Optimized generic expf and exp2f with wrappers
Based on new expf and exp2f code from https://github.com/ARM-software/optimized-routines/ with wrapper on aarch64: expf reciprocal-throughput: 2.3x faster expf latency: 1.7x faster without wrapper on aarch64: expf reciprocal-throughput: 3.3x faster expf latency: 1.7x faster without wrapper on aarch64: exp2f reciprocal-throughput: 2.8x faster exp2f latency: 1.3x faster libm.so size on aarch64: .text size: -152 bytes .rodata size: -1740 bytes expf/exp2f worst case nearest rounding error: 0.502 ulp worst case non-nearest rounding error: 1 ulp Error checks are inline and errno setting is in separate tail called functions, but the wrappers are kept in this patch to handle the _LIB_VERSION==_SVID_ case. (So e.g. errno is set twice for expf calls and once for __expf_finite calls on targets where the new code is used.) Double precision arithmetics is used which is expected to be faster on most targets (including soft-float) than using single precision and it is easier to get good precision result with it. Const data is kept in a separate translation unit which complicates maintenance a bit, but is expected to give good code for literal loads on most targets and allows sharing data across expf, exp2f and powf. (This data is disabled on i386, m68k and ia64 which have their own expf, exp2f and powf code.) Some details may need target specific tweaks: - best convert and round to int operation in the arg reduction may be different across targets. - code was optimized on fma target, optimal polynomial eval may be different without fma. - gcc does not always generate good code for fp bit representation access via unions or it may be inherently slow on some targets. The libm-test-ulps will need adjustment because.. - The argument reduction ideally uses nearest rounded rint, but that is not efficient on most targets, so the polynomial can get evaluated on a wider interval in non-nearest rounding mode making 1 ulp errors common in that case. - The polynomial is evaluated such that it may have 1 ulp error on negative tiny inputs with upward rounding. * math/Makefile (type-float-routines): Add math_errf and e_exp2f_data. * sysdeps/aarch64/fpu/math_private.h (TOINT_INTRINSICS): Define. (roundtoint, converttoint): Likewise. * sysdeps/ieee754/flt-32/e_expf.c: New implementation. * sysdeps/ieee754/flt-32/e_exp2f.c: New implementation. * sysdeps/ieee754/flt-32/e_exp2f_data.c: New file. * sysdeps/ieee754/flt-32/math_config.h: New file. * sysdeps/ieee754/flt-32/math_errf.c: New file. * sysdeps/ieee754/flt-32/t_exp2f.h: Remove. * sysdeps/i386/fpu/e_exp2f_data.c: New file. * sysdeps/i386/fpu/math_errf.c: New file. * sysdeps/ia64/fpu/e_exp2f_data.c: New file. * sysdeps/ia64/fpu/math_errf.c: New file. * sysdeps/m68k/m680x0/fpu/e_exp2f_data.c: New file. * sysdeps/m68k/m680x0/fpu/math_errf.c: New file.
This commit is contained in:
Szabolcs Nagy
parent
fcafcd162c
commit
72aa623345
18
ChangeLog
18
ChangeLog
|
|
@ -1,3 +1,21 @@
|
|||
2017-09-25 Szabolcs Nagy <szabolcs.nagy@arm.com>
|
||||
|
||||
* math/Makefile (type-float-routines): Add math_errf and e_exp2f_data.
|
||||
* sysdeps/aarch64/fpu/math_private.h (TOINT_INTRINSICS): Define.
|
||||
(roundtoint, converttoint): Likewise.
|
||||
* sysdeps/ieee754/flt-32/e_expf.c: New implementation.
|
||||
* sysdeps/ieee754/flt-32/e_exp2f.c: New implementation.
|
||||
* sysdeps/ieee754/flt-32/e_exp2f_data.c: New file.
|
||||
* sysdeps/ieee754/flt-32/math_config.h: New file.
|
||||
* sysdeps/ieee754/flt-32/math_errf.c: New file.
|
||||
* sysdeps/ieee754/flt-32/t_exp2f.h: Remove.
|
||||
* sysdeps/i386/fpu/e_exp2f_data.c: New file.
|
||||
* sysdeps/i386/fpu/math_errf.c: New file.
|
||||
* sysdeps/ia64/fpu/e_exp2f_data.c: New file.
|
||||
* sysdeps/ia64/fpu/math_errf.c: New file.
|
||||
* sysdeps/m68k/m680x0/fpu/e_exp2f_data.c: New file.
|
||||
* sysdeps/m68k/m680x0/fpu/math_errf.c: New file.
|
||||
|
||||
2017-09-24 Samuel Thibault <samuel.thibault@ens-lyon.org>
|
||||
|
||||
* sysdeps/mach/hurd/dl-sysdep.c (check_no_hidden): New macro.
|
||||
|
|
|
|||
2
NEWS
2
NEWS
|
|
@ -14,6 +14,8 @@ Major new features:
|
|||
|
||||
* Optimized x86-64 trunc and truncf for processors with SSE4.1.
|
||||
|
||||
* Optimized generic expf and exp2f.
|
||||
|
||||
* In order to support faster and safer process termination the malloc API
|
||||
family of functions will no longer print a failure address and stack
|
||||
backtrace after detecting heap corruption. The goal is to minimize the
|
||||
|
|
|
|||
|
|
@ -115,7 +115,7 @@ type-double-routines := branred doasin dosincos halfulp mpa mpatan2 \
|
|||
|
||||
# float support
|
||||
type-float-suffix := f
|
||||
type-float-routines := k_rem_pio2f
|
||||
type-float-routines := k_rem_pio2f math_errf e_exp2f_data
|
||||
|
||||
# _Float128 support
|
||||
type-float128-suffix := f128
|
||||
|
|
|
|||
|
|
@ -319,6 +319,26 @@ libc_feresetround_noex_aarch64_ctx (struct rm_ctx *ctx)
|
|||
#define libc_feresetround_noexf_ctx libc_feresetround_noex_aarch64_ctx
|
||||
#define libc_feresetround_noexl_ctx libc_feresetround_noex_aarch64_ctx
|
||||
|
||||
/* Hack: only include the large arm_neon.h when needed. */
|
||||
#ifdef _MATH_CONFIG_H
|
||||
# include <arm_neon.h>
|
||||
|
||||
/* ACLE intrinsics for frintn and fcvtns instructions. */
|
||||
# define TOINT_INTRINSICS 1
|
||||
|
||||
static inline double_t
|
||||
roundtoint (double_t x)
|
||||
{
|
||||
return vget_lane_f64 (vrndn_f64 (vld1_f64 (&x)), 0);
|
||||
}
|
||||
|
||||
static inline uint64_t
|
||||
converttoint (double_t x)
|
||||
{
|
||||
return vcvtnd_s64_f64 (x);
|
||||
}
|
||||
#endif
|
||||
|
||||
#include_next <math_private.h>
|
||||
|
||||
#endif
|
||||
|
|
|
|||
|
|
@ -0,0 +1 @@
|
|||
/* Not needed. */
|
||||
|
|
@ -0,0 +1 @@
|
|||
/* Not needed. */
|
||||
|
|
@ -0,0 +1 @@
|
|||
/* Not needed. */
|
||||
|
|
@ -0,0 +1 @@
|
|||
/* Not needed. */
|
||||
|
|
@ -1,7 +1,6 @@
|
|||
/* Single-precision floating point 2^x.
|
||||
Copyright (C) 1997-2017 Free Software Foundation, Inc.
|
||||
/* Single-precision 2^x function.
|
||||
Copyright (C) 2017 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
|
||||
|
||||
The GNU C Library is free software; you can redistribute it and/or
|
||||
modify it under the terms of the GNU Lesser General Public
|
||||
|
|
@ -17,116 +16,73 @@
|
|||
License along with the GNU C Library; if not, see
|
||||
<http://www.gnu.org/licenses/>. */
|
||||
|
||||
/* The basic design here is from
|
||||
Shmuel Gal and Boris Bachelis, "An Accurate Elementary Mathematical
|
||||
Library for the IEEE Floating Point Standard", ACM Trans. Math. Soft.,
|
||||
17 (1), March 1991, pp. 26-45.
|
||||
It has been slightly modified to compute 2^x instead of e^x, and for
|
||||
single-precision.
|
||||
*/
|
||||
#ifndef _GNU_SOURCE
|
||||
# define _GNU_SOURCE
|
||||
#endif
|
||||
#include <stdlib.h>
|
||||
#include <float.h>
|
||||
#include <ieee754.h>
|
||||
#include <math.h>
|
||||
#include <fenv.h>
|
||||
#include <inttypes.h>
|
||||
#include <math_private.h>
|
||||
#include <stdint.h>
|
||||
#include "math_config.h"
|
||||
|
||||
#include "t_exp2f.h"
|
||||
/*
|
||||
EXP2F_TABLE_BITS = 5
|
||||
EXP2F_POLY_ORDER = 3
|
||||
|
||||
static const float TWOM100 = 7.88860905e-31;
|
||||
static const float TWO127 = 1.7014118346e+38;
|
||||
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
|
||||
__ieee754_exp2f (float x)
|
||||
{
|
||||
static const float himark = (float) FLT_MAX_EXP;
|
||||
static const float lomark = (float) (FLT_MIN_EXP - FLT_MANT_DIG - 1);
|
||||
uint32_t abstop;
|
||||
uint64_t ki, t;
|
||||
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
|
||||
double_t kd, xd, z, r, r2, y, s;
|
||||
|
||||
/* Check for usual case. */
|
||||
if (isless (x, himark) && isgreaterequal (x, lomark))
|
||||
xd = (double_t) x;
|
||||
abstop = top12 (x) & 0x7ff;
|
||||
if (__glibc_unlikely (abstop >= top12 (128.0f)))
|
||||
{
|
||||
static const float THREEp14 = 49152.0;
|
||||
int tval, unsafe;
|
||||
float rx, x22, result;
|
||||
union ieee754_float ex2_u, scale_u;
|
||||
|
||||
if (fabsf (x) < FLT_EPSILON / 4.0f)
|
||||
return 1.0f + x;
|
||||
|
||||
{
|
||||
SET_RESTORE_ROUND_NOEXF (FE_TONEAREST);
|
||||
|
||||
/* 1. Argument reduction.
|
||||
Choose integers ex, -128 <= t < 128, and some real
|
||||
-1/512 <= x1 <= 1/512 so that
|
||||
x = ex + t/512 + x1.
|
||||
|
||||
First, calculate rx = ex + t/256. */
|
||||
rx = x + THREEp14;
|
||||
rx -= THREEp14;
|
||||
x -= rx; /* Compute x=x1. */
|
||||
/* Compute tval = (ex*256 + t)+128.
|
||||
Now, t = (tval mod 256)-128 and ex=tval/256 [that's mod, NOT %;
|
||||
and /-round-to-nearest not the usual c integer /]. */
|
||||
tval = (int) (rx * 256.0f + 128.0f);
|
||||
|
||||
/* 2. Adjust for accurate table entry.
|
||||
Find e so that
|
||||
x = ex + t/256 + e + x2
|
||||
where -7e-4 < e < 7e-4, and
|
||||
(float)(2^(t/256+e))
|
||||
is accurate to one part in 2^-64. */
|
||||
|
||||
/* 'tval & 255' is the same as 'tval%256' except that it's always
|
||||
positive.
|
||||
Compute x = x2. */
|
||||
x -= __exp2f_deltatable[tval & 255];
|
||||
|
||||
/* 3. Compute ex2 = 2^(t/255+e+ex). */
|
||||
ex2_u.f = __exp2f_atable[tval & 255];
|
||||
tval >>= 8;
|
||||
/* x2 is an integer multiple of 2^-30; avoid intermediate
|
||||
underflow from the calculation of x22 * x. */
|
||||
unsafe = abs(tval) >= -FLT_MIN_EXP - 32;
|
||||
ex2_u.ieee.exponent += tval >> unsafe;
|
||||
scale_u.f = 1.0;
|
||||
scale_u.ieee.exponent += tval - (tval >> unsafe);
|
||||
|
||||
/* 4. Approximate 2^x2 - 1, using a second-degree polynomial,
|
||||
with maximum error in [-2^-9 - 2^-14, 2^-9 + 2^-14]
|
||||
less than 1.3e-10. */
|
||||
|
||||
x22 = (.24022656679f * x + .69314736128f) * ex2_u.f;
|
||||
}
|
||||
|
||||
/* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex). */
|
||||
result = x22 * x + ex2_u.f;
|
||||
|
||||
if (!unsafe)
|
||||
return result;
|
||||
else
|
||||
{
|
||||
result *= scale_u.f;
|
||||
math_check_force_underflow_nonneg (result);
|
||||
return result;
|
||||
}
|
||||
/* |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);
|
||||
#if WANT_ERRNO_UFLOW
|
||||
if (x < -149.0f)
|
||||
return __math_may_uflowf (0);
|
||||
#endif
|
||||
}
|
||||
/* Exceptional cases: */
|
||||
else if (isless (x, himark))
|
||||
{
|
||||
if (isinf (x))
|
||||
/* e^-inf == 0, with no error. */
|
||||
return 0;
|
||||
else
|
||||
/* Underflow */
|
||||
return TWOM100 * TWOM100;
|
||||
}
|
||||
else
|
||||
/* Return x, if x is a NaN or Inf; or overflow, otherwise. */
|
||||
return TWO127*x;
|
||||
|
||||
/* x = k/N + r with r in [-1/(2N), 1/(2N)] and int k. */
|
||||
kd = math_narrow_eval ((double) (xd + SHIFT)); /* Needs to be double. */
|
||||
ki = asuint64 (kd);
|
||||
kd -= SHIFT; /* k/N for int k. */
|
||||
r = xd - kd;
|
||||
|
||||
/* 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 (float) y;
|
||||
}
|
||||
strong_alias (__ieee754_exp2f, __exp2f_finite)
|
||||
|
|
|
|||
|
|
@ -0,0 +1,44 @@
|
|||
/* Shared data between expf, exp2f and powf.
|
||||
Copyright (C) 2017 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
|
||||
The GNU C Library is free software; you can redistribute it and/or
|
||||
modify it under the terms of the GNU Lesser General Public
|
||||
License as published by the Free Software Foundation; either
|
||||
version 2.1 of the License, or (at your option) any later version.
|
||||
|
||||
The GNU C Library is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
||||
Lesser General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU Lesser General Public
|
||||
License along with the GNU C Library; if not, see
|
||||
<http://www.gnu.org/licenses/>. */
|
||||
|
||||
#include "math_config.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
|
||||
},
|
||||
};
|
||||
|
|
@ -1,7 +1,6 @@
|
|||
/* Single-precision floating point e^x.
|
||||
Copyright (C) 1997-2017 Free Software Foundation, Inc.
|
||||
/* Single-precision e^x function.
|
||||
Copyright (C) 2017 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
|
||||
|
||||
The GNU C Library is free software; you can redistribute it and/or
|
||||
modify it under the terms of the GNU Lesser General Public
|
||||
|
|
@ -17,117 +16,87 @@
|
|||
License along with the GNU C Library; if not, see
|
||||
<http://www.gnu.org/licenses/>. */
|
||||
|
||||
/* How this works:
|
||||
|
||||
The input value, x, is written as
|
||||
|
||||
x = n * ln(2) + t/512 + delta[t] + x;
|
||||
|
||||
where:
|
||||
- n is an integer, 127 >= n >= -150;
|
||||
- t is an integer, 177 >= t >= -177
|
||||
- delta is based on a table entry, delta[t] < 2^-28
|
||||
- x is whatever is left, |x| < 2^-10
|
||||
|
||||
Then e^x is approximated as
|
||||
|
||||
e^x = 2^n ( e^(t/512 + delta[t])
|
||||
+ ( e^(t/512 + delta[t])
|
||||
* ( p(x + delta[t] + n * ln(2)) - delta ) ) )
|
||||
|
||||
where
|
||||
- p(x) is a polynomial approximating e(x)-1;
|
||||
- e^(t/512 + delta[t]) is obtained from a table.
|
||||
|
||||
The table used is the same one as for the double precision version;
|
||||
since we have the table, we might as well use it.
|
||||
|
||||
It turns out to be faster to do calculations in double precision than
|
||||
to perform an 'accurate table method' expf, because of the range reduction
|
||||
overhead (compare exp2f).
|
||||
*/
|
||||
#include <float.h>
|
||||
#include <ieee754.h>
|
||||
#include <math.h>
|
||||
#include <fenv.h>
|
||||
#include <inttypes.h>
|
||||
#include <math_private.h>
|
||||
#include <stdint.h>
|
||||
#include "math_config.h"
|
||||
|
||||
extern const float __exp_deltatable[178];
|
||||
extern const double __exp_atable[355] /* __attribute__((mode(DF))) */;
|
||||
/*
|
||||
EXP2F_TABLE_BITS = 5
|
||||
EXP2F_POLY_ORDER = 3
|
||||
|
||||
static const float TWOM100 = 7.88860905e-31;
|
||||
static const float TWO127 = 1.7014118346e+38;
|
||||
ULP error: 0.502 (nearest rounding.)
|
||||
Relative error: 1.69 * 2^-34 in [-ln2/64, ln2/64] (before rounding.)
|
||||
Wrong count: 170635 (all nearest rounding wrong results with fma.)
|
||||
Non-nearest ULP error: 1 (rounded ULP error)
|
||||
*/
|
||||
|
||||
#define N (1 << EXP2F_TABLE_BITS)
|
||||
#define InvLn2N __exp2f_data.invln2_scaled
|
||||
#define T __exp2f_data.tab
|
||||
#define C __exp2f_data.poly_scaled
|
||||
|
||||
static inline uint32_t
|
||||
top12 (float x)
|
||||
{
|
||||
return asuint (x) >> 20;
|
||||
}
|
||||
|
||||
float
|
||||
__ieee754_expf (float x)
|
||||
{
|
||||
static const float himark = 88.72283935546875;
|
||||
static const float lomark = -103.972084045410;
|
||||
/* Check for usual case. */
|
||||
if (isless (x, himark) && isgreater (x, lomark))
|
||||
uint32_t abstop;
|
||||
uint64_t ki, t;
|
||||
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
|
||||
double_t kd, xd, z, r, r2, y, s;
|
||||
|
||||
xd = (double_t) x;
|
||||
abstop = top12 (x) & 0x7ff;
|
||||
if (__glibc_unlikely (abstop >= top12 (88.0f)))
|
||||
{
|
||||
static const float THREEp42 = 13194139533312.0;
|
||||
static const float THREEp22 = 12582912.0;
|
||||
/* 1/ln(2). */
|
||||
#undef M_1_LN2
|
||||
static const float M_1_LN2 = 1.44269502163f;
|
||||
/* ln(2) */
|
||||
#undef M_LN2
|
||||
static const double M_LN2 = .6931471805599452862;
|
||||
|
||||
int tval;
|
||||
double x22, t, result, dx;
|
||||
float n, delta;
|
||||
union ieee754_double ex2_u;
|
||||
|
||||
{
|
||||
SET_RESTORE_ROUND_NOEXF (FE_TONEAREST);
|
||||
|
||||
/* Calculate n. */
|
||||
n = x * M_1_LN2 + THREEp22;
|
||||
n -= THREEp22;
|
||||
dx = x - n*M_LN2;
|
||||
|
||||
/* Calculate t/512. */
|
||||
t = dx + THREEp42;
|
||||
t -= THREEp42;
|
||||
dx -= t;
|
||||
|
||||
/* Compute tval = t. */
|
||||
tval = (int) (t * 512.0);
|
||||
|
||||
if (t >= 0)
|
||||
delta = - __exp_deltatable[tval];
|
||||
else
|
||||
delta = __exp_deltatable[-tval];
|
||||
|
||||
/* Compute ex2 = 2^n e^(t/512+delta[t]). */
|
||||
ex2_u.d = __exp_atable[tval+177];
|
||||
ex2_u.ieee.exponent += (int) n;
|
||||
|
||||
/* Approximate e^(dx+delta) - 1, using a second-degree polynomial,
|
||||
with maximum error in [-2^-10-2^-28,2^-10+2^-28]
|
||||
less than 5e-11. */
|
||||
x22 = (0.5000000496709180453 * dx + 1.0000001192102037084) * dx + delta;
|
||||
}
|
||||
|
||||
/* Return result. */
|
||||
result = x22 * ex2_u.d + ex2_u.d;
|
||||
return (float) result;
|
||||
/* |x| >= 88 or x is nan. */
|
||||
if (asuint (x) == asuint (-INFINITY))
|
||||
return 0.0f;
|
||||
if (abstop >= top12 (INFINITY))
|
||||
return x + x;
|
||||
if (x > 0x1.62e42ep6f) /* x > log(0x1p128) ~= 88.72 */
|
||||
return __math_oflowf (0);
|
||||
if (x < -0x1.9fe368p6f) /* x < log(0x1p-150) ~= -103.97 */
|
||||
return __math_uflowf (0);
|
||||
#if WANT_ERRNO_UFLOW
|
||||
if (x < -0x1.9d1d9ep6f) /* x < log(0x1p-149) ~= -103.28 */
|
||||
return __math_may_uflowf (0);
|
||||
#endif
|
||||
}
|
||||
/* Exceptional cases: */
|
||||
else if (isless (x, himark))
|
||||
{
|
||||
if (isinf (x))
|
||||
/* e^-inf == 0, with no error. */
|
||||
return 0;
|
||||
else
|
||||
/* Underflow */
|
||||
return TWOM100 * TWOM100;
|
||||
}
|
||||
else
|
||||
/* Return x, if x is a NaN or Inf; or overflow, otherwise. */
|
||||
return TWO127*x;
|
||||
|
||||
/* x*N/Ln2 = k + r with r in [-1/2, 1/2] and int k. */
|
||||
z = InvLn2N * xd;
|
||||
|
||||
/* Round and convert z to int, the result is in [-150*N, 128*N] and
|
||||
ideally ties-to-even rule is used, otherwise the magnitude of r
|
||||
can be bigger which gives larger approximation error. */
|
||||
#if TOINT_INTRINSICS
|
||||
kd = roundtoint (z);
|
||||
ki = converttoint (z);
|
||||
#elif TOINT_RINT
|
||||
kd = rint (z);
|
||||
ki = (long) kd;
|
||||
#elif TOINT_SHIFT
|
||||
# define SHIFT __exp2f_data.shift
|
||||
kd = math_narrow_eval ((double) (z + SHIFT)); /* Needs to be double. */
|
||||
ki = asuint64 (kd);
|
||||
kd -= SHIFT;
|
||||
#endif
|
||||
r = z - kd;
|
||||
|
||||
/* exp(x) = 2^(k/N) * 2^(r/N) ~= 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 (float) y;
|
||||
}
|
||||
strong_alias (__ieee754_expf, __expf_finite)
|
||||
|
|
|
|||
|
|
@ -0,0 +1,114 @@
|
|||
/* Configuration for math routines.
|
||||
Copyright (C) 2017 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
|
||||
The GNU C Library is free software; you can redistribute it and/or
|
||||
modify it under the terms of the GNU Lesser General Public
|
||||
License as published by the Free Software Foundation; either
|
||||
version 2.1 of the License, or (at your option) any later version.
|
||||
|
||||
The GNU C Library is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
||||
Lesser General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU Lesser General Public
|
||||
License along with the GNU C Library; if not, see
|
||||
<http://www.gnu.org/licenses/>. */
|
||||
|
||||
#ifndef _MATH_CONFIG_H
|
||||
#define _MATH_CONFIG_H
|
||||
|
||||
#include <math.h>
|
||||
#include <math_private.h>
|
||||
#include <stdint.h>
|
||||
|
||||
#ifndef WANT_ROUNDING
|
||||
/* Correct special case results in non-nearest rounding modes. */
|
||||
# define WANT_ROUNDING 1
|
||||
#endif
|
||||
#ifndef WANT_ERRNO
|
||||
/* Set errno according to ISO C with (math_errhandling & MATH_ERRNO) != 0. */
|
||||
# define WANT_ERRNO 1
|
||||
#endif
|
||||
#ifndef WANT_ERRNO_UFLOW
|
||||
/* Set errno to ERANGE if result underflows to 0 (in all rounding modes). */
|
||||
# define WANT_ERRNO_UFLOW (WANT_ROUNDING && WANT_ERRNO)
|
||||
#endif
|
||||
|
||||
#ifndef TOINT_INTRINSICS
|
||||
# define TOINT_INTRINSICS 0
|
||||
#endif
|
||||
#ifndef TOINT_RINT
|
||||
# define TOINT_RINT 0
|
||||
#endif
|
||||
#ifndef TOINT_SHIFT
|
||||
# define TOINT_SHIFT 1
|
||||
#endif
|
||||
|
||||
static inline uint32_t
|
||||
asuint (float f)
|
||||
{
|
||||
union
|
||||
{
|
||||
float f;
|
||||
uint32_t i;
|
||||
} u = {f};
|
||||
return u.i;
|
||||
}
|
||||
|
||||
static inline float
|
||||
asfloat (uint32_t i)
|
||||
{
|
||||
union
|
||||
{
|
||||
uint32_t i;
|
||||
float f;
|
||||
} u = {i};
|
||||
return u.f;
|
||||
}
|
||||
|
||||
static inline uint64_t
|
||||
asuint64 (double f)
|
||||
{
|
||||
union
|
||||
{
|
||||
double f;
|
||||
uint64_t i;
|
||||
} u = {f};
|
||||
return u.i;
|
||||
}
|
||||
|
||||
static inline double
|
||||
asdouble (uint64_t i)
|
||||
{
|
||||
union
|
||||
{
|
||||
uint64_t i;
|
||||
double f;
|
||||
} u = {i};
|
||||
return u.f;
|
||||
}
|
||||
|
||||
#define NOINLINE __attribute__ ((noinline))
|
||||
|
||||
attribute_hidden float __math_oflowf (unsigned long);
|
||||
attribute_hidden float __math_uflowf (unsigned long);
|
||||
attribute_hidden float __math_may_uflowf (unsigned long);
|
||||
attribute_hidden float __math_divzerof (unsigned long);
|
||||
attribute_hidden float __math_invalidf (float);
|
||||
|
||||
/* Shared between expf, exp2f and powf. */
|
||||
#define EXP2F_TABLE_BITS 5
|
||||
#define EXP2F_POLY_ORDER 3
|
||||
extern 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 attribute_hidden;
|
||||
|
||||
#endif
|
||||
|
|
@ -0,0 +1,76 @@
|
|||
/* Single-precision math error handling.
|
||||
Copyright (C) 2017 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
|
||||
The GNU C Library is free software; you can redistribute it and/or
|
||||
modify it under the terms of the GNU Lesser General Public
|
||||
License as published by the Free Software Foundation; either
|
||||
version 2.1 of the License, or (at your option) any later version.
|
||||
|
||||
The GNU C Library is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
||||
Lesser General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU Lesser General Public
|
||||
License along with the GNU C Library; if not, see
|
||||
<http://www.gnu.org/licenses/>. */
|
||||
|
||||
#include "math_config.h"
|
||||
|
||||
#if WANT_ERRNO
|
||||
# include <errno.h>
|
||||
/* NOINLINE reduces code size. */
|
||||
NOINLINE static float
|
||||
with_errnof (float y, int e)
|
||||
{
|
||||
errno = e;
|
||||
return y;
|
||||
}
|
||||
#else
|
||||
# define with_errnof(x, e) (x)
|
||||
#endif
|
||||
|
||||
/* NOINLINE prevents fenv semantics breaking optimizations. */
|
||||
NOINLINE static float
|
||||
xflowf (unsigned long sign, float y)
|
||||
{
|
||||
y = (sign ? -y : y) * y;
|
||||
return with_errnof (y, ERANGE);
|
||||
}
|
||||
|
||||
attribute_hidden float
|
||||
__math_uflowf (unsigned long sign)
|
||||
{
|
||||
return xflowf (sign, 0x1p-95f);
|
||||
}
|
||||
|
||||
#if WANT_ERRNO_UFLOW
|
||||
/* Underflows to zero in some non-nearest rounding mode, setting errno
|
||||
is valid even if the result is non-zero, but in the subnormal range. */
|
||||
attribute_hidden float
|
||||
__math_may_uflowf (unsigned long sign)
|
||||
{
|
||||
return xflowf (sign, 0x1.4p-75f);
|
||||
}
|
||||
#endif
|
||||
|
||||
attribute_hidden float
|
||||
__math_oflowf (unsigned long sign)
|
||||
{
|
||||
return xflowf (sign, 0x1p97f);
|
||||
}
|
||||
|
||||
attribute_hidden float
|
||||
__math_divzerof (unsigned long sign)
|
||||
{
|
||||
float y = 0;
|
||||
return with_errnof ((sign ? -1 : 1) / y, ERANGE);
|
||||
}
|
||||
|
||||
attribute_hidden float
|
||||
__math_invalidf (float x)
|
||||
{
|
||||
float y = (x - x) / (x - x);
|
||||
return isnan (x) ? y : with_errnof (y, EDOM);
|
||||
}
|
||||
|
|
@ -1,351 +0,0 @@
|
|||
/* Accurate tables for exp2f().
|
||||
Copyright (C) 1998-2017 Free Software Foundation, Inc.
|
||||
This file is part of the GNU C Library.
|
||||
Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
|
||||
|
||||
The GNU C Library is free software; you can redistribute it and/or
|
||||
modify it under the terms of the GNU Lesser General Public
|
||||
License as published by the Free Software Foundation; either
|
||||
version 2.1 of the License, or (at your option) any later version.
|
||||
|
||||
The GNU C Library is distributed in the hope that it will be useful,
|
||||
but WITHOUT ANY WARRANTY; without even the implied warranty of
|
||||
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
|
||||
Lesser General Public License for more details.
|
||||
|
||||
You should have received a copy of the GNU Lesser General Public
|
||||
License along with the GNU C Library; if not, see
|
||||
<http://www.gnu.org/licenses/>. */
|
||||
|
||||
/* This table has the property that, for all integers -128 <= i <= 127,
|
||||
exp(i/256.0 + __exp2f_deltatable[i-128]) == __exp2f_atable[i+128] + r
|
||||
for some -2^-35 < r < 2^-35 (abs(r) < 2^-36 if i <= 0); and that
|
||||
__exp2f_deltatable[i+128] == t * 2^-30
|
||||
for integer t so that abs(t) <= 43447 * 2^0. */
|
||||
|
||||
#define W30 (9.31322575e-10)
|
||||
static const float __exp2f_deltatable[256] = {
|
||||
-810*W30, 283*W30, -1514*W30, 1304*W30,
|
||||
-1148*W30, -98*W30, -744*W30, -156*W30,
|
||||
-419*W30, -155*W30, 474*W30, 167*W30,
|
||||
-1984*W30, -826*W30, 692*W30, 781*W30,
|
||||
-578*W30, -411*W30, -129*W30, -1500*W30,
|
||||
654*W30, -141*W30, -816*W30, -53*W30,
|
||||
148*W30, 493*W30, -2214*W30, 760*W30,
|
||||
260*W30, 750*W30, -1300*W30, 1424*W30,
|
||||
-1445*W30, -339*W30, -680*W30, -349*W30,
|
||||
-922*W30, 531*W30, 193*W30, -2892*W30,
|
||||
290*W30, -2145*W30, -276*W30, 485*W30,
|
||||
-695*W30, 215*W30, -7093*W30, 412*W30,
|
||||
-4596*W30, 367*W30, 592*W30, -615*W30,
|
||||
-97*W30, -1066*W30, 972*W30, -226*W30,
|
||||
-625*W30, -374*W30, -5647*W30, -180*W30,
|
||||
20349*W30, -447*W30, 111*W30, -4164*W30,
|
||||
-87*W30, -21*W30, -251*W30, 66*W30,
|
||||
-517*W30, 2093*W30, -263*W30, 182*W30,
|
||||
-601*W30, 475*W30, -483*W30, -1251*W30,
|
||||
-373*W30, 1471*W30, -92*W30, -215*W30,
|
||||
-97*W30, -190*W30, 0*W30, -290*W30,
|
||||
-2647*W30, 1940*W30, -582*W30, 28*W30,
|
||||
833*W30, 1493*W30, 34*W30, 321*W30,
|
||||
3327*W30, -35*W30, 177*W30, -135*W30,
|
||||
-796*W30, -428*W30, 129*W30, 9332*W30,
|
||||
-12*W30, -69*W30, -1743*W30, 6508*W30,
|
||||
-60*W30, 359*W30, 43447*W30, 15*W30,
|
||||
-23*W30, -305*W30, -375*W30, -652*W30,
|
||||
667*W30, 269*W30, -1575*W30, 185*W30,
|
||||
-329*W30, 200*W30, 6002*W30, 163*W30,
|
||||
-647*W30, 19*W30, -603*W30, -755*W30,
|
||||
742*W30, -438*W30, 3587*W30, 2560*W30,
|
||||
0*W30, -520*W30, -241*W30, -299*W30,
|
||||
-1270*W30, -991*W30, -1138*W30, 255*W30,
|
||||
-1192*W30, 1722*W30, 1023*W30, 3700*W30,
|
||||
-1388*W30, -1551*W30, -2549*W30, 27*W30,
|
||||
282*W30, 673*W30, 113*W30, 1561*W30,
|
||||
72*W30, 873*W30, 87*W30, -395*W30,
|
||||
-433*W30, 629*W30, 3440*W30, -284*W30,
|
||||
-592*W30, -103*W30, -46*W30, -3844*W30,
|
||||
1712*W30, 303*W30, 1555*W30, -631*W30,
|
||||
-1400*W30, -961*W30, -854*W30, -276*W30,
|
||||
407*W30, 833*W30, -345*W30, -1501*W30,
|
||||
121*W30, -1581*W30, 400*W30, 150*W30,
|
||||
1224*W30, -139*W30, -563*W30, 879*W30,
|
||||
933*W30, 2939*W30, 788*W30, 211*W30,
|
||||
530*W30, -192*W30, 706*W30, -13347*W30,
|
||||
1065*W30, 3*W30, 111*W30, -208*W30,
|
||||
-360*W30, -532*W30, -291*W30, 483*W30,
|
||||
987*W30, -33*W30, -1373*W30, -166*W30,
|
||||
-1174*W30, -3955*W30, 1601*W30, -280*W30,
|
||||
1405*W30, 600*W30, -1659*W30, -23*W30,
|
||||
390*W30, 449*W30, 570*W30, -13143*W30,
|
||||
-9*W30, -1646*W30, 1201*W30, 294*W30,
|
||||
2181*W30, -1173*W30, 1388*W30, -4504*W30,
|
||||
190*W30, -2304*W30, 211*W30, 239*W30,
|
||||
48*W30, -817*W30, 1018*W30, 1828*W30,
|
||||
-663*W30, 1408*W30, 408*W30, -36*W30,
|
||||
1295*W30, -230*W30, 1341*W30, 9*W30,
|
||||
40*W30, 705*W30, 186*W30, 376*W30,
|
||||
557*W30, 5866*W30, 363*W30, -1558*W30,
|
||||
718*W30, 669*W30, 1369*W30, -2972*W30,
|
||||
-468*W30, -121*W30, -219*W30, 667*W30,
|
||||
29954*W30, 366*W30, 48*W30, -203*W30
|
||||
};
|
||||
|
||||
static const float __exp2f_atable[256] /* __attribute__((mode(SF))) */ = {
|
||||
0.707106411447, /* 0x0.b504ecfff */
|
||||
0.709024071690, /* 0x0.b58299fff */
|
||||
0.710945606239, /* 0x0.b60088000 */
|
||||
0.712874472142, /* 0x0.b67ef1000 */
|
||||
0.714806139464, /* 0x0.b6fd88fff */
|
||||
0.716744661340, /* 0x0.b77c94000 */
|
||||
0.718687653549, /* 0x0.b7fbea000 */
|
||||
0.720636486992, /* 0x0.b87ba1fff */
|
||||
0.722590208040, /* 0x0.b8fbabfff */
|
||||
0.724549472323, /* 0x0.b97c12fff */
|
||||
0.726514220228, /* 0x0.b9fcd5fff */
|
||||
0.728483855735, /* 0x0.ba7deb000 */
|
||||
0.730457961549, /* 0x0.baff4afff */
|
||||
0.732438981522, /* 0x0.bb811efff */
|
||||
0.734425544748, /* 0x0.bc0350000 */
|
||||
0.736416816713, /* 0x0.bc85d0000 */
|
||||
0.738412797450, /* 0x0.bd089efff */
|
||||
0.740414917465, /* 0x0.bd8bd4fff */
|
||||
0.742422521111, /* 0x0.be0f66fff */
|
||||
0.744434773914, /* 0x0.be9346fff */
|
||||
0.746454179287, /* 0x0.bf179f000 */
|
||||
0.748477637755, /* 0x0.bf9c3afff */
|
||||
0.750506639473, /* 0x0.c02133fff */
|
||||
0.752541840064, /* 0x0.c0a694fff */
|
||||
0.754582285889, /* 0x0.c12c4e000 */
|
||||
0.756628334525, /* 0x0.c1b265000 */
|
||||
0.758678436269, /* 0x0.c238bffff */
|
||||
0.760736882681, /* 0x0.c2bfa6fff */
|
||||
0.762799203401, /* 0x0.c346cf000 */
|
||||
0.764867603790, /* 0x0.c3ce5d000 */
|
||||
0.766940355298, /* 0x0.c45633fff */
|
||||
0.769021093841, /* 0x0.c4de90fff */
|
||||
0.771104693409, /* 0x0.c5671dfff */
|
||||
0.773195922364, /* 0x0.c5f02afff */
|
||||
0.775292098512, /* 0x0.c6798afff */
|
||||
0.777394294745, /* 0x0.c70350000 */
|
||||
0.779501736166, /* 0x0.c78d6d000 */
|
||||
0.781615912910, /* 0x0.c817fafff */
|
||||
0.783734917628, /* 0x0.c8a2d9fff */
|
||||
0.785858273516, /* 0x0.c92e02000 */
|
||||
0.787990570071, /* 0x0.c9b9c0000 */
|
||||
0.790125787245, /* 0x0.ca45aefff */
|
||||
0.792268991467, /* 0x0.cad223fff */
|
||||
0.794417440881, /* 0x0.cb5ef0fff */
|
||||
0.796570718287, /* 0x0.cbec0efff */
|
||||
0.798730909811, /* 0x0.cc79a0fff */
|
||||
0.800892710672, /* 0x0.cd074dfff */
|
||||
0.803068041795, /* 0x0.cd95ddfff */
|
||||
0.805242776881, /* 0x0.ce2464000 */
|
||||
0.807428598393, /* 0x0.ceb3a3fff */
|
||||
0.809617877002, /* 0x0.cf431dfff */
|
||||
0.811812341211, /* 0x0.cfd2eefff */
|
||||
0.814013659956, /* 0x0.d06333000 */
|
||||
0.816220164311, /* 0x0.d0f3ce000 */
|
||||
0.818434238424, /* 0x0.d184e7fff */
|
||||
0.820652604094, /* 0x0.d21649fff */
|
||||
0.822877407074, /* 0x0.d2a818000 */
|
||||
0.825108587751, /* 0x0.d33a51000 */
|
||||
0.827342867839, /* 0x0.d3ccbdfff */
|
||||
0.829588949684, /* 0x0.d45ff1000 */
|
||||
0.831849217401, /* 0x0.d4f411fff */
|
||||
0.834093391880, /* 0x0.d58724fff */
|
||||
0.836355149750, /* 0x0.d61b5f000 */
|
||||
0.838620424257, /* 0x0.d6afd3fff */
|
||||
0.840896368027, /* 0x0.d744fc000 */
|
||||
0.843176305293, /* 0x0.d7da66fff */
|
||||
0.845462262643, /* 0x0.d87037000 */
|
||||
0.847754716864, /* 0x0.d90673fff */
|
||||
0.850052893157, /* 0x0.d99d10fff */
|
||||
0.852359056469, /* 0x0.da3433fff */
|
||||
0.854668736446, /* 0x0.dacb91fff */
|
||||
0.856986224651, /* 0x0.db6373000 */
|
||||
0.859309315673, /* 0x0.dbfbb1fff */
|
||||
0.861639738080, /* 0x0.dc946bfff */
|
||||
0.863975346095, /* 0x0.dd2d7d000 */
|
||||
0.866317391394, /* 0x0.ddc6f9fff */
|
||||
0.868666708472, /* 0x0.de60f1000 */
|
||||
0.871022939695, /* 0x0.defb5c000 */
|
||||
0.873383641229, /* 0x0.df9611fff */
|
||||
0.875751554968, /* 0x0.e03141000 */
|
||||
0.878126025200, /* 0x0.e0ccde000 */
|
||||
0.880506813521, /* 0x0.e168e4fff */
|
||||
0.882894217966, /* 0x0.e2055afff */
|
||||
0.885287821299, /* 0x0.e2a239000 */
|
||||
0.887686729423, /* 0x0.e33f6ffff */
|
||||
0.890096127973, /* 0x0.e3dd56fff */
|
||||
0.892507970338, /* 0x0.e47b67000 */
|
||||
0.894928157336, /* 0x0.e51a03000 */
|
||||
0.897355020043, /* 0x0.e5b90efff */
|
||||
0.899788379682, /* 0x0.e65888000 */
|
||||
0.902227103705, /* 0x0.e6f85afff */
|
||||
0.904673457151, /* 0x0.e798ae000 */
|
||||
0.907128036008, /* 0x0.e8398afff */
|
||||
0.909585535528, /* 0x0.e8da99000 */
|
||||
0.912051796915, /* 0x0.e97c3a000 */
|
||||
0.914524436003, /* 0x0.ea1e46000 */
|
||||
0.917003571999, /* 0x0.eac0bf000 */
|
||||
0.919490039339, /* 0x0.eb63b2fff */
|
||||
0.921983361257, /* 0x0.ec071a000 */
|
||||
0.924488604054, /* 0x0.ecab48fff */
|
||||
0.926989555360, /* 0x0.ed4f30000 */
|
||||
0.929502844812, /* 0x0.edf3e6000 */
|
||||
0.932021975503, /* 0x0.ee98fdfff */
|
||||
0.934553921208, /* 0x0.ef3eecfff */
|
||||
0.937083780759, /* 0x0.efe4b8fff */
|
||||
0.939624726786, /* 0x0.f08b3f000 */
|
||||
0.942198514924, /* 0x0.f133ebfff */
|
||||
0.944726586343, /* 0x0.f1d99a000 */
|
||||
0.947287976728, /* 0x0.f28176fff */
|
||||
0.949856162070, /* 0x0.f329c5fff */
|
||||
0.952431440345, /* 0x0.f3d28bfff */
|
||||
0.955013573175, /* 0x0.f47bc5000 */
|
||||
0.957603693021, /* 0x0.f52584000 */
|
||||
0.960199773321, /* 0x0.f5cfa7000 */
|
||||
0.962801992906, /* 0x0.f67a31000 */
|
||||
0.965413510788, /* 0x0.f72556fff */
|
||||
0.968030691152, /* 0x0.f7d0dc000 */
|
||||
0.970655620084, /* 0x0.f87ce2fff */
|
||||
0.973290979849, /* 0x0.f92998fff */
|
||||
0.975926160805, /* 0x0.f9d64bfff */
|
||||
0.978571653370, /* 0x0.fa83ac000 */
|
||||
0.981225252139, /* 0x0.fb3193fff */
|
||||
0.983885228626, /* 0x0.fbdfe6fff */
|
||||
0.986552715296, /* 0x0.fc8eb7fff */
|
||||
0.989228487027, /* 0x0.fd3e14000 */
|
||||
0.991909801964, /* 0x0.fdedcd000 */
|
||||
0.994601726545, /* 0x0.fe9e38000 */
|
||||
0.997297704209, /* 0x0.ff4ee6fff */
|
||||
1.000000000000, /* 0x1.000000000 */
|
||||
1.002710938457, /* 0x1.00b1aa000 */
|
||||
1.005429744692, /* 0x1.0163d7ffe */
|
||||
1.008155703526, /* 0x1.02167dffe */
|
||||
1.010888457284, /* 0x1.02c995fff */
|
||||
1.013629436498, /* 0x1.037d38000 */
|
||||
1.016377568250, /* 0x1.043152000 */
|
||||
1.019134163841, /* 0x1.04e5f9ffe */
|
||||
1.021896362316, /* 0x1.059b00000 */
|
||||
1.024668931945, /* 0x1.0650b3ffe */
|
||||
1.027446627635, /* 0x1.0706be001 */
|
||||
1.030234098408, /* 0x1.07bd6bffe */
|
||||
1.033023953416, /* 0x1.087441ffe */
|
||||
1.035824656494, /* 0x1.092bce000 */
|
||||
1.038632392900, /* 0x1.09e3d0001 */
|
||||
1.041450142840, /* 0x1.0a9c79ffe */
|
||||
1.044273972530, /* 0x1.0b558a001 */
|
||||
1.047105550795, /* 0x1.0c0f1c001 */
|
||||
1.049944162390, /* 0x1.0cc924001 */
|
||||
1.052791833895, /* 0x1.0d83c4001 */
|
||||
1.055645227426, /* 0x1.0e3ec3fff */
|
||||
1.058507919326, /* 0x1.0efa60001 */
|
||||
1.061377286898, /* 0x1.0fb66bfff */
|
||||
1.064254641510, /* 0x1.1072fdffe */
|
||||
1.067140102389, /* 0x1.113018000 */
|
||||
1.070034146304, /* 0x1.11edc1fff */
|
||||
1.072937250162, /* 0x1.12ac04001 */
|
||||
1.075843691823, /* 0x1.136a7dfff */
|
||||
1.078760385496, /* 0x1.1429a3ffe */
|
||||
1.081685543070, /* 0x1.14e958000 */
|
||||
1.084618330005, /* 0x1.15a98c000 */
|
||||
1.087556362176, /* 0x1.166a18001 */
|
||||
1.090508937863, /* 0x1.172b98001 */
|
||||
1.093464612954, /* 0x1.17ed4bfff */
|
||||
1.096430182434, /* 0x1.18afa5ffe */
|
||||
1.099401354802, /* 0x1.19725e000 */
|
||||
1.102381587017, /* 0x1.1a35adfff */
|
||||
1.105370759965, /* 0x1.1af994000 */
|
||||
1.108367800686, /* 0x1.1bbdfdffe */
|
||||
1.111373305331, /* 0x1.1c82f6000 */
|
||||
1.114387035385, /* 0x1.1d4878001 */
|
||||
1.117408752440, /* 0x1.1e0e7ffff */
|
||||
1.120437502874, /* 0x1.1ed4fe000 */
|
||||
1.123474478729, /* 0x1.1f9c06000 */
|
||||
1.126521706601, /* 0x1.2063ba001 */
|
||||
1.129574775716, /* 0x1.212bd0001 */
|
||||
1.132638812065, /* 0x1.21f49e000 */
|
||||
1.135709524130, /* 0x1.22bddbffe */
|
||||
1.138789534565, /* 0x1.2387b5fff */
|
||||
1.141876101508, /* 0x1.2451fe000 */
|
||||
1.144971728301, /* 0x1.251cddffe */
|
||||
1.148077130296, /* 0x1.25e861ffe */
|
||||
1.151189923305, /* 0x1.26b462001 */
|
||||
1.154312610610, /* 0x1.278107ffe */
|
||||
1.157440662410, /* 0x1.284e08001 */
|
||||
1.160578370109, /* 0x1.291baa001 */
|
||||
1.163725256932, /* 0x1.29e9e6000 */
|
||||
1.166879892324, /* 0x1.2ab8a3ffe */
|
||||
1.170044302935, /* 0x1.2b8805fff */
|
||||
1.173205971694, /* 0x1.2c5739ffe */
|
||||
1.176397800428, /* 0x1.2d2867ffe */
|
||||
1.179586529747, /* 0x1.2df962001 */
|
||||
1.182784795737, /* 0x1.2ecafbffe */
|
||||
1.185991406414, /* 0x1.2f9d21ffe */
|
||||
1.189206838636, /* 0x1.306fdc001 */
|
||||
1.192430973067, /* 0x1.314328000 */
|
||||
1.195664167430, /* 0x1.32170c001 */
|
||||
1.198906540890, /* 0x1.32eb8a001 */
|
||||
1.202157497408, /* 0x1.33c098000 */
|
||||
1.205416083326, /* 0x1.349625fff */
|
||||
1.208683252332, /* 0x1.356c43fff */
|
||||
1.211961269402, /* 0x1.364318001 */
|
||||
1.215246438983, /* 0x1.371a64000 */
|
||||
1.218539118740, /* 0x1.37f22dffe */
|
||||
1.221847295770, /* 0x1.38cafc000 */
|
||||
1.225158572187, /* 0x1.39a3fdfff */
|
||||
1.228481650325, /* 0x1.3a7dc5ffe */
|
||||
1.231811761846, /* 0x1.3b5803fff */
|
||||
1.235149741144, /* 0x1.3c32c5ffe */
|
||||
1.238499879811, /* 0x1.3d0e53ffe */
|
||||
1.241858124726, /* 0x1.3dea69fff */
|
||||
1.245225191102, /* 0x1.3ec713fff */
|
||||
1.248601436624, /* 0x1.3fa458000 */
|
||||
1.251975655584, /* 0x1.40817a001 */
|
||||
1.255380749731, /* 0x1.4160a2001 */
|
||||
1.258783102010, /* 0x1.423f9bffe */
|
||||
1.262198328973, /* 0x1.431f6e000 */
|
||||
1.265619754780, /* 0x1.43ffa7fff */
|
||||
1.269052743928, /* 0x1.44e0a4001 */
|
||||
1.272490739830, /* 0x1.45c1f4000 */
|
||||
1.275942921659, /* 0x1.46a432001 */
|
||||
1.279397487615, /* 0x1.478697ffe */
|
||||
1.282870173427, /* 0x1.486a2dffe */
|
||||
1.286346316319, /* 0x1.494dfdffe */
|
||||
1.289836049094, /* 0x1.4a32b2001 */
|
||||
1.293333172770, /* 0x1.4b17e1ffe */
|
||||
1.296839594835, /* 0x1.4bfdadfff */
|
||||
1.300354957560, /* 0x1.4ce40fffe */
|
||||
1.303882122055, /* 0x1.4dcb38001 */
|
||||
1.307417988757, /* 0x1.4eb2f1ffe */
|
||||
1.310960650439, /* 0x1.4f9b1dfff */
|
||||
1.314516782746, /* 0x1.50842bfff */
|
||||
1.318079948424, /* 0x1.516daffff */
|
||||
1.321653246888, /* 0x1.5257de000 */
|
||||
1.325237751030, /* 0x1.5342c8001 */
|
||||
1.328829526907, /* 0x1.542e2c000 */
|
||||
1.332433700535, /* 0x1.551a5fffe */
|
||||
1.336045145966, /* 0x1.56070dffe */
|
||||
1.339667558645, /* 0x1.56f473ffe */
|
||||
1.343300342533, /* 0x1.57e287ffe */
|
||||
1.346941947961, /* 0x1.58d130001 */
|
||||
1.350594043714, /* 0x1.59c087ffe */
|
||||
1.354256033883, /* 0x1.5ab085fff */
|
||||
1.357932448365, /* 0x1.5ba175ffe */
|
||||
1.361609339707, /* 0x1.5c926dfff */
|
||||
1.365299344044, /* 0x1.5d8441ffe */
|
||||
1.369003057507, /* 0x1.5e76fc001 */
|
||||
1.372714757920, /* 0x1.5f6a3c000 */
|
||||
1.376437187179, /* 0x1.605e2fffe */
|
||||
1.380165219333, /* 0x1.615282001 */
|
||||
1.383909463864, /* 0x1.6247e3ffe */
|
||||
1.387661933907, /* 0x1.633dd0000 */
|
||||
1.391424179060, /* 0x1.64345fffe */
|
||||
1.395197510706, /* 0x1.652ba9fff */
|
||||
1.399006724329, /* 0x1.66254dffe */
|
||||
1.402773022651, /* 0x1.671c22000 */
|
||||
1.406576037403, /* 0x1.68155dfff */
|
||||
1.410389423392, /* 0x1.690f48001 */
|
||||
};
|
||||
|
|
@ -0,0 +1 @@
|
|||
/* Not needed. */
|
||||
|
|
@ -0,0 +1 @@
|
|||
/* Not needed. */
|
||||
Loading…
Reference in New Issue