding-math in gcc). Then it can be
used to evaluate an expression for its fenv side-effects only. */
#ifndef fp_force_evalf
#define fp_force_evalf fp_force_evalf
static inline void fp_force_evalf(float x)
{
volatile float y;
y = x;
}
#endif
#ifndef fp_force_eval
#define fp_force_eval fp_force_eval
static inline void fp_force_eval(double x)
{
volatile double y;
y = x;
}
#endif
#ifndef fp_force_evall
#define fp_force_evall fp_force_evall
static inline void fp_force_evall(long double x)
{
volatile long double y;
y = x;
}
#endif
#define FORCE_EVAL(x) do { \
if (sizeof(x) == sizeof(float)) { \
fp_force_evalf(x); \
} else if (sizeof(x) == sizeof(double)) { \
fp_force_eval(x); \
} else { \
fp_force_evall(x); \
} \
} while(0)
#define asuint(f) ((union{float _f; uint32_t _i;}){f})._i
#define asfloat(i) ((union{uint32_t _i; float _f;}){i})._f
#define asuint64(f) ((union{double _f; uint64_t _i;}){f})._i
#define asdouble(i) ((union{uint64_t _i; double _f;}){i})._f
#define EXTRACT_WORDS(hi,lo,d) \
do { \
uint64_t __u = asuint64(d); \
(hi) = __u >> 32; \
(lo) = (uint32_t)__u; \
} while (0)
#define GET_HIGH_WORD(hi,d) \
do { \
(hi) = asuint64(d) >> 32; \
} while (0)
#define GET_LOW_WORD(lo,d) \
do { \
(lo) = (uint32_t)asuint64(d); \
} while (0)
#define INSERT_WORDS(d,hi,lo) \
do { \
(d) = asdouble(((uint64_t)(hi)<<32) | (uint32_t)(lo)); \
} while (0)
#define SET_HIGH_WORD(d,hi) \
INSERT_WORDS(d, hi, (uint32_t)asuint64(d))
#define SET_LOW_WORD(d,lo) \
INSERT_WORDS(d, asuint64(d)>>32, lo)
#define GET_FLOAT_WORD(w,d) \
do { \
(w) = asuint(d); \
} while (0)
#define SET_FLOAT_WORD(d,w) \
do { \
(d) = asfloat(w); \
} while (0)
extern int __signgam;
hidden double __lgamma_r(double, int *);
hidden float __lgammaf_r(float, int *);
/* error handling functions */
hidden float __math_xflowf(uint32_t, float);
hidden float __math_uflowf(uint32_t);
hidden float __math_oflowf(uint32_t);
hidden float __math_divzerof(uint32_t);
hidden float __math_invalidf(float);
hidden double __math_xflow(uint32_t, double);
hidden double __math_uflow(uint32_t);
hidden double __math_oflow(uint32_t);
hidden double __math_divzero(uint32_t);
hidden double __math_invalid(double);
#endif

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/*
* Double-precision log(x) function.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "log_data.h"
#define T __log_data.tab
#define T2 __log_data.tab2
#define B __log_data.poly1
#define A __log_data.poly
#define Ln2hi __log_data.ln2hi
#define Ln2lo __log_data.ln2lo
#define N (1 << LOG_TABLE_BITS)
#define OFF 0x3fe6000000000000
/* Top 16 bits of a double. */
static inline uint32_t top16(double x)
{
return asuint64(x) >> 48;
}
double log(double x)
{
double_t w, z, r, r2, r3, y, invc, logc, kd, hi, lo;
uint64_t ix, iz, tmp;
uint32_t top;
int k, i;
ix = asuint64(x);
top = top16(x);
#define LO asuint64(1.0 - 0x1p-4)
#define HI asuint64(1.0 + 0x1.09p-4)
if (predict_false(ix - LO < HI - LO)) {
/* Handle close to 1.0 inputs separately. */
/* Fix sign of zero with downward rounding when x==1. */
if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
return 0;
r = x - 1.0;
r2 = r * r;
r3 = r * r2;
y = r3 *
(B[1] + r * B[2] + r2 * B[3] +
r3 * (B[4] + r * B[5] + r2 * B[6] +
r3 * (B[7] + r * B[8] + r2 * B[9] + r3 * B[10])));
/* Worst-case error is around 0.507 ULP. */
w = r * 0x1p27;
double_t rhi = r + w - w;
double_t rlo = r - rhi;
w = rhi * rhi * B[0]; /* B[0] == -0.5. */
hi = r + w;
lo = r - hi + w;
lo += B[0] * rlo * (rhi + r);
y += lo;
y += hi;
return eval_as_double(y);
}
if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
/* x < 0x1p-1022 or inf or nan. */
if (ix * 2 == 0)
return __math_divzero(1);
if (ix == asuint64(INFINITY)) /* log(inf) == inf. */
return x;
if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
return __math_invalid(x);
/* x is subnormal, normalize it. */
ix = asuint64(x * 0x1p52);
ix -= 52ULL << 52;
}
/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (52 - LOG_TABLE_BITS)) % N;
k = (int64_t)tmp >> 52; /* arithmetic shift */
iz = ix - (tmp & 0xfffULL << 52);
invc = T[i].invc;
logc = T[i].logc;
z = asdouble(iz);
/* log(x) = log1p(z/c-1) + log(c) + k*Ln2. */
/* r ~= z/c - 1, |r| < 1/(2*N). */
#if __FP_FAST_FMA
/* rounding error: 0x1p-55/N. */
r = __builtin_fma(z, invc, -1.0);
#else
/* rounding error: 0x1p-55/N + 0x1p-66. */
r = (z - T2[i].chi - T2[i].clo) * invc;
#endif
kd = (double_t)k;
/* hi + lo = r + log(c) + k*Ln2. */
w = kd * Ln2hi + logc;
hi = w + r;
lo = w - hi + r + kd * Ln2lo;
/* log(x) = lo + (log1p(r) - r) + hi. */
r2 = r * r; /* rounding error: 0x1p-54/N^2. */
/* Worst case error if |y| > 0x1p-5:
0.5 + 4.13/N + abs-poly-error*2^57 ULP (+ 0.002 ULP without fma)
Worst case error if |y| > 0x1p-4:
0.5 + 2.06/N + abs-poly-error*2^56 ULP (+ 0.001 ULP without fma). */
y = lo + r2 * A[0] +
r * r2 * (A[1] + r * A[2] + r2 * (A[3] + r * A[4])) + hi;
return eval_as_double(y);
}

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/*
* Double-precision log2(x) function.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "log2_data.h"
#define T __log2_data.tab
#define T2 __log2_data.tab2
#define B __log2_data.poly1
#define A __log2_data.poly
#define InvLn2hi __log2_data.invln2hi
#define InvLn2lo __log2_data.invln2lo
#define N (1 << LOG2_TABLE_BITS)
#define OFF 0x3fe6000000000000
/* Top 16 bits of a double. */
static inline uint32_t top16(double x)
{
return asuint64(x) >> 48;
}
double log2(double x)
{
double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
uint64_t ix, iz, tmp;
uint32_t top;
int k, i;
ix = asuint64(x);
top = top16(x);
#define LO asuint64(1.0 - 0x1.5b51p-5)
#define HI asuint64(1.0 + 0x1.6ab2p-5)
if (predict_false(ix - LO < HI - LO)) {
/* Handle close to 1.0 inputs separately. */
/* Fix sign of zero with downward rounding when x==1. */
if (WANT_ROUNDING && predict_false(ix == asuint64(1.0)))
return 0;
r = x - 1.0;
#if __FP_FAST_FMA
hi = r * InvLn2hi;
lo = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -hi);
#else
double_t rhi, rlo;
rhi = asdouble(asuint64(r) & -1ULL << 32);
rlo = r - rhi;
hi = rhi * InvLn2hi;
lo = rlo * InvLn2hi + r * InvLn2lo;
#endif
r2 = r * r; /* rounding error: 0x1p-62. */
r4 = r2 * r2;
/* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
p = r2 * (B[0] + r * B[1]);
y = hi + p;
lo += hi - y + p;
lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5]) +
r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
y += lo;
return eval_as_double(y);
}
if (predict_false(top - 0x0010 >= 0x7ff0 - 0x0010)) {
/* x < 0x1p-1022 or inf or nan. */
if (ix * 2 == 0)
return __math_divzero(1);
if (ix == asuint64(INFINITY)) /* log(inf) == inf. */
return x;
if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
return __math_invalid(x);
/* x is subnormal, normalize it. */
ix = asuint64(x * 0x1p52);
ix -= 52ULL << 52;
}
/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
k = (int64_t)tmp >> 52; /* arithmetic shift */
iz = ix - (tmp & 0xfffULL << 52);
invc = T[i].invc;
logc = T[i].logc;
z = asdouble(iz);
kd = (double_t)k;
/* log2(x) = log2(z/c) + log2(c) + k. */
/* r ~= z/c - 1, |r| < 1/(2*N). */
#if __FP_FAST_FMA
/* rounding error: 0x1p-55/N. */
r = __builtin_fma(z, invc, -1.0);
t1 = r * InvLn2hi;
t2 = r * InvLn2lo + __builtin_fma(r, InvLn2hi, -t1);
#else
double_t rhi, rlo;
/* rounding error: 0x1p-55/N + 0x1p-65. */
r = (z - T2[i].chi - T2[i].clo) * invc;
rhi = asdouble(asuint64(r) & -1ULL << 32);
rlo = r - rhi;
t1 = rhi * InvLn2hi;
t2 = rlo * InvLn2hi + r * InvLn2lo;
#endif
/* hi + lo = r/ln2 + log2(c) + k. */
t3 = kd + logc;
hi = t3 + t1;
lo = t3 - hi + t1 + t2;
/* log2(r+1) = r/ln2 + r^2*poly(r). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r; /* rounding error: 0x1p-54/N^2. */
r4 = r2 * r2;
/* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
y = lo + r2 * p + hi;
return eval_as_double(y);
}

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/*
* Data for log2.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include "log2_data.h"
#define N (1 << LOG2_TABLE_BITS)
const struct log2_data __log2_data = {
// First coefficient: 0x1.71547652b82fe1777d0ffda0d24p0
.invln2hi = 0x1.7154765200000p+0,
.invln2lo = 0x1.705fc2eefa200p-33,
.poly1 = {
// relative error: 0x1.2fad8188p-63
// in -0x1.5b51p-5 0x1.6ab2p-5
-0x1.71547652b82fep-1,
0x1.ec709dc3a03f7p-2,
-0x1.71547652b7c3fp-2,
0x1.2776c50f05be4p-2,
-0x1.ec709dd768fe5p-3,
0x1.a61761ec4e736p-3,
-0x1.7153fbc64a79bp-3,
0x1.484d154f01b4ap-3,
-0x1.289e4a72c383cp-3,
0x1.0b32f285aee66p-3,
},
.poly = {
// relative error: 0x1.a72c2bf8p-58
// abs error: 0x1.67a552c8p-66
// in -0x1.f45p-8 0x1.f45p-8
-0x1.71547652b8339p-1,
0x1.ec709dc3a04bep-2,
-0x1.7154764702ffbp-2,
0x1.2776c50034c48p-2,
-0x1.ec7b328ea92bcp-3,
0x1.a6225e117f92ep-3,
},
/* Algorithm:
x = 2^k z
log2(x) = k + log2(c) + log2(z/c)
log2(z/c) = poly(z/c - 1)
where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls
into the ith one, then table entries are computed as
tab[i].invc = 1/c
tab[i].logc = (double)log2(c)
tab2[i].chi = (double)c
tab2[i].clo = (double)(c - (double)c)
where c is near the center of the subinterval and is chosen by trying +-2^29
floating point invc candidates around 1/center and selecting one for which
1) the rounding error in 0x1.8p10 + logc is 0,
2) the rounding error in z - chi - clo is < 0x1p-64 and
3) the rounding error in (double)log2(c) is minimized (< 0x1p-68).
Note: 1) ensures that k + logc can be computed without rounding error, 2)
ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to a
single rounding error when there is no fast fma for z*invc - 1, 3) ensures
that logc + poly(z/c - 1) has small error, however near x == 1 when
|log2(x)| < 0x1p-4, this is not enough so that is special cased. */
.tab = {
{0x1.724286bb1acf8p+0, -0x1.1095feecdb000p-1},
{0x1.6e1f766d2cca1p+0, -0x1.08494bd76d000p-1},
{0x1.6a13d0e30d48ap+0, -0x1.00143aee8f800p-1},
{0x1.661ec32d06c85p+0, -0x1.efec5360b4000p-2},
{0x1.623fa951198f8p+0, -0x1.dfdd91ab7e000p-2},
{0x1.5e75ba4cf026cp+0, -0x1.cffae0cc79000p-2},
{0x1.5ac055a214fb8p+0, -0x1.c043811fda000p-2},
{0x1.571ed0f166e1ep+0, -0x1.b0b67323ae000p-2},
{0x1.53909590bf835p+0, -0x1.a152f5a2db000p-2},
{0x1.5014fed61adddp+0, -0x1.9217f5af86000p-2},
{0x1.4cab88e487bd0p+0, -0x1.8304db0719000p-2},
{0x1.49539b4334feep+0, -0x1.74189f9a9e000p-2},
{0x1.460cbdfafd569p+0, -0x1.6552bb5199000p-2},
{0x1.42d664ee4b953p+0, -0x1.56b23a29b1000p-2},
{0x1.3fb01111dd8a6p+0, -0x1.483650f5fa000p-2},
{0x1.3c995b70c5836p+0, -0x1.39de937f6a000p-2},
{0x1.3991c4ab6fd4ap+0, -0x1.2baa1538d6000p-2},
{0x1.3698e0ce099b5p+0, -0x1.1d98340ca4000p-2},
{0x1.33ae48213e7b2p+0, -0x1.0fa853a40e000p-2},
{0x1.30d191985bdb1p+0, -0x1.01d9c32e73000p-2},
{0x1.2e025cab271d7p+0, -0x1.e857da2fa6000p-3},
{0x1.2b404cf13cd82p+0, -0x1.cd3c8633d8000p-3},
{0x1.288b02c7ccb50p+0, -0x1.b26034c14a000p-3},
{0x1.25e2263944de5p+0, -0x1.97c1c2f4fe000p-3},
{0x1.234563d8615b1p+0, -0x1.7d6023f800000p-3},
{0x1.20b46e33eaf38p+0, -0x1.633a71a05e000p-3},
{0x1.1e2eefdcda3ddp+0, -0x1.494f5e9570000p-3},
{0x1.1bb4a580b3930p+0, -0x1.2f9e424e0a000p-3},
{0x1.19453847f2200p+0, -0x1.162595afdc000p-3},
{0x1.16e06c0d5d73cp+0, -0x1.f9c9a75bd8000p-4},
{0x1.1485f47b7e4c2p+0, -0x1.c7b575bf9c000p-4},
{0x1.12358ad0085d1p+0, -0x1.960c60ff48000p-4},
{0x1.0fef00f532227p+0, -0x1.64ce247b60000p-4},
{0x1.0db2077d03a8fp+0, -0x1.33f78b2014000p-4},
{0x1.0b7e6d65980d9p+0, -0x1.0387d1a42c000p-4},
{0x1.0953efe7b408dp+0, -0x1.a6f9208b50000p-5},
{0x1.07325cac53b83p+0, -0x1.47a954f770000p-5},
{0x1.05197e40d1b5cp+0, -0x1.d23a8c50c0000p-6},
{0x1.03091c1208ea2p+0, -0x1.16a2629780000p-6},
{0x1.0101025b37e21p+0, -0x1.720f8d8e80000p-8},
{0x1.fc07ef9caa76bp-1, 0x1.6fe53b1500000p-7},
{0x1.f4465d3f6f184p-1, 0x1.11ccce10f8000p-5},
{0x1.ecc079f84107fp-1, 0x1.c4dfc8c8b8000p-5},
{0x1.e573a99975ae8p-1, 0x1.3aa321e574000p-4},
{0x1.de5d6f0bd3de6p-1, 0x1.918a0d08b8000p-4},
{0x1.d77b681ff38b3p-1, 0x1.e72e9da044000p-4},
{0x1.d0cb5724de943p-1, 0x1.1dcd2507f6000p-3},
{0x1.ca4b2dc0e7563p-1, 0x1.476ab03dea000p-3},
{0x1.c3f8ee8d6cb51p-1, 0x1.7074377e22000p-3},
{0x1.bdd2b4f020c4cp-1, 0x1.98ede8ba94000p-3},
{0x1.b7d6c006015cap-1, 0x1.c0db86ad2e000p-3},
{0x1.b20366e2e338fp-1, 0x1.e840aafcee000p-3},
{0x1.ac57026295039p-1, 0x1.0790ab4678000p-2},
{0x1.a6d01bc2731ddp-1, 0x1.1ac056801c000p-2},
{0x1.a16d3bc3ff18bp-1, 0x1.2db11d4fee000p-2},
{0x1.9c2d14967feadp-1, 0x1.406464ec58000p-2},
{0x1.970e4f47c9902p-1, 0x1.52dbe093af000p-2},
{0x1.920fb3982bcf2p-1, 0x1.651902050d000p-2},
{0x1.8d30187f759f1p-1, 0x1.771d2cdeaf000p-2},
{0x1.886e5ebb9f66dp-1, 0x1.88e9c857d9000p-2},
{0x1.83c97b658b994p-1, 0x1.9a80155e16000p-2},
{0x1.7f405ffc61022p-1, 0x1.abe186ed3d000p-2},
{0x1.7ad22181415cap-1, 0x1.bd0f2aea0e000p-2},
{0x1.767dcf99eff8cp-1, 0x1.ce0a43dbf4000p-2},
},
#if !__FP_FAST_FMA
.tab2 = {
{0x1.6200012b90a8ep-1, 0x1.904ab0644b605p-55},
{0x1.66000045734a6p-1, 0x1.1ff9bea62f7a9p-57},
{0x1.69fffc325f2c5p-1, 0x1.27ecfcb3c90bap-55},
{0x1.6e00038b95a04p-1, 0x1.8ff8856739326p-55},
{0x1.71fffe09994e3p-1, 0x1.afd40275f82b1p-55},
{0x1.7600015590e1p-1, -0x1.2fd75b4238341p-56},
{0x1.7a00012655bd5p-1, 0x1.808e67c242b76p-56},
{0x1.7e0003259e9a6p-1, -0x1.208e426f622b7p-57},
{0x1.81fffedb4b2d2p-1, -0x1.402461ea5c92fp-55},
{0x1.860002dfafcc3p-1, 0x1.df7f4a2f29a1fp-57},
{0x1.89ffff78c6b5p-1, -0x1.e0453094995fdp-55},
{0x1.8e00039671566p-1, -0x1.a04f3bec77b45p-55},
{0x1.91fffe2bf1745p-1, -0x1.7fa34400e203cp-56},
{0x1.95fffcc5c9fd1p-1, -0x1.6ff8005a0695dp-56},
{0x1.9a0003bba4767p-1, 0x1.0f8c4c4ec7e03p-56},
{0x1.9dfffe7b92da5p-1, 0x1.e7fd9478c4602p-55},
{0x1.a1fffd72efdafp-1, -0x1.a0c554dcdae7ep-57},
{0x1.a5fffde04ff95p-1, 0x1.67da98ce9b26bp-55},
{0x1.a9fffca5e8d2bp-1, -0x1.284c9b54c13dep-55},
{0x1.adfffddad03eap-1, 0x1.812c8ea602e3cp-58},
{0x1.b1ffff10d3d4dp-1, -0x1.efaddad27789cp-55},
{0x1.b5fffce21165ap-1, 0x1.3cb1719c61237p-58},
{0x1.b9fffd950e674p-1, 0x1.3f7d94194cep-56},
{0x1.be000139ca8afp-1, 0x1.50ac4215d9bcp-56},
{0x1.c20005b46df99p-1, 0x1.beea653e9c1c9p-57},
{0x1.c600040b9f7aep-1, -0x1.c079f274a70d6p-56},
{0x1.ca0006255fd8ap-1, -0x1.a0b4076e84c1fp-56},
{0x1.cdfffd94c095dp-1, 0x1.8f933f99ab5d7p-55},
{0x1.d1ffff975d6cfp-1, -0x1.82c08665fe1bep-58},
{0x1.d5fffa2561c93p-1, -0x1.b04289bd295f3p-56},
{0x1.d9fff9d228b0cp-1, 0x1.70251340fa236p-55},
{0x1.de00065bc7e16p-1, -0x1.5011e16a4d80cp-56},
{0x1.e200002f64791p-1, 0x1.9802f09ef62ep-55},
{0x1.e600057d7a6d8p-1, -0x1.e0b75580cf7fap-56},
{0x1.ea00027edc00cp-1, -0x1.c848309459811p-55},
{0x1.ee0006cf5cb7cp-1, -0x1.f8027951576f4p-55},
{0x1.f2000782b7dccp-1, -0x1.f81d97274538fp-55},
{0x1.f6000260c450ap-1, -0x1.071002727ffdcp-59},
{0x1.f9fffe88cd533p-1, -0x1.81bdce1fda8bp-58},
{0x1.fdfffd50f8689p-1, 0x1.7f91acb918e6ep-55},
{0x1.0200004292367p+0, 0x1.b7ff365324681p-54},
{0x1.05fffe3e3d668p+0, 0x1.6fa08ddae957bp-55},
{0x1.0a0000a85a757p+0, -0x1.7e2de80d3fb91p-58},
{0x1.0e0001a5f3fccp+0, -0x1.1823305c5f014p-54},
{0x1.11ffff8afbaf5p+0, -0x1.bfabb6680bac2p-55},
{0x1.15fffe54d91adp+0, -0x1.d7f121737e7efp-54},
{0x1.1a00011ac36e1p+0, 0x1.c000a0516f5ffp-54},
{0x1.1e00019c84248p+0, -0x1.082fbe4da5dap-54},
{0x1.220000ffe5e6ep+0, -0x1.8fdd04c9cfb43p-55},
{0x1.26000269fd891p+0, 0x1.cfe2a7994d182p-55},
{0x1.2a00029a6e6dap+0, -0x1.00273715e8bc5p-56},
{0x1.2dfffe0293e39p+0, 0x1.b7c39dab2a6f9p-54},
{0x1.31ffff7dcf082p+0, 0x1.df1336edc5254p-56},
{0x1.35ffff05a8b6p+0, -0x1.e03564ccd31ebp-54},
{0x1.3a0002e0eaeccp+0, 0x1.5f0e74bd3a477p-56},
{0x1.3e000043bb236p+0, 0x1.c7dcb149d8833p-54},
{0x1.4200002d187ffp+0, 0x1.e08afcf2d3d28p-56},
{0x1.460000d387cb1p+0, 0x1.20837856599a6p-55},
{0x1.4a00004569f89p+0, -0x1.9fa5c904fbcd2p-55},
{0x1.4e000043543f3p+0, -0x1.81125ed175329p-56},
{0x1.51fffcc027f0fp+0, 0x1.883d8847754dcp-54},
{0x1.55ffffd87b36fp+0, -0x1.709e731d02807p-55},
{0x1.59ffff21df7bap+0, 0x1.7f79f68727b02p-55},
{0x1.5dfffebfc3481p+0, -0x1.180902e30e93ep-54},
},
#endif
};

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@ -0,0 +1,28 @@
/*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#ifndef _LOG2_DATA_H
#define _LOG2_DATA_H
#include "musl_features.h"
#define LOG2_TABLE_BITS 6
#define LOG2_POLY_ORDER 7
#define LOG2_POLY1_ORDER 11
extern hidden const struct log2_data {
double invln2hi;
double invln2lo;
double poly[LOG2_POLY_ORDER - 1];
double poly1[LOG2_POLY1_ORDER - 1];
struct {
double invc, logc;
} tab[1 << LOG2_TABLE_BITS];
#if !__FP_FAST_FMA
struct {
double chi, clo;
} tab2[1 << LOG2_TABLE_BITS];
#endif
} __log2_data;
#endif

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@ -0,0 +1,328 @@
/*
* Data for log.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include "log_data.h"
#define N (1 << LOG_TABLE_BITS)
const struct log_data __log_data = {
.ln2hi = 0x1.62e42fefa3800p-1,
.ln2lo = 0x1.ef35793c76730p-45,
.poly1 = {
// relative error: 0x1.c04d76cp-63
// in -0x1p-4 0x1.09p-4 (|log(1+x)| > 0x1p-4 outside the interval)
-0x1p-1,
0x1.5555555555577p-2,
-0x1.ffffffffffdcbp-3,
0x1.999999995dd0cp-3,
-0x1.55555556745a7p-3,
0x1.24924a344de3p-3,
-0x1.fffffa4423d65p-4,
0x1.c7184282ad6cap-4,
-0x1.999eb43b068ffp-4,
0x1.78182f7afd085p-4,
-0x1.5521375d145cdp-4,
},
.poly = {
// relative error: 0x1.926199e8p-56
// abs error: 0x1.882ff33p-65
// in -0x1.fp-9 0x1.fp-9
-0x1.0000000000001p-1,
0x1.555555551305bp-2,
-0x1.fffffffeb459p-3,
0x1.999b324f10111p-3,
-0x1.55575e506c89fp-3,
},
/* Algorithm:
x = 2^k z
log(x) = k ln2 + log(c) + log(z/c)
log(z/c) = poly(z/c - 1)
where z is in [1.6p-1; 1.6p0] which is split into N subintervals and z falls
into the ith one, then table entries are computed as
tab[i].invc = 1/c
tab[i].logc = (double)log(c)
tab2[i].chi = (double)c
tab2[i].clo = (double)(c - (double)c)
where c is near the center of the subinterval and is chosen by trying +-2^29
floating point invc candidates around 1/center and selecting one for which
1) the rounding error in 0x1.8p9 + logc is 0,
2) the rounding error in z - chi - clo is < 0x1p-66 and
3) the rounding error in (double)log(c) is minimized (< 0x1p-66).
Note: 1) ensures that k*ln2hi + logc can be computed without rounding error,
2) ensures that z/c - 1 can be computed as (z - chi - clo)*invc with close to
a single rounding error when there is no fast fma for z*invc - 1, 3) ensures
that logc + poly(z/c - 1) has small error, however near x == 1 when
|log(x)| < 0x1p-4, this is not enough so that is special cased. */
.tab = {
{0x1.734f0c3e0de9fp+0, -0x1.7cc7f79e69000p-2},
{0x1.713786a2ce91fp+0, -0x1.76feec20d0000p-2},
{0x1.6f26008fab5a0p+0, -0x1.713e31351e000p-2},
{0x1.6d1a61f138c7dp+0, -0x1.6b85b38287800p-2},
{0x1.6b1490bc5b4d1p+0, -0x1.65d5590807800p-2},
{0x1.69147332f0cbap+0, -0x1.602d076180000p-2},
{0x1.6719f18224223p+0, -0x1.5a8ca86909000p-2},
{0x1.6524f99a51ed9p+0, -0x1.54f4356035000p-2},
{0x1.63356aa8f24c4p+0, -0x1.4f637c36b4000p-2},
{0x1.614b36b9ddc14p+0, -0x1.49da7fda85000p-2},
{0x1.5f66452c65c4cp+0, -0x1.445923989a800p-2},
{0x1.5d867b5912c4fp+0, -0x1.3edf439b0b800p-2},
{0x1.5babccb5b90dep+0, -0x1.396ce448f7000p-2},
{0x1.59d61f2d91a78p+0, -0x1.3401e17bda000p-2},
{0x1.5805612465687p+0, -0x1.2e9e2ef468000p-2},
{0x1.56397cee76bd3p+0, -0x1.2941b3830e000p-2},
{0x1.54725e2a77f93p+0, -0x1.23ec58cda8800p-2},
{0x1.52aff42064583p+0, -0x1.1e9e129279000p-2},
{0x1.50f22dbb2bddfp+0, -0x1.1956d2b48f800p-2},
{0x1.4f38f4734ded7p+0, -0x1.141679ab9f800p-2},
{0x1.4d843cfde2840p+0, -0x1.0edd094ef9800p-2},
{0x1.4bd3ec078a3c8p+0, -0x1.09aa518db1000p-2},
{0x1.4a27fc3e0258ap+0, -0x1.047e65263b800p-2},
{0x1.4880524d48434p+0, -0x1.feb224586f000p-3},
{0x1.46dce1b192d0bp+0, -0x1.f474a7517b000p-3},
{0x1.453d9d3391854p+0, -0x1.ea4443d103000p-3},
{0x1.43a2744b4845ap+0, -0x1.e020d44e9b000p-3},
{0x1.420b54115f8fbp+0, -0x1.d60a22977f000p-3},
{0x1.40782da3ef4b1p+0, -0x1.cc00104959000p-3},
{0x1.3ee8f5d57fe8fp+0, -0x1.c202956891000p-3},
{0x1.3d5d9a00b4ce9p+0, -0x1.b81178d811000p-3},
{0x1.3bd60c010c12bp+0, -0x1.ae2c9ccd3d000p-3},
{0x1.3a5242b75dab8p+0, -0x1.a45402e129000p-3},
{0x1.38d22cd9fd002p+0, -0x1.9a877681df000p-3},
{0x1.3755bc5847a1cp+0, -0x1.90c6d69483000p-3},
{0x1.35dce49ad36e2p+0, -0x1.87120a645c000p-3},
{0x1.34679984dd440p+0, -0x1.7d68fb4143000p-3},
{0x1.32f5cceffcb24p+0, -0x1.73cb83c627000p-3},
{0x1.3187775a10d49p+0, -0x1.6a39a9b376000p-3},
{0x1.301c8373e3990p+0, -0x1.60b3154b7a000p-3},
{0x1.2eb4ebb95f841p+0, -0x1.5737d76243000p-3},
{0x1.2d50a0219a9d1p+0, -0x1.4dc7b8fc23000p-3},
{0x1.2bef9a8b7fd2ap+0, -0x1.4462c51d20000p-3},
{0x1.2a91c7a0c1babp+0, -0x1.3b08abc830000p-3},
{0x1.293726014b530p+0, -0x1.31b996b490000p-3},
{0x1.27dfa5757a1f5p+0, -0x1.2875490a44000p-3},
{0x1.268b39b1d3bbfp+0, -0x1.1f3b9f879a000p-3},
{0x1.2539d838ff5bdp+0, -0x1.160c8252ca000p-3},
{0x1.23eb7aac9083bp+0, -0x1.0ce7f57f72000p-3},
{0x1.22a012ba940b6p+0, -0x1.03cdc49fea000p-3},
{0x1.2157996cc4132p+0, -0x1.f57bdbc4b8000p-4},
{0x1.201201dd2fc9bp+0, -0x1.e370896404000p-4},
{0x1.1ecf4494d480bp+0, -0x1.d17983ef94000p-4},
{0x1.1d8f5528f6569p+0, -0x1.bf9674ed8a000p-4},
{0x1.1c52311577e7cp+0, -0x1.adc79202f6000p-4},
{0x1.1b17c74cb26e9p+0, -0x1.9c0c3e7288000p-4},
{0x1.19e010c2c1ab6p+0, -0x1.8a646b372c000p-4},
{0x1.18ab07bb670bdp+0, -0x1.78d01b3ac0000p-4},
{0x1.1778a25efbcb6p+0, -0x1.674f145380000p-4},
{0x1.1648d354c31dap+0, -0x1.55e0e6d878000p-4},
{0x1.151b990275fddp+0, -0x1.4485cdea1e000p-4},
{0x1.13f0ea432d24cp+0, -0x1.333d94d6aa000p-4},
{0x1.12c8b7210f9dap+0, -0x1.22079f8c56000p-4},
{0x1.11a3028ecb531p+0, -0x1.10e4698622000p-4},
{0x1.107fbda8434afp+0, -0x1.ffa6c6ad20000p-5},
{0x1.0f5ee0f4e6bb3p+0, -0x1.dda8d4a774000p-5},
{0x1.0e4065d2a9fcep+0, -0x1.bbcece4850000p-5},
{0x1.0d244632ca521p+0, -0x1.9a1894012c000p-5},
{0x1.0c0a77ce2981ap+0, -0x1.788583302c000p-5},
{0x1.0af2f83c636d1p+0, -0x1.5715e67d68000p-5},
{0x1.09ddb98a01339p+0, -0x1.35c8a49658000p-5},
{0x1.08cabaf52e7dfp+0, -0x1.149e364154000p-5},
{0x1.07b9f2f4e28fbp+0, -0x1.e72c082eb8000p-6},
{0x1.06ab58c358f19p+0, -0x1.a55f152528000p-6},
{0x1.059eea5ecf92cp+0, -0x1.63d62cf818000p-6},
{0x1.04949cdd12c90p+0, -0x1.228fb8caa0000p-6},
{0x1.038c6c6f0ada9p+0, -0x1.c317b20f90000p-7},
{0x1.02865137932a9p+0, -0x1.419355daa0000p-7},
{0x1.0182427ea7348p+0, -0x1.81203c2ec0000p-8},
{0x1.008040614b195p+0, -0x1.0040979240000p-9},
{0x1.fe01ff726fa1ap-1, 0x1.feff384900000p-9},
{0x1.fa11cc261ea74p-1, 0x1.7dc41353d0000p-7},
{0x1.f6310b081992ep-1, 0x1.3cea3c4c28000p-6},
{0x1.f25f63ceeadcdp-1, 0x1.b9fc114890000p-6},
{0x1.ee9c8039113e7p-1, 0x1.1b0d8ce110000p-5},
{0x1.eae8078cbb1abp-1, 0x1.58a5bd001c000p-5},
{0x1.e741aa29d0c9bp-1, 0x1.95c8340d88000p-5},
{0x1.e3a91830a99b5p-1, 0x1.d276aef578000p-5},
{0x1.e01e009609a56p-1, 0x1.07598e598c000p-4},
{0x1.dca01e577bb98p-1, 0x1.253f5e30d2000p-4},
{0x1.d92f20b7c9103p-1, 0x1.42edd8b380000p-4},
{0x1.d5cac66fb5ccep-1, 0x1.606598757c000p-4},
{0x1.d272caa5ede9dp-1, 0x1.7da76356a0000p-4},
{0x1.cf26e3e6b2ccdp-1, 0x1.9ab434e1c6000p-4},
{0x1.cbe6da2a77902p-1, 0x1.b78c7bb0d6000p-4},
{0x1.c8b266d37086dp-1, 0x1.d431332e72000p-4},
{0x1.c5894bd5d5804p-1, 0x1.f0a3171de6000p-4},
{0x1.c26b533bb9f8cp-1, 0x1.067152b914000p-3},
{0x1.bf583eeece73fp-1, 0x1.147858292b000p-3},
{0x1.bc4fd75db96c1p-1, 0x1.2266ecdca3000p-3},
{0x1.b951e0c864a28p-1, 0x1.303d7a6c55000p-3},
{0x1.b65e2c5ef3e2cp-1, 0x1.3dfc33c331000p-3},
{0x1.b374867c9888bp-1, 0x1.4ba366b7a8000p-3},
{0x1.b094b211d304ap-1, 0x1.5933928d1f000p-3},
{0x1.adbe885f2ef7ep-1, 0x1.66acd2418f000p-3},
{0x1.aaf1d31603da2p-1, 0x1.740f8ec669000p-3},
{0x1.a82e63fd358a7p-1, 0x1.815c0f51af000p-3},
{0x1.a5740ef09738bp-1, 0x1.8e92954f68000p-3},
{0x1.a2c2a90ab4b27p-1, 0x1.9bb3602f84000p-3},
{0x1.a01a01393f2d1p-1, 0x1.a8bed1c2c0000p-3},
{0x1.9d79f24db3c1bp-1, 0x1.b5b515c01d000p-3},
{0x1.9ae2505c7b190p-1, 0x1.c2967ccbcc000p-3},
{0x1.9852ef297ce2fp-1, 0x1.cf635d5486000p-3},
{0x1.95cbaeea44b75p-1, 0x1.dc1bd3446c000p-3},
{0x1.934c69de74838p-1, 0x1.e8c01b8cfe000p-3},
{0x1.90d4f2f6752e6p-1, 0x1.f5509c0179000p-3},
{0x1.8e6528effd79dp-1, 0x1.00e6c121fb800p-2},
{0x1.8bfce9fcc007cp-1, 0x1.071b80e93d000p-2},
{0x1.899c0dabec30ep-1, 0x1.0d46b9e867000p-2},
{0x1.87427aa2317fbp-1, 0x1.13687334bd000p-2},
{0x1.84f00acb39a08p-1, 0x1.1980d67234800p-2},
{0x1.82a49e8653e55p-1, 0x1.1f8ffe0cc8000p-2},
{0x1.8060195f40260p-1, 0x1.2595fd7636800p-2},
{0x1.7e22563e0a329p-1, 0x1.2b9300914a800p-2},
{0x1.7beb377dcb5adp-1, 0x1.3187210436000p-2},
{0x1.79baa679725c2p-1, 0x1.377266dec1800p-2},
{0x1.77907f2170657p-1, 0x1.3d54ffbaf3000p-2},
{0x1.756cadbd6130cp-1, 0x1.432eee32fe000p-2},
},
#if !__FP_FAST_FMA
.tab2 = {
{0x1.61000014fb66bp-1, 0x1.e026c91425b3cp-56},
{0x1.63000034db495p-1, 0x1.dbfea48005d41p-55},
{0x1.650000d94d478p-1, 0x1.e7fa786d6a5b7p-55},
{0x1.67000074e6fadp-1, 0x1.1fcea6b54254cp-57},
{0x1.68ffffedf0faep-1, -0x1.c7e274c590efdp-56},
{0x1.6b0000763c5bcp-1, -0x1.ac16848dcda01p-55},
{0x1.6d0001e5cc1f6p-1, 0x1.33f1c9d499311p-55},
{0x1.6efffeb05f63ep-1, -0x1.e80041ae22d53p-56},
{0x1.710000e86978p-1, 0x1.bff6671097952p-56},
{0x1.72ffffc67e912p-1, 0x1.c00e226bd8724p-55},
{0x1.74fffdf81116ap-1, -0x1.e02916ef101d2p-57},
{0x1.770000f679c9p-1, -0x1.7fc71cd549c74p-57},
{0x1.78ffffa7ec835p-1, 0x1.1bec19ef50483p-55},
{0x1.7affffe20c2e6p-1, -0x1.07e1729cc6465p-56},
{0x1.7cfffed3fc9p-1, -0x1.08072087b8b1cp-55},
{0x1.7efffe9261a76p-1, 0x1.dc0286d9df9aep-55},
{0x1.81000049ca3e8p-1, 0x1.97fd251e54c33p-55},
{0x1.8300017932c8fp-1, -0x1.afee9b630f381p-55},
{0x1.850000633739cp-1, 0x1.9bfbf6b6535bcp-55},
{0x1.87000204289c6p-1, -0x1.bbf65f3117b75p-55},
{0x1.88fffebf57904p-1, -0x1.9006ea23dcb57p-55},
{0x1.8b00022bc04dfp-1, -0x1.d00df38e04b0ap-56},
{0x1.8cfffe50c1b8ap-1, -0x1.8007146ff9f05p-55},
{0x1.8effffc918e43p-1, 0x1.3817bd07a7038p-55},
{0x1.910001efa5fc7p-1, 0x1.93e9176dfb403p-55},
{0x1.9300013467bb9p-1, 0x1.f804e4b980276p-56},
{0x1.94fffe6ee076fp-1, -0x1.f7ef0d9ff622ep-55},
{0x1.96fffde3c12d1p-1, -0x1.082aa962638bap-56},
{0x1.98ffff4458a0dp-1, -0x1.7801b9164a8efp-55},
{0x1.9afffdd982e3ep-1, -0x1.740e08a5a9337p-55},
{0x1.9cfffed49fb66p-1, 0x1.fce08c19bep-60},
{0x1.9f00020f19c51p-1, -0x1.a3faa27885b0ap-55},
{0x1.a10001145b006p-1, 0x1.4ff489958da56p-56},
{0x1.a300007bbf6fap-1, 0x1.cbeab8a2b6d18p-55},
{0x1.a500010971d79p-1, 0x1.8fecadd78793p-55},
{0x1.a70001df52e48p-1, -0x1.f41763dd8abdbp-55},
{0x1.a90001c593352p-1, -0x1.ebf0284c27612p-55},
{0x1.ab0002a4f3e4bp-1, -0x1.9fd043cff3f5fp-57},
{0x1.acfffd7ae1ed1p-1, -0x1.23ee7129070b4p-55},
{0x1.aefffee510478p-1, 0x1.a063ee00edea3p-57},
{0x1.b0fffdb650d5bp-1, 0x1.a06c8381f0ab9p-58},
{0x1.b2ffffeaaca57p-1, -0x1.9011e74233c1dp-56},
{0x1.b4fffd995badcp-1, -0x1.9ff1068862a9fp-56},
{0x1.b7000249e659cp-1, 0x1.aff45d0864f3ep-55},
{0x1.b8ffff987164p-1, 0x1.cfe7796c2c3f9p-56},
{0x1.bafffd204cb4fp-1, -0x1.3ff27eef22bc4p-57},
{0x1.bcfffd2415c45p-1, -0x1.cffb7ee3bea21p-57},
{0x1.beffff86309dfp-1, -0x1.14103972e0b5cp-55},
{0x1.c0fffe1b57653p-1, 0x1.bc16494b76a19p-55},
{0x1.c2ffff1fa57e3p-1, -0x1.4feef8d30c6edp-57},
{0x1.c4fffdcbfe424p-1, -0x1.43f68bcec4775p-55},
{0x1.c6fffed54b9f7p-1, 0x1.47ea3f053e0ecp-55},
{0x1.c8fffeb998fd5p-1, 0x1.383068df992f1p-56},
{0x1.cb0002125219ap-1, -0x1.8fd8e64180e04p-57},
{0x1.ccfffdd94469cp-1, 0x1.e7ebe1cc7ea72p-55},
{0x1.cefffeafdc476p-1, 0x1.ebe39ad9f88fep-55},
{0x1.d1000169af82bp-1, 0x1.57d91a8b95a71p-56},
{0x1.d30000d0ff71dp-1, 0x1.9c1906970c7dap-55},
{0x1.d4fffea790fc4p-1, -0x1.80e37c558fe0cp-58},
{0x1.d70002edc87e5p-1, -0x1.f80d64dc10f44p-56},
{0x1.d900021dc82aap-1, -0x1.47c8f94fd5c5cp-56},
{0x1.dafffd86b0283p-1, 0x1.c7f1dc521617ep-55},
{0x1.dd000296c4739p-1, 0x1.8019eb2ffb153p-55},
{0x1.defffe54490f5p-1, 0x1.e00d2c652cc89p-57},
{0x1.e0fffcdabf694p-1, -0x1.f8340202d69d2p-56},
{0x1.e2fffdb52c8ddp-1, 0x1.b00c1ca1b0864p-56},
{0x1.e4ffff24216efp-1, 0x1.2ffa8b094ab51p-56},
{0x1.e6fffe88a5e11p-1, -0x1.7f673b1efbe59p-58},
{0x1.e9000119eff0dp-1, -0x1.4808d5e0bc801p-55},
{0x1.eafffdfa51744p-1, 0x1.80006d54320b5p-56},
{0x1.ed0001a127fa1p-1, -0x1.002f860565c92p-58},
{0x1.ef00007babcc4p-1, -0x1.540445d35e611p-55},
{0x1.f0ffff57a8d02p-1, -0x1.ffb3139ef9105p-59},
{0x1.f30001ee58ac7p-1, 0x1.a81acf2731155p-55},
{0x1.f4ffff5823494p-1, 0x1.a3f41d4d7c743p-55},
{0x1.f6ffffca94c6bp-1, -0x1.202f41c987875p-57},
{0x1.f8fffe1f9c441p-1, 0x1.77dd1f477e74bp-56},
{0x1.fafffd2e0e37ep-1, -0x1.f01199a7ca331p-57},
{0x1.fd0001c77e49ep-1, 0x1.181ee4bceacb1p-56},
{0x1.feffff7e0c331p-1, -0x1.e05370170875ap-57},
{0x1.00ffff465606ep+0, -0x1.a7ead491c0adap-55},
{0x1.02ffff3867a58p+0, -0x1.77f69c3fcb2ep-54},
{0x1.04ffffdfc0d17p+0, 0x1.7bffe34cb945bp-54},
{0x1.0700003cd4d82p+0, 0x1.20083c0e456cbp-55},
{0x1.08ffff9f2cbe8p+0, -0x1.dffdfbe37751ap-57},
{0x1.0b000010cda65p+0, -0x1.13f7faee626ebp-54},
{0x1.0d00001a4d338p+0, 0x1.07dfa79489ff7p-55},
{0x1.0effffadafdfdp+0, -0x1.7040570d66bcp-56},
{0x1.110000bbafd96p+0, 0x1.e80d4846d0b62p-55},
{0x1.12ffffae5f45dp+0, 0x1.dbffa64fd36efp-54},
{0x1.150000dd59ad9p+0, 0x1.a0077701250aep-54},
{0x1.170000f21559ap+0, 0x1.dfdf9e2e3deeep-55},
{0x1.18ffffc275426p+0, 0x1.10030dc3b7273p-54},
{0x1.1b000123d3c59p+0, 0x1.97f7980030188p-54},
{0x1.1cffff8299eb7p+0, -0x1.5f932ab9f8c67p-57},
{0x1.1effff48ad4p+0, 0x1.37fbf9da75bebp-54},
{0x1.210000c8b86a4p+0, 0x1.f806b91fd5b22p-54},
{0x1.2300003854303p+0, 0x1.3ffc2eb9fbf33p-54},
{0x1.24fffffbcf684p+0, 0x1.601e77e2e2e72p-56},
{0x1.26ffff52921d9p+0, 0x1.ffcbb767f0c61p-56},
{0x1.2900014933a3cp+0, -0x1.202ca3c02412bp-56},
{0x1.2b00014556313p+0, -0x1.2808233f21f02p-54},
{0x1.2cfffebfe523bp+0, -0x1.8ff7e384fdcf2p-55},
{0x1.2f0000bb8ad96p+0, -0x1.5ff51503041c5p-55},
{0x1.30ffffb7ae2afp+0, -0x1.10071885e289dp-55},
{0x1.32ffffeac5f7fp+0, -0x1.1ff5d3fb7b715p-54},
{0x1.350000ca66756p+0, 0x1.57f82228b82bdp-54},
{0x1.3700011fbf721p+0, 0x1.000bac40dd5ccp-55},
{0x1.38ffff9592fb9p+0, -0x1.43f9d2db2a751p-54},
{0x1.3b00004ddd242p+0, 0x1.57f6b707638e1p-55},
{0x1.3cffff5b2c957p+0, 0x1.a023a10bf1231p-56},
{0x1.3efffeab0b418p+0, 0x1.87f6d66b152bp-54},
{0x1.410001532aff4p+0, 0x1.7f8375f198524p-57},
{0x1.4300017478b29p+0, 0x1.301e672dc5143p-55},
{0x1.44fffe795b463p+0, 0x1.9ff69b8b2895ap-55},
{0x1.46fffe80475ep+0, -0x1.5c0b19bc2f254p-54},
{0x1.48fffef6fc1e7p+0, 0x1.b4009f23a2a72p-54},
{0x1.4afffe5bea704p+0, -0x1.4ffb7bf0d7d45p-54},
{0x1.4d000171027dep+0, -0x1.9c06471dc6a3dp-54},
{0x1.4f0000ff03ee2p+0, 0x1.77f890b85531cp-54},
{0x1.5100012dc4bd1p+0, 0x1.004657166a436p-57},
{0x1.530001605277ap+0, -0x1.6bfcece233209p-54},
{0x1.54fffecdb704cp+0, -0x1.902720505a1d7p-55},
{0x1.56fffef5f54a9p+0, 0x1.bbfe60ec96412p-54},
{0x1.5900017e61012p+0, 0x1.87ec581afef9p-55},
{0x1.5b00003c93e92p+0, -0x1.f41080abf0ccp-54},
{0x1.5d0001d4919bcp+0, -0x1.8812afb254729p-54},
{0x1.5efffe7b87a89p+0, -0x1.47eb780ed6904p-54},
},
#endif
};

View File

@ -0,0 +1,28 @@
/*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#ifndef _LOG_DATA_H
#define _LOG_DATA_H
#include "musl_features.h"
#define LOG_TABLE_BITS 7
#define LOG_POLY_ORDER 6
#define LOG_POLY1_ORDER 12
extern hidden const struct log_data {
double ln2hi;
double ln2lo;
double poly[LOG_POLY_ORDER - 1]; /* First coefficient is 1. */
double poly1[LOG_POLY1_ORDER - 1];
struct {
double invc, logc;
} tab[1 << LOG_TABLE_BITS];
#if !__FP_FAST_FMA
struct {
double chi, clo;
} tab2[1 << LOG_TABLE_BITS];
#endif
} __log_data;
#endif

View File

@ -7,11 +7,9 @@
#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "logf_data.h"
float __math_invalidf(float);
float __math_divzerof(uint32_t);
/*
LOGF_TABLE_BITS = 4
LOGF_POLY_ORDER = 4
@ -25,21 +23,6 @@ Relative error: 1.957 * 2^-26 (before rounding.)
#define Ln2 __logf_data.ln2
#define N (1 << LOGF_TABLE_BITS)
#define OFF 0x3f330000
#define WANT_ROUNDING 1
#define asuint(f) ((union{float _f; uint32_t _i;}){f})._i
#define asfloat(i) ((union{uint32_t _i; float _f;}){i})._f
/* Evaluate an expression as the specified type. With standard excess
precision handling a type cast or assignment is enough (with
-ffloat-store an assignment is required, in old compilers argument
passing and return statement may not drop excess precision). */
static inline float eval_as_float(float x)
{
float y = x;
return y;
}
float logf(float x)
{
@ -49,9 +32,9 @@ float logf(float x)
ix = asuint(x);
/* Fix sign of zero with downward rounding when x==1. */
if (WANT_ROUNDING && __builtin_expect(ix == 0x3f800000, 0))
if (WANT_ROUNDING && predict_false(ix == 0x3f800000))
return 0;
if (__builtin_expect(ix - 0x00800000 >= 0x7f800000 - 0x00800000, 0)) {
if (predict_false(ix - 0x00800000 >= 0x7f800000 - 0x00800000)) {
/* x < 0x1p-126 or inf or nan. */
if (ix * 2 == 0)
return __math_divzerof(1);

View File

@ -5,9 +5,11 @@
#ifndef _LOGF_DATA_H
#define _LOGF_DATA_H
#include "musl_features.h"
#define LOGF_TABLE_BITS 4
#define LOGF_POLY_ORDER 4
extern __attribute__((__visibility__("hidden"))) const struct logf_data {
extern hidden const struct logf_data {
struct {
double invc, logc;
} tab[1 << LOGF_TABLE_BITS];

View File

@ -0,0 +1,8 @@
#pragma once
#define weak __attribute__((__weak__))
#define hidden __attribute__((__visibility__("hidden")))
#define weak_alias(old, new) \
extern __typeof(old) new __attribute__((__weak__, __alias__(#old)))
#define predict_false(x) __builtin_expect(x, 0)

View File

@ -0,0 +1,343 @@
/*
* Double-precision x^y function.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include <math.h>
#include <stdint.h>
#include "libm.h"
#include "exp_data.h"
#include "pow_data.h"
/*
Worst-case error: 0.54 ULP (~= ulperr_exp + 1024*Ln2*relerr_log*2^53)
relerr_log: 1.3 * 2^-68 (Relative error of log, 1.5 * 2^-68 without fma)
ulperr_exp: 0.509 ULP (ULP error of exp, 0.511 ULP without fma)
*/
#define T __pow_log_data.tab
#define A __pow_log_data.poly
#define Ln2hi __pow_log_data.ln2hi
#define Ln2lo __pow_log_data.ln2lo
#define N (1 << POW_LOG_TABLE_BITS)
#define OFF 0x3fe6955500000000
/* Top 12 bits of a double (sign and exponent bits). */
static inline uint32_t top12(double x)
{
return asuint64(x) >> 52;
}
/* Compute y+TAIL = log(x) where the rounded result is y and TAIL has about
additional 15 bits precision. IX is the bit representation of x, but
normalized in the subnormal range using the sign bit for the exponent. */
static inline double_t log_inline(uint64_t ix, double_t *tail)
{
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t z, r, y, invc, logc, logctail, kd, hi, t1, t2, lo, lo1, lo2, p;
uint64_t iz, tmp;
int k, i;
/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (52 - POW_LOG_TABLE_BITS)) % N;
k = (int64_t)tmp >> 52; /* arithmetic shift */
iz = ix - (tmp & 0xfffULL << 52);
z = asdouble(iz);
kd = (double_t)k;
/* log(x) = k*Ln2 + log(c) + log1p(z/c-1). */
invc = T[i].invc;
logc = T[i].logc;
logctail = T[i].logctail;
/* Note: 1/c is j/N or j/N/2 where j is an integer in [N,2N) and
|z/c - 1| < 1/N, so r = z/c - 1 is exactly representible. */
#if __FP_FAST_FMA
r = __builtin_fma(z, invc, -1.0);
#else
/* Split z such that rhi, rlo and rhi*rhi are exact and |rlo| <= |r|. */
double_t zhi = asdouble((iz + (1ULL << 31)) & (-1ULL << 32));
double_t zlo = z - zhi;
double_t rhi = zhi * invc - 1.0;
double_t rlo = zlo * invc;
r = rhi + rlo;
#endif
/* k*Ln2 + log(c) + r. */
t1 = kd * Ln2hi + logc;
t2 = t1 + r;
lo1 = kd * Ln2lo + logctail;
lo2 = t1 - t2 + r;
/* Evaluation is optimized assuming superscalar pipelined execution. */
double_t ar, ar2, ar3, lo3, lo4;
ar = A[0] * r; /* A[0] = -0.5. */
ar2 = r * ar;
ar3 = r * ar2;
/* k*Ln2 + log(c) + r + A[0]*r*r. */
#if __FP_FAST_FMA
hi = t2 + ar2;
lo3 = __builtin_fma(ar, r, -ar2);
lo4 = t2 - hi + ar2;
#else
double_t arhi = A[0] * rhi;
double_t arhi2 = rhi * arhi;
hi = t2 + arhi2;
lo3 = rlo * (ar + arhi);
lo4 = t2 - hi + arhi2;
#endif
/* p = log1p(r) - r - A[0]*r*r. */
p = (ar3 * (A[1] + r * A[2] +
ar2 * (A[3] + r * A[4] + ar2 * (A[5] + r * A[6]))));
lo = lo1 + lo2 + lo3 + lo4 + p;
y = hi + lo;
*tail = hi - y + lo;
return y;
}
#undef N
#undef T
#define N (1 << EXP_TABLE_BITS)
#define InvLn2N __exp_data.invln2N
#define NegLn2hiN __exp_data.negln2hiN
#define NegLn2loN __exp_data.negln2loN
#define Shift __exp_data.shift
#define T __exp_data.tab
#define C2 __exp_data.poly[5 - EXP_POLY_ORDER]
#define C3 __exp_data.poly[6 - EXP_POLY_ORDER]
#define C4 __exp_data.poly[7 - EXP_POLY_ORDER]
#define C5 __exp_data.poly[8 - EXP_POLY_ORDER]
#define C6 __exp_data.poly[9 - EXP_POLY_ORDER]
/* Handle cases that may overflow or underflow when computing the result that
is scale*(1+TMP) without intermediate rounding. The bit representation of
scale is in SBITS, however it has a computed exponent that may have
overflown into the sign bit so that needs to be adjusted before using it as
a double. (int32_t)KI is the k used in the argument reduction and exponent
adjustment of scale, positive k here means the result may overflow and
negative k means the result may underflow. */
static inline double specialcase(double_t tmp, uint64_t sbits, uint64_t ki)
{
double_t scale, y;
if ((ki & 0x80000000) == 0) {
/* k > 0, the exponent of scale might have overflowed by <= 460. */
sbits -= 1009ull << 52;
scale = asdouble(sbits);
y = 0x1p1009 * (scale + scale * tmp);
return eval_as_double(y);
}
/* k < 0, need special care in the subnormal range. */
sbits += 1022ull << 52;
/* Note: sbits is signed scale. */
scale = asdouble(sbits);
y = scale + scale * tmp;
if (fabs(y) < 1.0) {
/* Round y to the right precision before scaling it into the subnormal
range to avoid double rounding that can cause 0.5+E/2 ulp error where
E is the worst-case ulp error outside the subnormal range. So this
is only useful if the goal is better than 1 ulp worst-case error. */
double_t hi, lo, one = 1.0;
if (y < 0.0)
one = -1.0;
lo = scale - y + scale * tmp;
hi = one + y;
lo = one - hi + y + lo;
y = eval_as_double(hi + lo) - one;
/* Fix the sign of 0. */
if (y == 0.0)
y = asdouble(sbits & 0x8000000000000000);
/* The underflow exception needs to be signaled explicitly. */
fp_force_eval(fp_barrier(0x1p-1022) * 0x1p-1022);
}
y = 0x1p-1022 * y;
return eval_as_double(y);
}
#define SIGN_BIAS (0x800 << EXP_TABLE_BITS)
/* Computes sign*exp(x+xtail) where |xtail| < 2^-8/N and |xtail| <= |x|.
The sign_bias argument is SIGN_BIAS or 0 and sets the sign to -1 or 1. */
static inline double exp_inline(double_t x, double_t xtail, uint32_t sign_bias)
{
uint32_t abstop;
uint64_t ki, idx, top, sbits;
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t kd, z, r, r2, scale, tail, tmp;
abstop = top12(x) & 0x7ff;
if (predict_false(abstop - top12(0x1p-54) >=
top12(512.0) - top12(0x1p-54))) {
if (abstop - top12(0x1p-54) >= 0x80000000) {
/* Avoid spurious underflow for tiny x. */
/* Note: 0 is common input. */
double_t one = WANT_ROUNDING ? 1.0 + x : 1.0;
return sign_bias ? -one : one;
}
if (abstop >= top12(1024.0)) {
/* Note: inf and nan are already handled. */
if (asuint64(x) >> 63)
return __math_uflow(sign_bias);
else
return __math_oflow(sign_bias);
}
/* Large x is special cased below. */
abstop = 0;
}
/* exp(x) = 2^(k/N) * exp(r), with exp(r) in [2^(-1/2N),2^(1/2N)]. */
/* x = ln2/N*k + r, with int k and r in [-ln2/2N, ln2/2N]. */
z = InvLn2N * x;
#if TOINT_INTRINSICS
kd = roundtoint(z);
ki = converttoint(z);
#elif EXP_USE_TOINT_NARROW
/* z - kd is in [-0.5-2^-16, 0.5] in all rounding modes. */
kd = eval_as_double(z + Shift);
ki = asuint64(kd) >> 16;
kd = (double_t)(int32_t)ki;
#else
/* z - kd is in [-1, 1] in non-nearest rounding modes. */
kd = eval_as_double(z + Shift);
ki = asuint64(kd);
kd -= Shift;
#endif
r = x + kd * NegLn2hiN + kd * NegLn2loN;
/* The code assumes 2^-200 < |xtail| < 2^-8/N. */
r += xtail;
/* 2^(k/N) ~= scale * (1 + tail). */
idx = 2 * (ki % N);
top = (ki + sign_bias) << (52 - EXP_TABLE_BITS);
tail = asdouble(T[idx]);
/* This is only a valid scale when -1023*N < k < 1024*N. */
sbits = T[idx + 1] + top;
/* exp(x) = 2^(k/N) * exp(r) ~= scale + scale * (tail + exp(r) - 1). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r;
/* Without fma the worst case error is 0.25/N ulp larger. */
/* Worst case error is less than 0.5+1.11/N+(abs poly error * 2^53) ulp. */
tmp = tail + r + r2 * (C2 + r * C3) + r2 * r2 * (C4 + r * C5);
if (predict_false(abstop == 0))
return specialcase(tmp, sbits, ki);
scale = asdouble(sbits);
/* Note: tmp == 0 or |tmp| > 2^-200 and scale > 2^-739, so there
is no spurious underflow here even without fma. */
return eval_as_double(scale + scale * tmp);
}
/* Returns 0 if not int, 1 if odd int, 2 if even int. The argument is
the bit representation of a non-zero finite floating-point value. */
static inline int checkint(uint64_t iy)
{
int e = iy >> 52 & 0x7ff;
if (e < 0x3ff)
return 0;
if (e > 0x3ff + 52)
return 2;
if (iy & ((1ULL << (0x3ff + 52 - e)) - 1))
return 0;
if (iy & (1ULL << (0x3ff + 52 - e)))
return 1;
return 2;
}
/* Returns 1 if input is the bit representation of 0, infinity or nan. */
static inline int zeroinfnan(uint64_t i)
{
return 2 * i - 1 >= 2 * asuint64(INFINITY) - 1;
}
double pow(double x, double y)
{
uint32_t sign_bias = 0;
uint64_t ix, iy;
uint32_t topx, topy;
ix = asuint64(x);
iy = asuint64(y);
topx = top12(x);
topy = top12(y);
if (predict_false(topx - 0x001 >= 0x7ff - 0x001 ||
(topy & 0x7ff) - 0x3be >= 0x43e - 0x3be)) {
/* Note: if |y| > 1075 * ln2 * 2^53 ~= 0x1.749p62 then pow(x,y) = inf/0
and if |y| < 2^-54 / 1075 ~= 0x1.e7b6p-65 then pow(x,y) = +-1. */
/* Special cases: (x < 0x1p-126 or inf or nan) or
(|y| < 0x1p-65 or |y| >= 0x1p63 or nan). */
if (predict_false(zeroinfnan(iy))) {
if (2 * iy == 0)
return issignaling_inline(x) ? x + y : 1.0;
if (ix == asuint64(1.0))
return issignaling_inline(y) ? x + y : 1.0;
if (2 * ix > 2 * asuint64(INFINITY) ||
2 * iy > 2 * asuint64(INFINITY))
return x + y;
if (2 * ix == 2 * asuint64(1.0))
return 1.0;
if ((2 * ix < 2 * asuint64(1.0)) == !(iy >> 63))
return 0.0; /* |x|<1 && y==inf or |x|>1 && y==-inf. */
return y * y;
}
if (predict_false(zeroinfnan(ix))) {
double_t x2 = x * x;
if (ix >> 63 && checkint(iy) == 1)
x2 = -x2;
/* Without the barrier some versions of clang hoist the 1/x2 and
thus division by zero exception can be signaled spuriously. */
return iy >> 63 ? fp_barrier(1 / x2) : x2;
}
/* Here x and y are non-zero finite. */
if (ix >> 63) {
/* Finite x < 0. */
int yint = checkint(iy);
if (yint == 0)
return __math_invalid(x);
if (yint == 1)
sign_bias = SIGN_BIAS;
ix &= 0x7fffffffffffffff;
topx &= 0x7ff;
}
if ((topy & 0x7ff) - 0x3be >= 0x43e - 0x3be) {
/* Note: sign_bias == 0 here because y is not odd. */
if (ix == asuint64(1.0))
return 1.0;
if ((topy & 0x7ff) < 0x3be) {
/* |y| < 2^-65, x^y ~= 1 + y*log(x). */
if (WANT_ROUNDING)
return ix > asuint64(1.0) ? 1.0 + y :
1.0 - y;
else
return 1.0;
}
return (ix > asuint64(1.0)) == (topy < 0x800) ?
__math_oflow(0) :
__math_uflow(0);
}
if (topx == 0) {
/* Normalize subnormal x so exponent becomes negative. */
ix = asuint64(x * 0x1p52);
ix &= 0x7fffffffffffffff;
ix -= 52ULL << 52;
}
}
double_t lo;
double_t hi = log_inline(ix, &lo);
double_t ehi, elo;
#if __FP_FAST_FMA
ehi = y * hi;
elo = y * lo + __builtin_fma(y, hi, -ehi);
#else
double_t yhi = asdouble(iy & -1ULL << 27);
double_t ylo = y - yhi;
double_t lhi = asdouble(asuint64(hi) & -1ULL << 27);
double_t llo = hi - lhi + lo;
ehi = yhi * lhi;
elo = ylo * lhi + y * llo; /* |elo| < |ehi| * 2^-25. */
#endif
return exp_inline(ehi, elo, sign_bias);
}

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@ -0,0 +1,180 @@
/*
* Data for the log part of pow.
*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#include "pow_data.h"
#define N (1 << POW_LOG_TABLE_BITS)
const struct pow_log_data __pow_log_data = {
.ln2hi = 0x1.62e42fefa3800p-1,
.ln2lo = 0x1.ef35793c76730p-45,
.poly = {
// relative error: 0x1.11922ap-70
// in -0x1.6bp-8 0x1.6bp-8
// Coefficients are scaled to match the scaling during evaluation.
-0x1p-1,
0x1.555555555556p-2 * -2,
-0x1.0000000000006p-2 * -2,
0x1.999999959554ep-3 * 4,
-0x1.555555529a47ap-3 * 4,
0x1.2495b9b4845e9p-3 * -8,
-0x1.0002b8b263fc3p-3 * -8,
},
/* Algorithm:
x = 2^k z
log(x) = k ln2 + log(c) + log(z/c)
log(z/c) = poly(z/c - 1)
where z is in [0x1.69555p-1; 0x1.69555p0] which is split into N subintervals
and z falls into the ith one, then table entries are computed as
tab[i].invc = 1/c
tab[i].logc = round(0x1p43*log(c))/0x1p43
tab[i].logctail = (double)(log(c) - logc)
where c is chosen near the center of the subinterval such that 1/c has only a
few precision bits so z/c - 1 is exactly representible as double:
1/c = center < 1 ? round(N/center)/N : round(2*N/center)/N/2
Note: |z/c - 1| < 1/N for the chosen c, |log(c) - logc - logctail| < 0x1p-97,
the last few bits of logc are rounded away so k*ln2hi + logc has no rounding
error and the interval for z is selected such that near x == 1, where log(x)
is tiny, large cancellation error is avoided in logc + poly(z/c - 1). */
.tab = {
#define A(a, b, c) {a, 0, b, c},
A(0x1.6a00000000000p+0, -0x1.62c82f2b9c800p-2, 0x1.ab42428375680p-48)
A(0x1.6800000000000p+0, -0x1.5d1bdbf580800p-2, -0x1.ca508d8e0f720p-46)
A(0x1.6600000000000p+0, -0x1.5767717455800p-2, -0x1.362a4d5b6506dp-45)
A(0x1.6400000000000p+0, -0x1.51aad872df800p-2, -0x1.684e49eb067d5p-49)
A(0x1.6200000000000p+0, -0x1.4be5f95777800p-2, -0x1.41b6993293ee0p-47)
A(0x1.6000000000000p+0, -0x1.4618bc21c6000p-2, 0x1.3d82f484c84ccp-46)
A(0x1.5e00000000000p+0, -0x1.404308686a800p-2, 0x1.c42f3ed820b3ap-50)
A(0x1.5c00000000000p+0, -0x1.3a64c55694800p-2, 0x1.0b1c686519460p-45)
A(0x1.5a00000000000p+0, -0x1.347dd9a988000p-2, 0x1.5594dd4c58092p-45)
A(0x1.5800000000000p+0, -0x1.2e8e2bae12000p-2, 0x1.67b1e99b72bd8p-45)
A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46)
A(0x1.5600000000000p+0, -0x1.2895a13de8800p-2, 0x1.5ca14b6cfb03fp-46)
A(0x1.5400000000000p+0, -0x1.22941fbcf7800p-2, -0x1.65a242853da76p-46)
A(0x1.5200000000000p+0, -0x1.1c898c1699800p-2, -0x1.fafbc68e75404p-46)
A(0x1.5000000000000p+0, -0x1.1675cababa800p-2, 0x1.f1fc63382a8f0p-46)
A(0x1.4e00000000000p+0, -0x1.1058bf9ae4800p-2, -0x1.6a8c4fd055a66p-45)
A(0x1.4c00000000000p+0, -0x1.0a324e2739000p-2, -0x1.c6bee7ef4030ep-47)
A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48)
A(0x1.4a00000000000p+0, -0x1.0402594b4d000p-2, -0x1.036b89ef42d7fp-48)
A(0x1.4800000000000p+0, -0x1.fb9186d5e4000p-3, 0x1.d572aab993c87p-47)
A(0x1.4600000000000p+0, -0x1.ef0adcbdc6000p-3, 0x1.b26b79c86af24p-45)
A(0x1.4400000000000p+0, -0x1.e27076e2af000p-3, -0x1.72f4f543fff10p-46)
A(0x1.4200000000000p+0, -0x1.d5c216b4fc000p-3, 0x1.1ba91bbca681bp-45)
A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45)
A(0x1.4000000000000p+0, -0x1.c8ff7c79aa000p-3, 0x1.7794f689f8434p-45)
A(0x1.3e00000000000p+0, -0x1.bc286742d9000p-3, 0x1.94eb0318bb78fp-46)
A(0x1.3c00000000000p+0, -0x1.af3c94e80c000p-3, 0x1.a4e633fcd9066p-52)
A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45)
A(0x1.3a00000000000p+0, -0x1.a23bc1fe2b000p-3, -0x1.58c64dc46c1eap-45)
A(0x1.3800000000000p+0, -0x1.9525a9cf45000p-3, -0x1.ad1d904c1d4e3p-45)
A(0x1.3600000000000p+0, -0x1.87fa06520d000p-3, 0x1.bbdbf7fdbfa09p-45)
A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45)
A(0x1.3400000000000p+0, -0x1.7ab890210e000p-3, 0x1.bdb9072534a58p-45)
A(0x1.3200000000000p+0, -0x1.6d60fe719d000p-3, -0x1.0e46aa3b2e266p-46)
A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46)
A(0x1.3000000000000p+0, -0x1.5ff3070a79000p-3, -0x1.e9e439f105039p-46)
A(0x1.2e00000000000p+0, -0x1.526e5e3a1b000p-3, -0x1.0de8b90075b8fp-45)
A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46)
A(0x1.2c00000000000p+0, -0x1.44d2b6ccb8000p-3, 0x1.70cc16135783cp-46)
A(0x1.2a00000000000p+0, -0x1.371fc201e9000p-3, 0x1.178864d27543ap-48)
A(0x1.2800000000000p+0, -0x1.29552f81ff000p-3, -0x1.48d301771c408p-45)
A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45)
A(0x1.2600000000000p+0, -0x1.1b72ad52f6000p-3, -0x1.e80a41811a396p-45)
A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47)
A(0x1.2400000000000p+0, -0x1.0d77e7cd09000p-3, 0x1.a699688e85bf4p-47)
A(0x1.2200000000000p+0, -0x1.fec9131dbe000p-4, -0x1.575545ca333f2p-45)
A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45)
A(0x1.2000000000000p+0, -0x1.e27076e2b0000p-4, 0x1.a342c2af0003cp-45)
A(0x1.1e00000000000p+0, -0x1.c5e548f5bc000p-4, -0x1.d0c57585fbe06p-46)
A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45)
A(0x1.1c00000000000p+0, -0x1.a926d3a4ae000p-4, 0x1.53935e85baac8p-45)
A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46)
A(0x1.1a00000000000p+0, -0x1.8c345d631a000p-4, 0x1.37c294d2f5668p-46)
A(0x1.1800000000000p+0, -0x1.6f0d28ae56000p-4, -0x1.69737c93373dap-45)
A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46)
A(0x1.1600000000000p+0, -0x1.51b073f062000p-4, 0x1.f025b61c65e57p-46)
A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45)
A(0x1.1400000000000p+0, -0x1.341d7961be000p-4, 0x1.c5edaccf913dfp-45)
A(0x1.1200000000000p+0, -0x1.16536eea38000p-4, 0x1.47c5e768fa309p-46)
A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45)
A(0x1.1000000000000p+0, -0x1.f0a30c0118000p-5, 0x1.d599e83368e91p-45)
A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46)
A(0x1.0e00000000000p+0, -0x1.b42dd71198000p-5, 0x1.c827ae5d6704cp-46)
A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45)
A(0x1.0c00000000000p+0, -0x1.77458f632c000p-5, -0x1.cfc4634f2a1eep-45)
A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48)
A(0x1.0a00000000000p+0, -0x1.39e87b9fec000p-5, 0x1.502b7f526feaap-48)
A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45)
A(0x1.0800000000000p+0, -0x1.f829b0e780000p-6, -0x1.980267c7e09e4p-45)
A(0x1.0600000000000p+0, -0x1.7b91b07d58000p-6, -0x1.88d5493faa639p-45)
A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50)
A(0x1.0400000000000p+0, -0x1.fc0a8b0fc0000p-7, -0x1.f1e7cf6d3a69cp-50)
A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46)
A(0x1.0200000000000p+0, -0x1.fe02a6b100000p-8, -0x1.9e23f0dda40e4p-46)
A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0)
A(0x1.0000000000000p+0, 0x0.0000000000000p+0, 0x0.0000000000000p+0)
A(0x1.fc00000000000p-1, 0x1.0101575890000p-7, -0x1.0c76b999d2be8p-46)
A(0x1.f800000000000p-1, 0x1.0205658938000p-6, -0x1.3dc5b06e2f7d2p-45)
A(0x1.f400000000000p-1, 0x1.8492528c90000p-6, -0x1.aa0ba325a0c34p-45)
A(0x1.f000000000000p-1, 0x1.0415d89e74000p-5, 0x1.111c05cf1d753p-47)
A(0x1.ec00000000000p-1, 0x1.466aed42e0000p-5, -0x1.c167375bdfd28p-45)
A(0x1.e800000000000p-1, 0x1.894aa149fc000p-5, -0x1.97995d05a267dp-46)
A(0x1.e400000000000p-1, 0x1.ccb73cdddc000p-5, -0x1.a68f247d82807p-46)
A(0x1.e200000000000p-1, 0x1.eea31c006c000p-5, -0x1.e113e4fc93b7bp-47)
A(0x1.de00000000000p-1, 0x1.1973bd1466000p-4, -0x1.5325d560d9e9bp-45)
A(0x1.da00000000000p-1, 0x1.3bdf5a7d1e000p-4, 0x1.cc85ea5db4ed7p-45)
A(0x1.d600000000000p-1, 0x1.5e95a4d97a000p-4, -0x1.c69063c5d1d1ep-45)
A(0x1.d400000000000p-1, 0x1.700d30aeac000p-4, 0x1.c1e8da99ded32p-49)
A(0x1.d000000000000p-1, 0x1.9335e5d594000p-4, 0x1.3115c3abd47dap-45)
A(0x1.cc00000000000p-1, 0x1.b6ac88dad6000p-4, -0x1.390802bf768e5p-46)
A(0x1.ca00000000000p-1, 0x1.c885801bc4000p-4, 0x1.646d1c65aacd3p-45)
A(0x1.c600000000000p-1, 0x1.ec739830a2000p-4, -0x1.dc068afe645e0p-45)
A(0x1.c400000000000p-1, 0x1.fe89139dbe000p-4, -0x1.534d64fa10afdp-45)
A(0x1.c000000000000p-1, 0x1.1178e8227e000p-3, 0x1.1ef78ce2d07f2p-45)
A(0x1.be00000000000p-1, 0x1.1aa2b7e23f000p-3, 0x1.ca78e44389934p-45)
A(0x1.ba00000000000p-1, 0x1.2d1610c868000p-3, 0x1.39d6ccb81b4a1p-47)
A(0x1.b800000000000p-1, 0x1.365fcb0159000p-3, 0x1.62fa8234b7289p-51)
A(0x1.b400000000000p-1, 0x1.4913d8333b000p-3, 0x1.5837954fdb678p-45)
A(0x1.b200000000000p-1, 0x1.527e5e4a1b000p-3, 0x1.633e8e5697dc7p-45)
A(0x1.ae00000000000p-1, 0x1.6574ebe8c1000p-3, 0x1.9cf8b2c3c2e78p-46)
A(0x1.ac00000000000p-1, 0x1.6f0128b757000p-3, -0x1.5118de59c21e1p-45)
A(0x1.aa00000000000p-1, 0x1.7898d85445000p-3, -0x1.c661070914305p-46)
A(0x1.a600000000000p-1, 0x1.8beafeb390000p-3, -0x1.73d54aae92cd1p-47)
A(0x1.a400000000000p-1, 0x1.95a5adcf70000p-3, 0x1.7f22858a0ff6fp-47)
A(0x1.a000000000000p-1, 0x1.a93ed3c8ae000p-3, -0x1.8724350562169p-45)
A(0x1.9e00000000000p-1, 0x1.b31d8575bd000p-3, -0x1.c358d4eace1aap-47)
A(0x1.9c00000000000p-1, 0x1.bd087383be000p-3, -0x1.d4bc4595412b6p-45)
A(0x1.9a00000000000p-1, 0x1.c6ffbc6f01000p-3, -0x1.1ec72c5962bd2p-48)
A(0x1.9600000000000p-1, 0x1.db13db0d49000p-3, -0x1.aff2af715b035p-45)
A(0x1.9400000000000p-1, 0x1.e530effe71000p-3, 0x1.212276041f430p-51)
A(0x1.9200000000000p-1, 0x1.ef5ade4dd0000p-3, -0x1.a211565bb8e11p-51)
A(0x1.9000000000000p-1, 0x1.f991c6cb3b000p-3, 0x1.bcbecca0cdf30p-46)
A(0x1.8c00000000000p-1, 0x1.07138604d5800p-2, 0x1.89cdb16ed4e91p-48)
A(0x1.8a00000000000p-1, 0x1.0c42d67616000p-2, 0x1.7188b163ceae9p-45)
A(0x1.8800000000000p-1, 0x1.1178e8227e800p-2, -0x1.c210e63a5f01cp-45)
A(0x1.8600000000000p-1, 0x1.16b5ccbacf800p-2, 0x1.b9acdf7a51681p-45)
A(0x1.8400000000000p-1, 0x1.1bf99635a6800p-2, 0x1.ca6ed5147bdb7p-45)
A(0x1.8200000000000p-1, 0x1.214456d0eb800p-2, 0x1.a87deba46baeap-47)
A(0x1.7e00000000000p-1, 0x1.2bef07cdc9000p-2, 0x1.a9cfa4a5004f4p-45)
A(0x1.7c00000000000p-1, 0x1.314f1e1d36000p-2, -0x1.8e27ad3213cb8p-45)
A(0x1.7a00000000000p-1, 0x1.36b6776be1000p-2, 0x1.16ecdb0f177c8p-46)
A(0x1.7800000000000p-1, 0x1.3c25277333000p-2, 0x1.83b54b606bd5cp-46)
A(0x1.7600000000000p-1, 0x1.419b423d5e800p-2, 0x1.8e436ec90e09dp-47)
A(0x1.7400000000000p-1, 0x1.4718dc271c800p-2, -0x1.f27ce0967d675p-45)
A(0x1.7200000000000p-1, 0x1.4c9e09e173000p-2, -0x1.e20891b0ad8a4p-45)
A(0x1.7000000000000p-1, 0x1.522ae0738a000p-2, 0x1.ebe708164c759p-45)
A(0x1.6e00000000000p-1, 0x1.57bf753c8d000p-2, 0x1.fadedee5d40efp-46)
A(0x1.6c00000000000p-1, 0x1.5d5bddf596000p-2, -0x1.a0b2a08a465dcp-47)
},
};

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@ -0,0 +1,22 @@
/*
* Copyright (c) 2018, Arm Limited.
* SPDX-License-Identifier: MIT
*/
#ifndef _POW_DATA_H
#define _POW_DATA_H
#include "musl_features.h"
#define POW_LOG_TABLE_BITS 7
#define POW_LOG_POLY_ORDER 8
extern hidden const struct pow_log_data {
double ln2hi;
double ln2lo;
double poly[POW_LOG_POLY_ORDER - 1]; /* First coefficient is 1. */
/* Note: the pad field is unused, but allows slightly faster indexing. */
struct {
double invc, pad, logc, logctail;
} tab[1 << POW_LOG_TABLE_BITS];
} __pow_log_data;
#endif