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lib: Add libm_dbl, a double-precision math library, from musl-1.1.16.

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32
lib/libm_dbl/README Normal file
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This directory contains source code for the standard double-precision math
functions.
The files lgamma.c, log10.c and tanh.c are too small to have a meaningful
copyright or license.
The rest of the files in this directory are copied from the musl library,
v1.1.16, and, unless otherwise stated in the individual file, have the
following copyright and MIT license:
----------------------------------------------------------------------
Copyright © 2005-2014 Rich Felker, et al.
Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
----------------------------------------------------------------------

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lib/libm_dbl/__cos.c Normal file
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/* origin: FreeBSD /usr/src/lib/msun/src/k_cos.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
* __cos( x, y )
* kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
*
* Algorithm
* 1. Since cos(-x) = cos(x), we need only to consider positive x.
* 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
* 3. cos(x) is approximated by a polynomial of degree 14 on
* [0,pi/4]
* 4 14
* cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
* where the remez error is
*
* | 2 4 6 8 10 12 14 | -58
* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
* | |
*
* 4 6 8 10 12 14
* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
* cos(x) ~ 1 - x*x/2 + r
* since cos(x+y) ~ cos(x) - sin(x)*y
* ~ cos(x) - x*y,
* a correction term is necessary in cos(x) and hence
* cos(x+y) = 1 - (x*x/2 - (r - x*y))
* For better accuracy, rearrange to
* cos(x+y) ~ w + (tmp + (r-x*y))
* where w = 1 - x*x/2 and tmp is a tiny correction term
* (1 - x*x/2 == w + tmp exactly in infinite precision).
* The exactness of w + tmp in infinite precision depends on w
* and tmp having the same precision as x. If they have extra
* precision due to compiler bugs, then the extra precision is
* only good provided it is retained in all terms of the final
* expression for cos(). Retention happens in all cases tested
* under FreeBSD, so don't pessimize things by forcibly clipping
* any extra precision in w.
*/
#include "libm.h"
static const double
C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
double __cos(double x, double y)
{
double_t hz,z,r,w;
z = x*x;
w = z*z;
r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
hz = 0.5*z;
w = 1.0-hz;
return w + (((1.0-w)-hz) + (z*r-x*y));
}

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#include "libm.h"
/* k is such that k*ln2 has minimal relative error and x - kln2 > log(DBL_MIN) */
static const int k = 2043;
static const double kln2 = 0x1.62066151add8bp+10;
/* exp(x)/2 for x >= log(DBL_MAX), slightly better than 0.5*exp(x/2)*exp(x/2) */
double __expo2(double x)
{
double scale;
/* note that k is odd and scale*scale overflows */
INSERT_WORDS(scale, (uint32_t)(0x3ff + k/2) << 20, 0);
/* exp(x - k ln2) * 2**(k-1) */
return exp(x - kln2) * scale * scale;
}

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#include <math.h>
#include <stdint.h>
int __fpclassifyd(double x)
{
union {double f; uint64_t i;} u = {x};
int e = u.i>>52 & 0x7ff;
if (!e) return u.i<<1 ? FP_SUBNORMAL : FP_ZERO;
if (e==0x7ff) return u.i<<12 ? FP_NAN : FP_INFINITE;
return FP_NORMAL;
}

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lib/libm_dbl/__rem_pio2.c Normal file
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/* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
* Optimized by Bruce D. Evans.
*/
/* __rem_pio2(x,y)
*
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __rem_pio2_large() for large x
*/
#include "libm.h"
#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
#define EPS DBL_EPSILON
#elif FLT_EVAL_METHOD==2
#define EPS LDBL_EPSILON
#endif
/*
* invpio2: 53 bits of 2/pi
* pio2_1: first 33 bit of pi/2
* pio2_1t: pi/2 - pio2_1
* pio2_2: second 33 bit of pi/2
* pio2_2t: pi/2 - (pio2_1+pio2_2)
* pio2_3: third 33 bit of pi/2
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
static const double
toint = 1.5/EPS,
invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
/* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
int __rem_pio2(double x, double *y)
{
union {double f; uint64_t i;} u = {x};
double_t z,w,t,r,fn;
double tx[3],ty[2];
uint32_t ix;
int sign, n, ex, ey, i;
sign = u.i>>63;
ix = u.i>>32 & 0x7fffffff;
if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
goto medium; /* cancellation -- use medium case */
if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
if (!sign) {
z = x - pio2_1; /* one round good to 85 bits */
y[0] = z - pio2_1t;
y[1] = (z-y[0]) - pio2_1t;
return 1;
} else {
z = x + pio2_1;
y[0] = z + pio2_1t;
y[1] = (z-y[0]) + pio2_1t;
return -1;
}
} else {
if (!sign) {
z = x - 2*pio2_1;
y[0] = z - 2*pio2_1t;
y[1] = (z-y[0]) - 2*pio2_1t;
return 2;
} else {
z = x + 2*pio2_1;
y[0] = z + 2*pio2_1t;
y[1] = (z-y[0]) + 2*pio2_1t;
return -2;
}
}
}
if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
goto medium;
if (!sign) {
z = x - 3*pio2_1;
y[0] = z - 3*pio2_1t;
y[1] = (z-y[0]) - 3*pio2_1t;
return 3;
} else {
z = x + 3*pio2_1;
y[0] = z + 3*pio2_1t;
y[1] = (z-y[0]) + 3*pio2_1t;
return -3;
}
} else {
if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
goto medium;
if (!sign) {
z = x - 4*pio2_1;
y[0] = z - 4*pio2_1t;
y[1] = (z-y[0]) - 4*pio2_1t;
return 4;
} else {
z = x + 4*pio2_1;
y[0] = z + 4*pio2_1t;
y[1] = (z-y[0]) + 4*pio2_1t;
return -4;
}
}
}
if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
medium:
/* rint(x/(pi/2)), Assume round-to-nearest. */
fn = (double_t)x*invpio2 + toint - toint;
n = (int32_t)fn;
r = x - fn*pio2_1;
w = fn*pio2_1t; /* 1st round, good to 85 bits */
y[0] = r - w;
u.f = y[0];
ey = u.i>>52 & 0x7ff;
ex = ix>>20;
if (ex - ey > 16) { /* 2nd round, good to 118 bits */
t = r;
w = fn*pio2_2;
r = t - w;
w = fn*pio2_2t - ((t-r)-w);
y[0] = r - w;
u.f = y[0];
ey = u.i>>52 & 0x7ff;
if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
t = r;
w = fn*pio2_3;
r = t - w;
w = fn*pio2_3t - ((t-r)-w);
y[0] = r - w;
}
}
y[1] = (r - y[0]) - w;
return n;
}
/*
* all other (large) arguments
*/
if (ix >= 0x7ff00000) { /* x is inf or NaN */
y[0] = y[1] = x - x;
return 0;
}
/* set z = scalbn(|x|,-ilogb(x)+23) */
u.f = x;
u.i &= (uint64_t)-1>>12;
u.i |= (uint64_t)(0x3ff + 23)<<52;
z = u.f;
for (i=0; i < 2; i++) {
tx[i] = (double)(int32_t)z;
z = (z-tx[i])*0x1p24;
}
tx[i] = z;
/* skip zero terms, first term is non-zero */
while (tx[i] == 0.0)
i--;
n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
if (sign) {
y[0] = -ty[0];
y[1] = -ty[1];
return -n;
}
y[0] = ty[0];
y[1] = ty[1];
return n;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/k_rem_pio2.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
* __rem_pio2_large(x,y,e0,nx,prec)
* double x[],y[]; int e0,nx,prec;
*
* __rem_pio2_large return the last three digits of N with
* y = x - N*pi/2
* so that |y| < pi/2.
*
* The method is to compute the integer (mod 8) and fraction parts of
* (2/pi)*x without doing the full multiplication. In general we
* skip the part of the product that are known to be a huge integer (
* more accurately, = 0 mod 8 ). Thus the number of operations are
* independent of the exponent of the input.
*
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
*
* Input parameters:
* x[] The input value (must be positive) is broken into nx
* pieces of 24-bit integers in double precision format.
* x[i] will be the i-th 24 bit of x. The scaled exponent
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
* match x's up to 24 bits.
*
* Example of breaking a double positive z into x[0]+x[1]+x[2]:
* e0 = ilogb(z)-23
* z = scalbn(z,-e0)
* for i = 0,1,2
* x[i] = floor(z)
* z = (z-x[i])*2**24
*
*
* y[] ouput result in an array of double precision numbers.
* The dimension of y[] is:
* 24-bit precision 1
* 53-bit precision 2
* 64-bit precision 2
* 113-bit precision 3
* The actual value is the sum of them. Thus for 113-bit
* precison, one may have to do something like:
*
* long double t,w,r_head, r_tail;
* t = (long double)y[2] + (long double)y[1];
* w = (long double)y[0];
* r_head = t+w;
* r_tail = w - (r_head - t);
*
* e0 The exponent of x[0]. Must be <= 16360 or you need to
* expand the ipio2 table.
*
* nx dimension of x[]
*
* prec an integer indicating the precision:
* 0 24 bits (single)
* 1 53 bits (double)
* 2 64 bits (extended)
* 3 113 bits (quad)
*
* External function:
* double scalbn(), floor();
*
*
* Here is the description of some local variables:
*
* jk jk+1 is the initial number of terms of ipio2[] needed
* in the computation. The minimum and recommended value
* for jk is 3,4,4,6 for single, double, extended, and quad.
* jk+1 must be 2 larger than you might expect so that our
* recomputation test works. (Up to 24 bits in the integer
* part (the 24 bits of it that we compute) and 23 bits in
* the fraction part may be lost to cancelation before we
* recompute.)
*
* jz local integer variable indicating the number of
* terms of ipio2[] used.
*
* jx nx - 1
*
* jv index for pointing to the suitable ipio2[] for the
* computation. In general, we want
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
* is an integer. Thus
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
* Hence jv = max(0,(e0-3)/24).
*
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
*
* q[] double array with integral value, representing the
* 24-bits chunk of the product of x and 2/pi.
*
* q0 the corresponding exponent of q[0]. Note that the
* exponent for q[i] would be q0-24*i.
*
* PIo2[] double precision array, obtained by cutting pi/2
* into 24 bits chunks.
*
* f[] ipio2[] in floating point
*
* iq[] integer array by breaking up q[] in 24-bits chunk.
*
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
*
* ih integer. If >0 it indicates q[] is >= 0.5, hence
* it also indicates the *sign* of the result.
*
*/
/*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include "libm.h"
static const int init_jk[] = {3,4,4,6}; /* initial value for jk */
/*
* Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
*
* integer array, contains the (24*i)-th to (24*i+23)-th
* bit of 2/pi after binary point. The corresponding
* floating value is
*
* ipio2[i] * 2^(-24(i+1)).
*
* NB: This table must have at least (e0-3)/24 + jk terms.
* For quad precision (e0 <= 16360, jk = 6), this is 686.
*/
static const int32_t ipio2[] = {
0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
#if LDBL_MAX_EXP > 1024
0x47C419, 0xC367CD, 0xDCE809, 0x2A8359, 0xC4768B, 0x961CA6,
0xDDAF44, 0xD15719, 0x053EA5, 0xFF0705, 0x3F7E33, 0xE832C2,
0xDE4F98, 0x327DBB, 0xC33D26, 0xEF6B1E, 0x5EF89F, 0x3A1F35,
0xCAF27F, 0x1D87F1, 0x21907C, 0x7C246A, 0xFA6ED5, 0x772D30,
0x433B15, 0xC614B5, 0x9D19C3, 0xC2C4AD, 0x414D2C, 0x5D000C,
0x467D86, 0x2D71E3, 0x9AC69B, 0x006233, 0x7CD2B4, 0x97A7B4,
0xD55537, 0xF63ED7, 0x1810A3, 0xFC764D, 0x2A9D64, 0xABD770,
0xF87C63, 0x57B07A, 0xE71517, 0x5649C0, 0xD9D63B, 0x3884A7,
0xCB2324, 0x778AD6, 0x23545A, 0xB91F00, 0x1B0AF1, 0xDFCE19,
0xFF319F, 0x6A1E66, 0x615799, 0x47FBAC, 0xD87F7E, 0xB76522,
0x89E832, 0x60BFE6, 0xCDC4EF, 0x09366C, 0xD43F5D, 0xD7DE16,
0xDE3B58, 0x929BDE, 0x2822D2, 0xE88628, 0x4D58E2, 0x32CAC6,
0x16E308, 0xCB7DE0, 0x50C017, 0xA71DF3, 0x5BE018, 0x34132E,
0x621283, 0x014883, 0x5B8EF5, 0x7FB0AD, 0xF2E91E, 0x434A48,
0xD36710, 0xD8DDAA, 0x425FAE, 0xCE616A, 0xA4280A, 0xB499D3,
0xF2A606, 0x7F775C, 0x83C2A3, 0x883C61, 0x78738A, 0x5A8CAF,
0xBDD76F, 0x63A62D, 0xCBBFF4, 0xEF818D, 0x67C126, 0x45CA55,
0x36D9CA, 0xD2A828, 0x8D61C2, 0x77C912, 0x142604, 0x9B4612,
0xC459C4, 0x44C5C8, 0x91B24D, 0xF31700, 0xAD43D4, 0xE54929,
0x10D5FD, 0xFCBE00, 0xCC941E, 0xEECE70, 0xF53E13, 0x80F1EC,
0xC3E7B3, 0x28F8C7, 0x940593, 0x3E71C1, 0xB3092E, 0xF3450B,
0x9C1288, 0x7B20AB, 0x9FB52E, 0xC29247, 0x2F327B, 0x6D550C,
0x90A772, 0x1FE76B, 0x96CB31, 0x4A1679, 0xE27941, 0x89DFF4,
0x9794E8, 0x84E6E2, 0x973199, 0x6BED88, 0x365F5F, 0x0EFDBB,
0xB49A48, 0x6CA467, 0x427271, 0x325D8D, 0xB8159F, 0x09E5BC,
0x25318D, 0x3974F7, 0x1C0530, 0x010C0D, 0x68084B, 0x58EE2C,
0x90AA47, 0x02E774, 0x24D6BD, 0xA67DF7, 0x72486E, 0xEF169F,
0xA6948E, 0xF691B4, 0x5153D1, 0xF20ACF, 0x339820, 0x7E4BF5,
0x6863B2, 0x5F3EDD, 0x035D40, 0x7F8985, 0x295255, 0xC06437,
0x10D86D, 0x324832, 0x754C5B, 0xD4714E, 0x6E5445, 0xC1090B,
0x69F52A, 0xD56614, 0x9D0727, 0x50045D, 0xDB3BB4, 0xC576EA,
0x17F987, 0x7D6B49, 0xBA271D, 0x296996, 0xACCCC6, 0x5414AD,
0x6AE290, 0x89D988, 0x50722C, 0xBEA404, 0x940777, 0x7030F3,
0x27FC00, 0xA871EA, 0x49C266, 0x3DE064, 0x83DD97, 0x973FA3,
0xFD9443, 0x8C860D, 0xDE4131, 0x9D3992, 0x8C70DD, 0xE7B717,
0x3BDF08, 0x2B3715, 0xA0805C, 0x93805A, 0x921110, 0xD8E80F,
0xAF806C, 0x4BFFDB, 0x0F9038, 0x761859, 0x15A562, 0xBBCB61,
0xB989C7, 0xBD4010, 0x04F2D2, 0x277549, 0xF6B6EB, 0xBB22DB,
0xAA140A, 0x2F2689, 0x768364, 0x333B09, 0x1A940E, 0xAA3A51,
0xC2A31D, 0xAEEDAF, 0x12265C, 0x4DC26D, 0x9C7A2D, 0x9756C0,
0x833F03, 0xF6F009, 0x8C402B, 0x99316D, 0x07B439, 0x15200C,
0x5BC3D8, 0xC492F5, 0x4BADC6, 0xA5CA4E, 0xCD37A7, 0x36A9E6,
0x9492AB, 0x6842DD, 0xDE6319, 0xEF8C76, 0x528B68, 0x37DBFC,
0xABA1AE, 0x3115DF, 0xA1AE00, 0xDAFB0C, 0x664D64, 0xB705ED,
0x306529, 0xBF5657, 0x3AFF47, 0xB9F96A, 0xF3BE75, 0xDF9328,
0x3080AB, 0xF68C66, 0x15CB04, 0x0622FA, 0x1DE4D9, 0xA4B33D,
0x8F1B57, 0x09CD36, 0xE9424E, 0xA4BE13, 0xB52333, 0x1AAAF0,
0xA8654F, 0xA5C1D2, 0x0F3F0B, 0xCD785B, 0x76F923, 0x048B7B,
0x721789, 0x53A6C6, 0xE26E6F, 0x00EBEF, 0x584A9B, 0xB7DAC4,
0xBA66AA, 0xCFCF76, 0x1D02D1, 0x2DF1B1, 0xC1998C, 0x77ADC3,
0xDA4886, 0xA05DF7, 0xF480C6, 0x2FF0AC, 0x9AECDD, 0xBC5C3F,
0x6DDED0, 0x1FC790, 0xB6DB2A, 0x3A25A3, 0x9AAF00, 0x9353AD,
0x0457B6, 0xB42D29, 0x7E804B, 0xA707DA, 0x0EAA76, 0xA1597B,
0x2A1216, 0x2DB7DC, 0xFDE5FA, 0xFEDB89, 0xFDBE89, 0x6C76E4,
0xFCA906, 0x70803E, 0x156E85, 0xFF87FD, 0x073E28, 0x336761,
0x86182A, 0xEABD4D, 0xAFE7B3, 0x6E6D8F, 0x396795, 0x5BBF31,
0x48D784, 0x16DF30, 0x432DC7, 0x356125, 0xCE70C9, 0xB8CB30,
0xFD6CBF, 0xA200A4, 0xE46C05, 0xA0DD5A, 0x476F21, 0xD21262,
0x845CB9, 0x496170, 0xE0566B, 0x015299, 0x375550, 0xB7D51E,
0xC4F133, 0x5F6E13, 0xE4305D, 0xA92E85, 0xC3B21D, 0x3632A1,
0xA4B708, 0xD4B1EA, 0x21F716, 0xE4698F, 0x77FF27, 0x80030C,
0x2D408D, 0xA0CD4F, 0x99A520, 0xD3A2B3, 0x0A5D2F, 0x42F9B4,
0xCBDA11, 0xD0BE7D, 0xC1DB9B, 0xBD17AB, 0x81A2CA, 0x5C6A08,
0x17552E, 0x550027, 0xF0147F, 0x8607E1, 0x640B14, 0x8D4196,
0xDEBE87, 0x2AFDDA, 0xB6256B, 0x34897B, 0xFEF305, 0x9EBFB9,
0x4F6A68, 0xA82A4A, 0x5AC44F, 0xBCF82D, 0x985AD7, 0x95C7F4,
0x8D4D0D, 0xA63A20, 0x5F57A4, 0xB13F14, 0x953880, 0x0120CC,
0x86DD71, 0xB6DEC9, 0xF560BF, 0x11654D, 0x6B0701, 0xACB08C,
0xD0C0B2, 0x485551, 0x0EFB1E, 0xC37295, 0x3B06A3, 0x3540C0,
0x7BDC06, 0xCC45E0, 0xFA294E, 0xC8CAD6, 0x41F3E8, 0xDE647C,
0xD8649B, 0x31BED9, 0xC397A4, 0xD45877, 0xC5E369, 0x13DAF0,
0x3C3ABA, 0x461846, 0x5F7555, 0xF5BDD2, 0xC6926E, 0x5D2EAC,
0xED440E, 0x423E1C, 0x87C461, 0xE9FD29, 0xF3D6E7, 0xCA7C22,
0x35916F, 0xC5E008, 0x8DD7FF, 0xE26A6E, 0xC6FDB0, 0xC10893,
0x745D7C, 0xB2AD6B, 0x9D6ECD, 0x7B723E, 0x6A11C6, 0xA9CFF7,
0xDF7329, 0xBAC9B5, 0x5100B7, 0x0DB2E2, 0x24BA74, 0x607DE5,
0x8AD874, 0x2C150D, 0x0C1881, 0x94667E, 0x162901, 0x767A9F,
0xBEFDFD, 0xEF4556, 0x367ED9, 0x13D9EC, 0xB9BA8B, 0xFC97C4,
0x27A831, 0xC36EF1, 0x36C594, 0x56A8D8, 0xB5A8B4, 0x0ECCCF,
0x2D8912, 0x34576F, 0x89562C, 0xE3CE99, 0xB920D6, 0xAA5E6B,
0x9C2A3E, 0xCC5F11, 0x4A0BFD, 0xFBF4E1, 0x6D3B8E, 0x2C86E2,
0x84D4E9, 0xA9B4FC, 0xD1EEEF, 0xC9352E, 0x61392F, 0x442138,
0xC8D91B, 0x0AFC81, 0x6A4AFB, 0xD81C2F, 0x84B453, 0x8C994E,
0xCC2254, 0xDC552A, 0xD6C6C0, 0x96190B, 0xB8701A, 0x649569,
0x605A26, 0xEE523F, 0x0F117F, 0x11B5F4, 0xF5CBFC, 0x2DBC34,
0xEEBC34, 0xCC5DE8, 0x605EDD, 0x9B8E67, 0xEF3392, 0xB817C9,
0x9B5861, 0xBC57E1, 0xC68351, 0x103ED8, 0x4871DD, 0xDD1C2D,
0xA118AF, 0x462C21, 0xD7F359, 0x987AD9, 0xC0549E, 0xFA864F,
0xFC0656, 0xAE79E5, 0x362289, 0x22AD38, 0xDC9367, 0xAAE855,
0x382682, 0x9BE7CA, 0xA40D51, 0xB13399, 0x0ED7A9, 0x480569,
0xF0B265, 0xA7887F, 0x974C88, 0x36D1F9, 0xB39221, 0x4A827B,
0x21CF98, 0xDC9F40, 0x5547DC, 0x3A74E1, 0x42EB67, 0xDF9DFE,
0x5FD45E, 0xA4677B, 0x7AACBA, 0xA2F655, 0x23882B, 0x55BA41,
0x086E59, 0x862A21, 0x834739, 0xE6E389, 0xD49EE5, 0x40FB49,
0xE956FF, 0xCA0F1C, 0x8A59C5, 0x2BFA94, 0xC5C1D3, 0xCFC50F,
0xAE5ADB, 0x86C547, 0x624385, 0x3B8621, 0x94792C, 0x876110,
0x7B4C2A, 0x1A2C80, 0x12BF43, 0x902688, 0x893C78, 0xE4C4A8,
0x7BDBE5, 0xC23AC4, 0xEAF426, 0x8A67F7, 0xBF920D, 0x2BA365,
0xB1933D, 0x0B7CBD, 0xDC51A4, 0x63DD27, 0xDDE169, 0x19949A,
0x9529A8, 0x28CE68, 0xB4ED09, 0x209F44, 0xCA984E, 0x638270,
0x237C7E, 0x32B90F, 0x8EF5A7, 0xE75614, 0x08F121, 0x2A9DB5,
0x4D7E6F, 0x5119A5, 0xABF9B5, 0xD6DF82, 0x61DD96, 0x023616,
0x9F3AC4, 0xA1A283, 0x6DED72, 0x7A8D39, 0xA9B882, 0x5C326B,
0x5B2746, 0xED3400, 0x7700D2, 0x55F4FC, 0x4D5901, 0x8071E0,
#endif
};
static const double PIo2[] = {
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
};
int __rem_pio2_large(double *x, double *y, int e0, int nx, int prec)
{
int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
double z,fw,f[20],fq[20],q[20];
/* initialize jk*/
jk = init_jk[prec];
jp = jk;
/* determine jx,jv,q0, note that 3>q0 */
jx = nx-1;
jv = (e0-3)/24; if(jv<0) jv=0;
q0 = e0-24*(jv+1);
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
j = jv-jx; m = jx+jk;
for (i=0; i<=m; i++,j++)
f[i] = j<0 ? 0.0 : (double)ipio2[j];
/* compute q[0],q[1],...q[jk] */
for (i=0; i<=jk; i++) {
for (j=0,fw=0.0; j<=jx; j++)
fw += x[j]*f[jx+i-j];
q[i] = fw;
}
jz = jk;
recompute:
/* distill q[] into iq[] reversingly */
for (i=0,j=jz,z=q[jz]; j>0; i++,j--) {
fw = (double)(int32_t)(0x1p-24*z);
iq[i] = (int32_t)(z - 0x1p24*fw);
z = q[j-1]+fw;
}
/* compute n */
z = scalbn(z,q0); /* actual value of z */
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
n = (int32_t)z;
z -= (double)n;
ih = 0;
if (q0 > 0) { /* need iq[jz-1] to determine n */
i = iq[jz-1]>>(24-q0); n += i;
iq[jz-1] -= i<<(24-q0);
ih = iq[jz-1]>>(23-q0);
}
else if (q0 == 0) ih = iq[jz-1]>>23;
else if (z >= 0.5) ih = 2;
if (ih > 0) { /* q > 0.5 */
n += 1; carry = 0;
for (i=0; i<jz; i++) { /* compute 1-q */
j = iq[i];
if (carry == 0) {
if (j != 0) {
carry = 1;
iq[i] = 0x1000000 - j;
}
} else
iq[i] = 0xffffff - j;
}
if (q0 > 0) { /* rare case: chance is 1 in 12 */
switch(q0) {
case 1:
iq[jz-1] &= 0x7fffff; break;
case 2:
iq[jz-1] &= 0x3fffff; break;
}
}
if (ih == 2) {
z = 1.0 - z;
if (carry != 0)
z -= scalbn(1.0,q0);
}
}
/* check if recomputation is needed */
if (z == 0.0) {
j = 0;
for (i=jz-1; i>=jk; i--) j |= iq[i];
if (j == 0) { /* need recomputation */
for (k=1; iq[jk-k]==0; k++); /* k = no. of terms needed */
for (i=jz+1; i<=jz+k; i++) { /* add q[jz+1] to q[jz+k] */
f[jx+i] = (double)ipio2[jv+i];
for (j=0,fw=0.0; j<=jx; j++)
fw += x[j]*f[jx+i-j];
q[i] = fw;
}
jz += k;
goto recompute;
}
}
/* chop off zero terms */
if (z == 0.0) {
jz -= 1;
q0 -= 24;
while (iq[jz] == 0) {
jz--;
q0 -= 24;
}
} else { /* break z into 24-bit if necessary */
z = scalbn(z,-q0);
if (z >= 0x1p24) {
fw = (double)(int32_t)(0x1p-24*z);
iq[jz] = (int32_t)(z - 0x1p24*fw);
jz += 1;
q0 += 24;
iq[jz] = (int32_t)fw;
} else
iq[jz] = (int32_t)z;
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbn(1.0,q0);
for (i=jz; i>=0; i--) {
q[i] = fw*(double)iq[i];
fw *= 0x1p-24;
}
/* compute PIo2[0,...,jp]*q[jz,...,0] */
for(i=jz; i>=0; i--) {
for (fw=0.0,k=0; k<=jp && k<=jz-i; k++)
fw += PIo2[k]*q[i+k];
fq[jz-i] = fw;
}
/* compress fq[] into y[] */
switch(prec) {
case 0:
fw = 0.0;
for (i=jz; i>=0; i--)
fw += fq[i];
y[0] = ih==0 ? fw : -fw;
break;
case 1:
case 2:
fw = 0.0;
for (i=jz; i>=0; i--)
fw += fq[i];
// TODO: drop excess precision here once double_t is used
fw = (double)fw;
y[0] = ih==0 ? fw : -fw;
fw = fq[0]-fw;
for (i=1; i<=jz; i++)
fw += fq[i];
y[1] = ih==0 ? fw : -fw;
break;
case 3: /* painful */
for (i=jz; i>0; i--) {
fw = fq[i-1]+fq[i];
fq[i] += fq[i-1]-fw;
fq[i-1] = fw;
}
for (i=jz; i>1; i--) {
fw = fq[i-1]+fq[i];
fq[i] += fq[i-1]-fw;
fq[i-1] = fw;
}
for (fw=0.0,i=jz; i>=2; i--)
fw += fq[i];
if (ih==0) {
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
} else {
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
}
}
return n&7;
}

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#include "libm.h"
int __signbitd(double x)
{
union {
double d;
uint64_t i;
} y = { x };
return y.i>>63;
}

64
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/* origin: FreeBSD /usr/src/lib/msun/src/k_sin.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __sin( x, y, iy)
* kernel sin function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
*
* Algorithm
* 1. Since sin(-x) = -sin(x), we need only to consider positive x.
* 2. Callers must return sin(-0) = -0 without calling here since our
* odd polynomial is not evaluated in a way that preserves -0.
* Callers may do the optimization sin(x) ~ x for tiny x.
* 3. sin(x) is approximated by a polynomial of degree 13 on
* [0,pi/4]
* 3 13
* sin(x) ~ x + S1*x + ... + S6*x
* where
*
* |sin(x) 2 4 6 8 10 12 | -58
* |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2
* | x |
*
* 4. sin(x+y) = sin(x) + sin'(x')*y
* ~ sin(x) + (1-x*x/2)*y
* For better accuracy, let
* 3 2 2 2 2
* r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6))))
* then 3 2
* sin(x) = x + (S1*x + (x *(r-y/2)+y))
*/
#include "libm.h"
static const double
S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */
S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */
S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */
S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */
S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */
S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */
double __sin(double x, double y, int iy)
{
double_t z,r,v,w;
z = x*x;
w = z*z;
r = S2 + z*(S3 + z*S4) + z*w*(S5 + z*S6);
v = z*x;
if (iy == 0)
return x + v*(S1 + z*r);
else
return x - ((z*(0.5*y - v*r) - y) - v*S1);
}

110
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/* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */
/*
* ====================================================
* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __tan( x, y, k )
* kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854
* Input x is assumed to be bounded by ~pi/4 in magnitude.
* Input y is the tail of x.
* Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is returned.
*
* Algorithm
* 1. Since tan(-x) = -tan(x), we need only to consider positive x.
* 2. Callers must return tan(-0) = -0 without calling here since our
* odd polynomial is not evaluated in a way that preserves -0.
* Callers may do the optimization tan(x) ~ x for tiny x.
* 3. tan(x) is approximated by a odd polynomial of degree 27 on
* [0,0.67434]
* 3 27
* tan(x) ~ x + T1*x + ... + T13*x
* where
*
* |tan(x) 2 4 26 | -59.2
* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
* | x |
*
* Note: tan(x+y) = tan(x) + tan'(x)*y
* ~ tan(x) + (1+x*x)*y
* Therefore, for better accuracy in computing tan(x+y), let
* 3 2 2 2 2
* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
* then
* 3 2
* tan(x+y) = x + (T1*x + (x *(r+y)+y))
*
* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
*/
#include "libm.h"
static const double T[] = {
3.33333333333334091986e-01, /* 3FD55555, 55555563 */
1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
},
pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
double __tan(double x, double y, int odd)
{
double_t z, r, v, w, s, a;
double w0, a0;
uint32_t hx;
int big, sign;
GET_HIGH_WORD(hx,x);
big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */
if (big) {
sign = hx>>31;
if (sign) {
x = -x;
y = -y;
}
x = (pio4 - x) + (pio4lo - y);
y = 0.0;
}
z = x * x;
w = z * z;
/*
* Break x^5*(T[1]+x^2*T[2]+...) into
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
*/
r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11]))));
v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12])))));
s = z * x;
r = y + z*(s*(r + v) + y) + s*T[0];
w = x + r;
if (big) {
s = 1 - 2*odd;
v = s - 2.0 * (x + (r - w*w/(w + s)));
return sign ? -v : v;
}
if (!odd)
return w;
/* -1.0/(x+r) has up to 2ulp error, so compute it accurately */
w0 = w;
SET_LOW_WORD(w0, 0);
v = r - (w0 - x); /* w0+v = r+x */
a0 = a = -1.0 / w;
SET_LOW_WORD(a0, 0);
return a0 + a*(1.0 + a0*w0 + a0*v);
}

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/* origin: FreeBSD /usr/src/lib/msun/src/e_acos.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* acos(x)
* Method :
* acos(x) = pi/2 - asin(x)
* acos(-x) = pi/2 + asin(x)
* For |x|<=0.5
* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
* For x>0.5
* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
* = 2asin(sqrt((1-x)/2))
* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
* = 2f + (2c + 2s*z*R(z))
* where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
* for f so that f+c ~ sqrt(z).
* For x<-0.5
* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
* Function needed: sqrt
*/
#include "libm.h"
static const double
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
static double R(double z)
{
double_t p, q;
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
return p/q;
}
double acos(double x)
{
double z,w,s,c,df;
uint32_t hx,ix;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7fffffff;
/* |x| >= 1 or nan */
if (ix >= 0x3ff00000) {
uint32_t lx;
GET_LOW_WORD(lx,x);
if (((ix-0x3ff00000) | lx) == 0) {
/* acos(1)=0, acos(-1)=pi */
if (hx >> 31)
return 2*pio2_hi + 0x1p-120f;
return 0;
}
return 0/(x-x);
}
/* |x| < 0.5 */
if (ix < 0x3fe00000) {
if (ix <= 0x3c600000) /* |x| < 2**-57 */
return pio2_hi + 0x1p-120f;
return pio2_hi - (x - (pio2_lo-x*R(x*x)));
}
/* x < -0.5 */
if (hx >> 31) {
z = (1.0+x)*0.5;
s = sqrt(z);
w = R(z)*s-pio2_lo;
return 2*(pio2_hi - (s+w));
}
/* x > 0.5 */
z = (1.0-x)*0.5;
s = sqrt(z);
df = s;
SET_LOW_WORD(df,0);
c = (z-df*df)/(s+df);
w = R(z)*s+c;
return 2*(df+w);
}

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#include "libm.h"
#if FLT_EVAL_METHOD==2
#undef sqrt
#define sqrt sqrtl
#endif
/* acosh(x) = log(x + sqrt(x*x-1)) */
double acosh(double x)
{
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
/* x < 1 domain error is handled in the called functions */
if (e < 0x3ff + 1)
/* |x| < 2, up to 2ulp error in [1,1.125] */
return log1p(x-1 + sqrt((x-1)*(x-1)+2*(x-1)));
if (e < 0x3ff + 26)
/* |x| < 0x1p26 */
return log(2*x - 1/(x+sqrt(x*x-1)));
/* |x| >= 0x1p26 or nan */
return log(x) + 0.693147180559945309417232121458176568;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/e_asin.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* asin(x)
* Method :
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we approximate asin(x) on [0,0.5] by
* asin(x) = x + x*x^2*R(x^2)
* where
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
* and its remez error is bounded by
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
*
* For x in [0.5,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
* then for x>0.98
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
* For x<=0.98, let pio4_hi = pio2_hi/2, then
* f = hi part of s;
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
* and
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
*/
#include "libm.h"
static const double
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
/* coefficients for R(x^2) */
pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
static double R(double z)
{
double_t p, q;
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
q = 1.0+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
return p/q;
}
double asin(double x)
{
double z,r,s;
uint32_t hx,ix;
GET_HIGH_WORD(hx, x);
ix = hx & 0x7fffffff;
/* |x| >= 1 or nan */
if (ix >= 0x3ff00000) {
uint32_t lx;
GET_LOW_WORD(lx, x);
if (((ix-0x3ff00000) | lx) == 0)
/* asin(1) = +-pi/2 with inexact */
return x*pio2_hi + 0x1p-120f;
return 0/(x-x);
}
/* |x| < 0.5 */
if (ix < 0x3fe00000) {
/* if 0x1p-1022 <= |x| < 0x1p-26, avoid raising underflow */
if (ix < 0x3e500000 && ix >= 0x00100000)
return x;
return x + x*R(x*x);
}
/* 1 > |x| >= 0.5 */
z = (1 - fabs(x))*0.5;
s = sqrt(z);
r = R(z);
if (ix >= 0x3fef3333) { /* if |x| > 0.975 */
x = pio2_hi-(2*(s+s*r)-pio2_lo);
} else {
double f,c;
/* f+c = sqrt(z) */
f = s;
SET_LOW_WORD(f,0);
c = (z-f*f)/(s+f);
x = 0.5*pio2_hi - (2*s*r - (pio2_lo-2*c) - (0.5*pio2_hi-2*f));
}
if (hx >> 31)
return -x;
return x;
}

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#include "libm.h"
/* asinh(x) = sign(x)*log(|x|+sqrt(x*x+1)) ~= x - x^3/6 + o(x^5) */
double asinh(double x)
{
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
unsigned s = u.i >> 63;
/* |x| */
u.i &= (uint64_t)-1/2;
x = u.f;
if (e >= 0x3ff + 26) {
/* |x| >= 0x1p26 or inf or nan */
x = log(x) + 0.693147180559945309417232121458176568;
} else if (e >= 0x3ff + 1) {
/* |x| >= 2 */
x = log(2*x + 1/(sqrt(x*x+1)+x));
} else if (e >= 0x3ff - 26) {
/* |x| >= 0x1p-26, up to 1.6ulp error in [0.125,0.5] */
x = log1p(x + x*x/(sqrt(x*x+1)+1));
} else {
/* |x| < 0x1p-26, raise inexact if x != 0 */
FORCE_EVAL(x + 0x1p120f);
}
return s ? -x : x;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/s_atan.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* atan(x)
* Method
* 1. Reduce x to positive by atan(x) = -atan(-x).
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
* is further reduced to one of the following intervals and the
* arctangent of t is evaluated by the corresponding formula:
*
* [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
* [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
* [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
* [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
* [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include "libm.h"
static const double atanhi[] = {
4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
};
static const double atanlo[] = {
2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
};
static const double aT[] = {
3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
-1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
-1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
-7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
-5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
-3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
};
double atan(double x)
{
double_t w,s1,s2,z;
uint32_t ix,sign;
int id;
GET_HIGH_WORD(ix, x);
sign = ix >> 31;
ix &= 0x7fffffff;
if (ix >= 0x44100000) { /* if |x| >= 2^66 */
if (isnan(x))
return x;
z = atanhi[3] + 0x1p-120f;
return sign ? -z : z;
}
if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
if (ix < 0x3e400000) { /* |x| < 2^-27 */
if (ix < 0x00100000)
/* raise underflow for subnormal x */
FORCE_EVAL((float)x);
return x;
}
id = -1;
} else {
x = fabs(x);
if (ix < 0x3ff30000) { /* |x| < 1.1875 */
if (ix < 0x3fe60000) { /* 7/16 <= |x| < 11/16 */
id = 0;
x = (2.0*x-1.0)/(2.0+x);
} else { /* 11/16 <= |x| < 19/16 */
id = 1;
x = (x-1.0)/(x+1.0);
}
} else {
if (ix < 0x40038000) { /* |x| < 2.4375 */
id = 2;
x = (x-1.5)/(1.0+1.5*x);
} else { /* 2.4375 <= |x| < 2^66 */
id = 3;
x = -1.0/x;
}
}
}
/* end of argument reduction */
z = x*x;
w = z*z;
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
if (id < 0)
return x - x*(s1+s2);
z = atanhi[id] - (x*(s1+s2) - atanlo[id] - x);
return sign ? -z : z;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/e_atan2.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
*/
/* atan2(y,x)
* Method :
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
* 2. Reduce x to positive by (if x and y are unexceptional):
* ARG (x+iy) = arctan(y/x) ... if x > 0,
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
*
* Special cases:
*
* ATAN2((anything), NaN ) is NaN;
* ATAN2(NAN , (anything) ) is NaN;
* ATAN2(+-0, +(anything but NaN)) is +-0 ;
* ATAN2(+-0, -(anything but NaN)) is +-pi ;
* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
* ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
* ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
* ATAN2(+-INF,+INF ) is +-pi/4 ;
* ATAN2(+-INF,-INF ) is +-3pi/4;
* ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include "libm.h"
static const double
pi = 3.1415926535897931160E+00, /* 0x400921FB, 0x54442D18 */
pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
double atan2(double y, double x)
{
double z;
uint32_t m,lx,ly,ix,iy;
if (isnan(x) || isnan(y))
return x+y;
EXTRACT_WORDS(ix, lx, x);
EXTRACT_WORDS(iy, ly, y);
if (((ix-0x3ff00000) | lx) == 0) /* x = 1.0 */
return atan(y);
m = ((iy>>31)&1) | ((ix>>30)&2); /* 2*sign(x)+sign(y) */
ix = ix & 0x7fffffff;
iy = iy & 0x7fffffff;
/* when y = 0 */
if ((iy|ly) == 0) {
switch(m) {
case 0:
case 1: return y; /* atan(+-0,+anything)=+-0 */
case 2: return pi; /* atan(+0,-anything) = pi */
case 3: return -pi; /* atan(-0,-anything) =-pi */
}
}
/* when x = 0 */
if ((ix|lx) == 0)
return m&1 ? -pi/2 : pi/2;
/* when x is INF */
if (ix == 0x7ff00000) {
if (iy == 0x7ff00000) {
switch(m) {
case 0: return pi/4; /* atan(+INF,+INF) */
case 1: return -pi/4; /* atan(-INF,+INF) */
case 2: return 3*pi/4; /* atan(+INF,-INF) */
case 3: return -3*pi/4; /* atan(-INF,-INF) */
}
} else {
switch(m) {
case 0: return 0.0; /* atan(+...,+INF) */
case 1: return -0.0; /* atan(-...,+INF) */
case 2: return pi; /* atan(+...,-INF) */
case 3: return -pi; /* atan(-...,-INF) */
}
}
}
/* |y/x| > 0x1p64 */
if (ix+(64<<20) < iy || iy == 0x7ff00000)
return m&1 ? -pi/2 : pi/2;
/* z = atan(|y/x|) without spurious underflow */
if ((m&2) && iy+(64<<20) < ix) /* |y/x| < 0x1p-64, x<0 */
z = 0;
else
z = atan(fabs(y/x));
switch (m) {
case 0: return z; /* atan(+,+) */
case 1: return -z; /* atan(-,+) */
case 2: return pi - (z-pi_lo); /* atan(+,-) */
default: /* case 3 */
return (z-pi_lo) - pi; /* atan(-,-) */
}
}

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#include "libm.h"
/* atanh(x) = log((1+x)/(1-x))/2 = log1p(2x/(1-x))/2 ~= x + x^3/3 + o(x^5) */
double atanh(double x)
{
union {double f; uint64_t i;} u = {.f = x};
unsigned e = u.i >> 52 & 0x7ff;
unsigned s = u.i >> 63;
double_t y;
/* |x| */
u.i &= (uint64_t)-1/2;
y = u.f;
if (e < 0x3ff - 1) {
if (e < 0x3ff - 32) {
/* handle underflow */
if (e == 0)
FORCE_EVAL((float)y);
} else {
/* |x| < 0.5, up to 1.7ulp error */
y = 0.5*log1p(2*y + 2*y*y/(1-y));
}
} else {
/* avoid overflow */
y = 0.5*log1p(2*(y/(1-y)));
}
return s ? -y : y;
}

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#include "libm.h"
#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
#define EPS DBL_EPSILON
#elif FLT_EVAL_METHOD==2
#define EPS LDBL_EPSILON
#endif
static const double_t toint = 1/EPS;
double ceil(double x)
{
union {double f; uint64_t i;} u = {x};
int e = u.i >> 52 & 0x7ff;
double_t y;
if (e >= 0x3ff+52 || x == 0)
return x;
/* y = int(x) - x, where int(x) is an integer neighbor of x */
if (u.i >> 63)
y = x - toint + toint - x;
else
y = x + toint - toint - x;
/* special case because of non-nearest rounding modes */
if (e <= 0x3ff-1) {
FORCE_EVAL(y);
return u.i >> 63 ? -0.0 : 1;
}
if (y < 0)
return x + y + 1;
return x + y;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/s_cos.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* cos(x)
* Return cosine function of x.
*
* kernel function:
* __sin ... sine function on [-pi/4,pi/4]
* __cos ... cosine function on [-pi/4,pi/4]
* __rem_pio2 ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
#include "libm.h"
double cos(double x)
{
double y[2];
uint32_t ix;
unsigned n;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
/* |x| ~< pi/4 */
if (ix <= 0x3fe921fb) {
if (ix < 0x3e46a09e) { /* |x| < 2**-27 * sqrt(2) */
/* raise inexact if x!=0 */
FORCE_EVAL(x + 0x1p120f);
return 1.0;
}
return __cos(x, 0);
}
/* cos(Inf or NaN) is NaN */
if (ix >= 0x7ff00000)
return x-x;
/* argument reduction */
n = __rem_pio2(x, y);
switch (n&3) {
case 0: return __cos(y[0], y[1]);
case 1: return -__sin(y[0], y[1], 1);
case 2: return -__cos(y[0], y[1]);
default:
return __sin(y[0], y[1], 1);
}
}

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#include "libm.h"
/* cosh(x) = (exp(x) + 1/exp(x))/2
* = 1 + 0.5*(exp(x)-1)*(exp(x)-1)/exp(x)
* = 1 + x*x/2 + o(x^4)
*/
double cosh(double x)
{
union {double f; uint64_t i;} u = {.f = x};
uint32_t w;
double t;
/* |x| */
u.i &= (uint64_t)-1/2;
x = u.f;
w = u.i >> 32;
/* |x| < log(2) */
if (w < 0x3fe62e42) {
if (w < 0x3ff00000 - (26<<20)) {
/* raise inexact if x!=0 */
FORCE_EVAL(x + 0x1p120f);
return 1;
}
t = expm1(x);
return 1 + t*t/(2*(1+t));
}
/* |x| < log(DBL_MAX) */
if (w < 0x40862e42) {
t = exp(x);
/* note: if x>log(0x1p26) then the 1/t is not needed */
return 0.5*(t + 1/t);
}
/* |x| > log(DBL_MAX) or nan */
/* note: the result is stored to handle overflow */
t = __expo2(x);
return t;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* double erf(double x)
* double erfc(double x)
* x
* 2 |\
* erf(x) = --------- | exp(-t*t)dt
* sqrt(pi) \|
* 0
*
* erfc(x) = 1-erf(x)
* Note that
* erf(-x) = -erf(x)
* erfc(-x) = 2 - erfc(x)
*
* Method:
* 1. For |x| in [0, 0.84375]
* erf(x) = x + x*R(x^2)
* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
* where R = P/Q where P is an odd poly of degree 8 and
* Q is an odd poly of degree 10.
* -57.90
* | R - (erf(x)-x)/x | <= 2
*
*
* Remark. The formula is derived by noting
* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
* and that
* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
* is close to one. The interval is chosen because the fix
* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
* near 0.6174), and by some experiment, 0.84375 is chosen to
* guarantee the error is less than one ulp for erf.
*
* 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
* c = 0.84506291151 rounded to single (24 bits)
* erf(x) = sign(x) * (c + P1(s)/Q1(s))
* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
* 1+(c+P1(s)/Q1(s)) if x < 0
* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
* Remark: here we use the taylor series expansion at x=1.
* erf(1+s) = erf(1) + s*Poly(s)
* = 0.845.. + P1(s)/Q1(s)
* That is, we use rational approximation to approximate
* erf(1+s) - (c = (single)0.84506291151)
* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
* where
* P1(s) = degree 6 poly in s
* Q1(s) = degree 6 poly in s
*
* 3. For x in [1.25,1/0.35(~2.857143)],
* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
* erf(x) = 1 - erfc(x)
* where
* R1(z) = degree 7 poly in z, (z=1/x^2)
* S1(z) = degree 8 poly in z
*
* 4. For x in [1/0.35,28]
* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
* = 2.0 - tiny (if x <= -6)
* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
* erf(x) = sign(x)*(1.0 - tiny)
* where
* R2(z) = degree 6 poly in z, (z=1/x^2)
* S2(z) = degree 7 poly in z
*
* Note1:
* To compute exp(-x*x-0.5625+R/S), let s be a single
* precision number and s := x; then
* -x*x = -s*s + (s-x)*(s+x)
* exp(-x*x-0.5626+R/S) =
* exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
* Note2:
* Here 4 and 5 make use of the asymptotic series
* exp(-x*x)
* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
* x*sqrt(pi)
* We use rational approximation to approximate
* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
* Here is the error bound for R1/S1 and R2/S2
* |R1/S1 - f(x)| < 2**(-62.57)
* |R2/S2 - f(x)| < 2**(-61.52)
*
* 5. For inf > x >= 28
* erf(x) = sign(x) *(1 - tiny) (raise inexact)
* erfc(x) = tiny*tiny (raise underflow) if x > 0
* = 2 - tiny if x<0
*
* 7. Special case:
* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
* erfc/erf(NaN) is NaN
*/
#include "libm.h"
static const double
erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
/*
* Coefficients for approximation to erf on [0,0.84375]
*/
efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
/*
* Coefficients for approximation to erf in [0.84375,1.25]
*/
pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
/*
* Coefficients for approximation to erfc in [1.25,1/0.35]
*/
ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
/*
* Coefficients for approximation to erfc in [1/.35,28]
*/
rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
static double erfc1(double x)
{
double_t s,P,Q;
s = fabs(x) - 1;
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
return 1 - erx - P/Q;
}
static double erfc2(uint32_t ix, double x)
{
double_t s,R,S;
double z;
if (ix < 0x3ff40000) /* |x| < 1.25 */
return erfc1(x);
x = fabs(x);
s = 1/(x*x);
if (ix < 0x4006db6d) { /* |x| < 1/.35 ~ 2.85714 */
R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
ra5+s*(ra6+s*ra7))))));
S = 1.0+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
sa5+s*(sa6+s*(sa7+s*sa8)))))));
} else { /* |x| > 1/.35 */
R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
rb5+s*rb6)))));
S = 1.0+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
sb5+s*(sb6+s*sb7))))));
}
z = x;
SET_LOW_WORD(z,0);
return exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)/x;
}
double erf(double x)
{
double r,s,z,y;
uint32_t ix;
int sign;
GET_HIGH_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x7ff00000) {
/* erf(nan)=nan, erf(+-inf)=+-1 */
return 1-2*sign + 1/x;
}
if (ix < 0x3feb0000) { /* |x| < 0.84375 */
if (ix < 0x3e300000) { /* |x| < 2**-28 */
/* avoid underflow */
return 0.125*(8*x + efx8*x);
}
z = x*x;
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
return x + x*y;
}
if (ix < 0x40180000) /* 0.84375 <= |x| < 6 */
y = 1 - erfc2(ix,x);
else
y = 1 - 0x1p-1022;
return sign ? -y : y;
}
double erfc(double x)
{
double r,s,z,y;
uint32_t ix;
int sign;
GET_HIGH_WORD(ix, x);
sign = ix>>31;
ix &= 0x7fffffff;
if (ix >= 0x7ff00000) {
/* erfc(nan)=nan, erfc(+-inf)=0,2 */
return 2*sign + 1/x;
}
if (ix < 0x3feb0000) { /* |x| < 0.84375 */
if (ix < 0x3c700000) /* |x| < 2**-56 */
return 1.0 - x;
z = x*x;
r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
y = r/s;
if (sign || ix < 0x3fd00000) { /* x < 1/4 */
return 1.0 - (x+x*y);
}
return 0.5 - (x - 0.5 + x*y);
}
if (ix < 0x403c0000) { /* 0.84375 <= |x| < 28 */
return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
}
return sign ? 2 - 0x1p-1022 : 0x1p-1022*0x1p-1022;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* exp(x)
* Returns the exponential of x.
*
* Method
* 1. Argument reduction:
* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| <= 0.5*ln2.
*
* Here r will be represented as r = hi-lo for better
* accuracy.
*
* 2. Approximation of exp(r) by a special rational function on
* the interval [0,0.34658]:
* Write
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
* We use a special Remez algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
* (where z=r*r, and the values of P1 to P5 are listed below)
* and
* | 5 | -59
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
* | |
* The computation of exp(r) thus becomes
* 2*r
* exp(r) = 1 + ----------
* R(r) - r
* r*c(r)
* = 1 + r + ----------- (for better accuracy)
* 2 - c(r)
* where
* 2 4 10
* c(r) = r - (P1*r + P2*r + ... + P5*r ).
*
* 3. Scale back to obtain exp(x):
* From step 1, we have
* exp(x) = 2^k * exp(r)
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) is 0, and
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Misc. info.
* For IEEE double
* if x > 709.782712893383973096 then exp(x) overflows
* if x < -745.133219101941108420 then exp(x) underflows
*/
#include "libm.h"
static const double
half[2] = {0.5,-0.5},
ln2hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
double exp(double x)
{
double_t hi, lo, c, xx, y;
int k, sign;
uint32_t hx;
GET_HIGH_WORD(hx, x);
sign = hx>>31;
hx &= 0x7fffffff; /* high word of |x| */
/* special cases */
if (hx >= 0x4086232b) { /* if |x| >= 708.39... */
if (isnan(x))
return x;
if (x > 709.782712893383973096) {
/* overflow if x!=inf */
x *= 0x1p1023;
return x;
}
if (x < -708.39641853226410622) {
/* underflow if x!=-inf */
FORCE_EVAL((float)(-0x1p-149/x));
if (x < -745.13321910194110842)
return 0;
}
}
/* argument reduction */
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if (hx >= 0x3ff0a2b2) /* if |x| >= 1.5 ln2 */
k = (int)(invln2*x + half[sign]);
else
k = 1 - sign - sign;
hi = x - k*ln2hi; /* k*ln2hi is exact here */
lo = k*ln2lo;
x = hi - lo;
} else if (hx > 0x3e300000) { /* if |x| > 2**-28 */
k = 0;
hi = x;
lo = 0;
} else {
/* inexact if x!=0 */
FORCE_EVAL(0x1p1023 + x);
return 1 + x;
}
/* x is now in primary range */
xx = x*x;
c = x - xx*(P1+xx*(P2+xx*(P3+xx*(P4+xx*P5))));
y = 1 + (x*c/(2-c) - lo + hi);
if (k == 0)
return y;
return scalbn(y, k);
}

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/* origin: FreeBSD /usr/src/lib/msun/src/s_expm1.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* expm1(x)
* Returns exp(x)-1, the exponential of x minus 1.
*
* Method
* 1. Argument reduction:
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
*
* Here a correction term c will be computed to compensate
* the error in r when rounded to a floating-point number.
*
* 2. Approximating expm1(r) by a special rational function on
* the interval [0,0.34658]:
* Since
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
* we define R1(r*r) by
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
* That is,
* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
* We use a special Remez algorithm on [0,0.347] to generate
* a polynomial of degree 5 in r*r to approximate R1. The
* maximum error of this polynomial approximation is bounded
* by 2**-61. In other words,
* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
* where Q1 = -1.6666666666666567384E-2,
* Q2 = 3.9682539681370365873E-4,
* Q3 = -9.9206344733435987357E-6,
* Q4 = 2.5051361420808517002E-7,
* Q5 = -6.2843505682382617102E-9;
* z = r*r,
* with error bounded by
* | 5 | -61
* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
* | |
*
* expm1(r) = exp(r)-1 is then computed by the following
* specific way which minimize the accumulation rounding error:
* 2 3
* r r [ 3 - (R1 + R1*r/2) ]
* expm1(r) = r + --- + --- * [--------------------]
* 2 2 [ 6 - r*(3 - R1*r/2) ]
*
* To compensate the error in the argument reduction, we use
* expm1(r+c) = expm1(r) + c + expm1(r)*c
* ~ expm1(r) + c + r*c
* Thus c+r*c will be added in as the correction terms for
* expm1(r+c). Now rearrange the term to avoid optimization
* screw up:
* ( 2 2 )
* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
* ( )
*
* = r - E
* 3. Scale back to obtain expm1(x):
* From step 1, we have
* expm1(x) = either 2^k*[expm1(r)+1] - 1
* = or 2^k*[expm1(r) + (1-2^-k)]
* 4. Implementation notes:
* (A). To save one multiplication, we scale the coefficient Qi
* to Qi*2^i, and replace z by (x^2)/2.
* (B). To achieve maximum accuracy, we compute expm1(x) by
* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
* (ii) if k=0, return r-E
* (iii) if k=-1, return 0.5*(r-E)-0.5
* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
* else return 1.0+2.0*(r-E);
* (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
* (vii) return 2^k(1-((E+2^-k)-r))
*
* Special cases:
* expm1(INF) is INF, expm1(NaN) is NaN;
* expm1(-INF) is -1, and
* for finite argument, only expm1(0)=0 is exact.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Misc. info.
* For IEEE double
* if x > 7.09782712893383973096e+02 then expm1(x) overflow
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include "libm.h"
static const double
o_threshold = 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
ln2_hi = 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
invln2 = 1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
/* Scaled Q's: Qn_here = 2**n * Qn_above, for R(2*z) where z = hxs = x*x/2: */
Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */
Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
double expm1(double x)
{
double_t y,hi,lo,c,t,e,hxs,hfx,r1,twopk;
union {double f; uint64_t i;} u = {x};
uint32_t hx = u.i>>32 & 0x7fffffff;
int k, sign = u.i>>63;
/* filter out huge and non-finite argument */
if (hx >= 0x4043687A) { /* if |x|>=56*ln2 */
if (isnan(x))
return x;
if (sign)
return -1;
if (x > o_threshold) {
x *= 0x1p1023;
return x;
}
}
/* argument reduction */
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if (hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
if (!sign) {
hi = x - ln2_hi;
lo = ln2_lo;
k = 1;
} else {
hi = x + ln2_hi;
lo = -ln2_lo;
k = -1;
}
} else {
k = invln2*x + (sign ? -0.5 : 0.5);
t = k;
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
lo = t*ln2_lo;
}
x = hi-lo;
c = (hi-x)-lo;
} else if (hx < 0x3c900000) { /* |x| < 2**-54, return x */
if (hx < 0x00100000)
FORCE_EVAL((float)x);
return x;
} else
k = 0;
/* x is now in primary range */
hfx = 0.5*x;
hxs = x*hfx;
r1 = 1.0+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
t = 3.0-r1*hfx;
e = hxs*((r1-t)/(6.0 - x*t));
if (k == 0) /* c is 0 */
return x - (x*e-hxs);
e = x*(e-c) - c;
e -= hxs;
/* exp(x) ~ 2^k (x_reduced - e + 1) */
if (k == -1)
return 0.5*(x-e) - 0.5;
if (k == 1) {
if (x < -0.25)
return -2.0*(e-(x+0.5));
return 1.0+2.0*(x-e);
}
u.i = (uint64_t)(0x3ff + k)<<52; /* 2^k */
twopk = u.f;
if (k < 0 || k > 56) { /* suffice to return exp(x)-1 */
y = x - e + 1.0;
if (k == 1024)
y = y*2.0*0x1p1023;
else
y = y*twopk;
return y - 1.0;
}
u.i = (uint64_t)(0x3ff - k)<<52; /* 2^-k */
if (k < 20)
y = (x-e+(1-u.f))*twopk;
else
y = (x-(e+u.f)+1)*twopk;
return y;
}

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#include "libm.h"
#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
#define EPS DBL_EPSILON
#elif FLT_EVAL_METHOD==2
#define EPS LDBL_EPSILON
#endif
static const double_t toint = 1/EPS;
double floor(double x)
{
union {double f; uint64_t i;} u = {x};
int e = u.i >> 52 & 0x7ff;
double_t y;
if (e >= 0x3ff+52 || x == 0)
return x;
/* y = int(x) - x, where int(x) is an integer neighbor of x */
if (u.i >> 63)
y = x - toint + toint - x;
else
y = x + toint - toint - x;
/* special case because of non-nearest rounding modes */
if (e <= 0x3ff-1) {
FORCE_EVAL(y);
return u.i >> 63 ? -1 : 0;
}
if (y > 0)
return x + y - 1;
return x + y;
}

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#include <math.h>
#include <stdint.h>
double fmod(double x, double y)
{
union {double f; uint64_t i;} ux = {x}, uy = {y};
int ex = ux.i>>52 & 0x7ff;
int ey = uy.i>>52 & 0x7ff;
int sx = ux.i>>63;
uint64_t i;
/* in the followings uxi should be ux.i, but then gcc wrongly adds */
/* float load/store to inner loops ruining performance and code size */
uint64_t uxi = ux.i;
if (uy.i<<1 == 0 || isnan(y) || ex == 0x7ff)
return (x*y)/(x*y);
if (uxi<<1 <= uy.i<<1) {
if (uxi<<1 == uy.i<<1)
return 0*x;
return x;
}
/* normalize x and y */
if (!ex) {
for (i = uxi<<12; i>>63 == 0; ex--, i <<= 1);
uxi <<= -ex + 1;
} else {
uxi &= -1ULL >> 12;
uxi |= 1ULL << 52;
}
if (!ey) {
for (i = uy.i<<12; i>>63 == 0; ey--, i <<= 1);
uy.i <<= -ey + 1;
} else {
uy.i &= -1ULL >> 12;
uy.i |= 1ULL << 52;
}
/* x mod y */
for (; ex > ey; ex--) {
i = uxi - uy.i;
if (i >> 63 == 0) {
if (i == 0)
return 0*x;
uxi = i;
}
uxi <<= 1;
}
i = uxi - uy.i;
if (i >> 63 == 0) {
if (i == 0)
return 0*x;
uxi = i;
}
for (; uxi>>52 == 0; uxi <<= 1, ex--);
/* scale result */
if (ex > 0) {
uxi -= 1ULL << 52;
uxi |= (uint64_t)ex << 52;
} else {
uxi >>= -ex + 1;
}
uxi |= (uint64_t)sx << 63;
ux.i = uxi;
return ux.f;
}

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#include <math.h>
#include <stdint.h>
double frexp(double x, int *e)
{
union { double d; uint64_t i; } y = { x };
int ee = y.i>>52 & 0x7ff;
if (!ee) {
if (x) {
x = frexp(x*0x1p64, e);
*e -= 64;
} else *e = 0;
return x;
} else if (ee == 0x7ff) {
return x;
}
*e = ee - 0x3fe;
y.i &= 0x800fffffffffffffull;
y.i |= 0x3fe0000000000000ull;
return y.d;
}

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#include <math.h>
double ldexp(double x, int n)
{
return scalbn(x, n);
}

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#include <math.h>
double __lgamma_r(double, int*);
double lgamma(double x) {
int sign;
return __lgamma_r(x, &sign);
}

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// Portions of this file are extracted from musl-1.1.16 src/internal/libm.h
/* origin: FreeBSD /usr/src/lib/msun/src/math_private.h */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#include <stdint.h>
#include <math.h>
#define FLT_EVAL_METHOD 0
#define FORCE_EVAL(x) do { \
if (sizeof(x) == sizeof(float)) { \
volatile float __x; \
__x = (x); \
(void)__x; \
} else if (sizeof(x) == sizeof(double)) { \
volatile double __x; \
__x = (x); \
(void)__x; \
} else { \
volatile long double __x; \
__x = (x); \
(void)__x; \
} \
} while(0)
/* Get two 32 bit ints from a double. */
#define EXTRACT_WORDS(hi,lo,d) \
do { \
union {double f; uint64_t i;} __u; \
__u.f = (d); \
(hi) = __u.i >> 32; \
(lo) = (uint32_t)__u.i; \
} while (0)
/* Get the more significant 32 bit int from a double. */
#define GET_HIGH_WORD(hi,d) \
do { \
union {double f; uint64_t i;} __u; \
__u.f = (d); \
(hi) = __u.i >> 32; \
} while (0)
/* Get the less significant 32 bit int from a double. */
#define GET_LOW_WORD(lo,d) \
do { \
union {double f; uint64_t i;} __u; \
__u.f = (d); \
(lo) = (uint32_t)__u.i; \
} while (0)
/* Set a double from two 32 bit ints. */
#define INSERT_WORDS(d,hi,lo) \
do { \
union {double f; uint64_t i;} __u; \
__u.i = ((uint64_t)(hi)<<32) | (uint32_t)(lo); \
(d) = __u.f; \
} while (0)
/* Set the more significant 32 bits of a double from an int. */
#define SET_HIGH_WORD(d,hi) \
do { \
union {double f; uint64_t i;} __u; \
__u.f = (d); \
__u.i &= 0xffffffff; \
__u.i |= (uint64_t)(hi) << 32; \
(d) = __u.f; \
} while (0)
/* Set the less significant 32 bits of a double from an int. */
#define SET_LOW_WORD(d,lo) \
do { \
union {double f; uint64_t i;} __u; \
__u.f = (d); \
__u.i &= 0xffffffff00000000ull; \
__u.i |= (uint32_t)(lo); \
(d) = __u.f; \
} while (0)
#define DBL_EPSILON 2.22044604925031308085e-16
int __rem_pio2(double, double*);
int __rem_pio2_large(double*, double*, int, int, int);
double __sin(double, double, int);
double __cos(double, double);
double __tan(double, double, int);
double __expo2(double);

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/* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* log(x)
* Return the logarithm of x
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+f).
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* = 2s + s*R
* We use a special Remez algorithm on [0,0.1716] to generate
* a polynomial of degree 14 to approximate R The maximum error
* of this polynomial approximation is bounded by 2**-58.45. In
* other words,
* 2 4 6 8 10 12 14
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lg1*s +...+Lg7*s - R(z) | <= 2
* | |
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
* In order to guarantee error in log below 1ulp, we compute log
* by
* log(1+f) = f - s*(f - R) (if f is not too large)
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k*ln2 + log(1+f).
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
* Here ln2 is split into two floating point number:
* ln2_hi + ln2_lo,
* where n*ln2_hi is always exact for |n| < 2000.
*
* Special cases:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(NaN) is that NaN with no signal.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include <math.h>
#include <stdint.h>
static const double
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
double log(double x)
{
union {double f; uint64_t i;} u = {x};
double_t hfsq,f,s,z,R,w,t1,t2,dk;
uint32_t hx;
int k;
hx = u.i>>32;
k = 0;
if (hx < 0x00100000 || hx>>31) {
if (u.i<<1 == 0)
return -1/(x*x); /* log(+-0)=-inf */
if (hx>>31)
return (x-x)/0.0; /* log(-#) = NaN */
/* subnormal number, scale x up */
k -= 54;
x *= 0x1p54;
u.f = x;
hx = u.i>>32;
} else if (hx >= 0x7ff00000) {
return x;
} else if (hx == 0x3ff00000 && u.i<<32 == 0)
return 0;
/* reduce x into [sqrt(2)/2, sqrt(2)] */
hx += 0x3ff00000 - 0x3fe6a09e;
k += (int)(hx>>20) - 0x3ff;
hx = (hx&0x000fffff) + 0x3fe6a09e;
u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
x = u.f;
f = x - 1.0;
hfsq = 0.5*f*f;
s = f/(2.0+f);
z = s*s;
w = z*z;
t1 = w*(Lg2+w*(Lg4+w*Lg6));
t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
R = t2 + t1;
dk = k;
return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
}

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#include <math.h>
static const double _M_LN10 = 2.302585092994046;
double log10(double x) {
return log(x) / (double)_M_LN10;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* double log1p(double x)
* Return the natural logarithm of 1+x.
*
* Method :
* 1. Argument Reduction: find k and f such that
* 1+x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* Note. If k=0, then f=x is exact. However, if k!=0, then f
* may not be representable exactly. In that case, a correction
* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
* and add back the correction term c/u.
* (Note: when x > 2**53, one can simply return log(x))
*
* 2. Approximation of log(1+f): See log.c
*
* 3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
*
* Special cases:
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
* log1p(NaN) is that NaN with no signal.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Constants:
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*
* Note: Assuming log() return accurate answer, the following
* algorithm can be used to compute log1p(x) to within a few ULP:
*
* u = 1+x;
* if(u==1.0) return x ; else
* return log(u)*(x/(u-1.0));
*
* See HP-15C Advanced Functions Handbook, p.193.
*/
#include "libm.h"
static const double
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
double log1p(double x)
{
union {double f; uint64_t i;} u = {x};
double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
uint32_t hx,hu;
int k;
hx = u.i>>32;
k = 1;
if (hx < 0x3fda827a || hx>>31) { /* 1+x < sqrt(2)+ */
if (hx >= 0xbff00000) { /* x <= -1.0 */
if (x == -1)
return x/0.0; /* log1p(-1) = -inf */
return (x-x)/0.0; /* log1p(x<-1) = NaN */
}
if (hx<<1 < 0x3ca00000<<1) { /* |x| < 2**-53 */
/* underflow if subnormal */
if ((hx&0x7ff00000) == 0)
FORCE_EVAL((float)x);
return x;
}
if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
k = 0;
c = 0;
f = x;
}
} else if (hx >= 0x7ff00000)
return x;
if (k) {
u.f = 1 + x;
hu = u.i>>32;
hu += 0x3ff00000 - 0x3fe6a09e;
k = (int)(hu>>20) - 0x3ff;
/* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
if (k < 54) {
c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
c /= u.f;
} else
c = 0;
/* reduce u into [sqrt(2)/2, sqrt(2)] */
hu = (hu&0x000fffff) + 0x3fe6a09e;
u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
f = u.f - 1;
}
hfsq = 0.5*f*f;
s = f/(2.0+f);
z = s*s;
w = z*z;
t1 = w*(Lg2+w*(Lg4+w*Lg6));
t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
R = t2 + t1;
dk = k;
return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
}

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#include "libm.h"
double modf(double x, double *iptr)
{
union {double f; uint64_t i;} u = {x};
uint64_t mask;
int e = (int)(u.i>>52 & 0x7ff) - 0x3ff;
/* no fractional part */
if (e >= 52) {
*iptr = x;
if (e == 0x400 && u.i<<12 != 0) /* nan */
return x;
u.i &= 1ULL<<63;
return u.f;
}
/* no integral part*/
if (e < 0) {
u.i &= 1ULL<<63;
*iptr = u.f;
return x;
}
mask = -1ULL>>12>>e;
if ((u.i & mask) == 0) {
*iptr = x;
u.i &= 1ULL<<63;
return u.f;
}
u.i &= ~mask;
*iptr = u.f;
return x - u.f;
}

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//#include <fenv.h>
#include <math.h>
/* nearbyint is the same as rint, but it must not raise the inexact exception */
double nearbyint(double x)
{
#ifdef FE_INEXACT
#pragma STDC FENV_ACCESS ON
int e;
e = fetestexcept(FE_INEXACT);
#endif
x = rint(x);
#ifdef FE_INEXACT
if (!e)
feclearexcept(FE_INEXACT);
#endif
return x;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/e_pow.c */
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* pow(x,y) return x**y
*
* n
* Method: Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53-24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
* arithmetic, where |y'|<=0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. 1 ** (anything) is 1
* 3. (anything except 1) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. -1 ** +-INF is 1
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF, raise divbyzero
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF, raise divbyzero
* 14. -0 ** (+odd integer) is -0
* 15. -0 ** (-odd integer) is -INF, raise divbyzero
* 16. +INF ** (+anything except 0,NAN) is +INF
* 17. +INF ** (-anything except 0,NAN) is +0
* 18. -INF ** (+odd integer) is -INF
* 19. -INF ** (anything) = -0 ** (-anything), (anything except odd integer)
* 20. (anything) ** 1 is (anything)
* 21. (anything) ** -1 is 1/(anything)
* 22. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 23. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular
* pow(integer,integer)
* always returns the correct integer provided it is
* representable.
*
* Constants :
* The hexadecimal values are the intended ones for the following
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/
#include "libm.h"
static const double
bp[] = {1.0, 1.5,},
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
huge = 1.0e300,
tiny = 1.0e-300,
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1 = 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
L2 = 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
L3 = 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
L4 = 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
L5 = 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
L6 = 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
P1 = 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2 = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3 = 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4 = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5 = 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
ovt = 8.0085662595372944372e-017, /* -(1024-log2(ovfl+.5ulp)) */
cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
ivln2_l = 1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
double pow(double x, double y)
{
double z,ax,z_h,z_l,p_h,p_l;
double y1,t1,t2,r,s,t,u,v,w;
int32_t i,j,k,yisint,n;
int32_t hx,hy,ix,iy;
uint32_t lx,ly;
EXTRACT_WORDS(hx, lx, x);
EXTRACT_WORDS(hy, ly, y);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* x**0 = 1, even if x is NaN */
if ((iy|ly) == 0)
return 1.0;
/* 1**y = 1, even if y is NaN */
if (hx == 0x3ff00000 && lx == 0)
return 1.0;
/* NaN if either arg is NaN */
if (ix > 0x7ff00000 || (ix == 0x7ff00000 && lx != 0) ||
iy > 0x7ff00000 || (iy == 0x7ff00000 && ly != 0))
return x + y;
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
yisint = 0;
if (hx < 0) {
if (iy >= 0x43400000)
yisint = 2; /* even integer y */
else if (iy >= 0x3ff00000) {
k = (iy>>20) - 0x3ff; /* exponent */
if (k > 20) {
uint32_t j = ly>>(52-k);
if ((j<<(52-k)) == ly)
yisint = 2 - (j&1);
} else if (ly == 0) {
uint32_t j = iy>>(20-k);
if ((j<<(20-k)) == iy)
yisint = 2 - (j&1);
}
}
}
/* special value of y */
if (ly == 0) {
if (iy == 0x7ff00000) { /* y is +-inf */
if (((ix-0x3ff00000)|lx) == 0) /* (-1)**+-inf is 1 */
return 1.0;
else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
return hy >= 0 ? y : 0.0;
else /* (|x|<1)**+-inf = 0,inf */
return hy >= 0 ? 0.0 : -y;
}
if (iy == 0x3ff00000) { /* y is +-1 */
if (hy >= 0)
return x;
y = 1/x;
#if FLT_EVAL_METHOD!=0
{
union {double f; uint64_t i;} u = {y};
uint64_t i = u.i & -1ULL/2;
if (i>>52 == 0 && (i&(i-1)))
FORCE_EVAL((float)y);
}
#endif
return y;
}
if (hy == 0x40000000) /* y is 2 */
return x*x;
if (hy == 0x3fe00000) { /* y is 0.5 */
if (hx >= 0) /* x >= +0 */
return sqrt(x);
}
}
ax = fabs(x);
/* special value of x */
if (lx == 0) {
if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) { /* x is +-0,+-inf,+-1 */
z = ax;
if (hy < 0) /* z = (1/|x|) */
z = 1.0/z;
if (hx < 0) {
if (((ix-0x3ff00000)|yisint) == 0) {
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
} else if (yisint == 1)
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
return z;
}
}
s = 1.0; /* sign of result */
if (hx < 0) {
if (yisint == 0) /* (x<0)**(non-int) is NaN */
return (x-x)/(x-x);
if (yisint == 1) /* (x<0)**(odd int) */
s = -1.0;
}
/* |y| is huge */
if (iy > 0x41e00000) { /* if |y| > 2**31 */
if (iy > 0x43f00000) { /* if |y| > 2**64, must o/uflow */
if (ix <= 0x3fefffff)
return hy < 0 ? huge*huge : tiny*tiny;
if (ix >= 0x3ff00000)
return hy > 0 ? huge*huge : tiny*tiny;
}
/* over/underflow if x is not close to one */
if (ix < 0x3fefffff)
return hy < 0 ? s*huge*huge : s*tiny*tiny;
if (ix > 0x3ff00000)
return hy > 0 ? s*huge*huge : s*tiny*tiny;
/* now |1-x| is tiny <= 2**-20, suffice to compute
log(x) by x-x^2/2+x^3/3-x^4/4 */
t = ax - 1.0; /* t has 20 trailing zeros */
w = (t*t)*(0.5 - t*(0.3333333333333333333333-t*0.25));
u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
v = t*ivln2_l - w*ivln2;
t1 = u + v;
SET_LOW_WORD(t1, 0);
t2 = v - (t1-u);
} else {
double ss,s2,s_h,s_l,t_h,t_l;
n = 0;
/* take care subnormal number */
if (ix < 0x00100000) {
ax *= two53;
n -= 53;
GET_HIGH_WORD(ix,ax);
}
n += ((ix)>>20) - 0x3ff;
j = ix & 0x000fffff;
/* determine interval */
ix = j | 0x3ff00000; /* normalize ix */
if (j <= 0x3988E) /* |x|<sqrt(3/2) */
k = 0;
else if (j < 0xBB67A) /* |x|<sqrt(3) */
k = 1;
else {
k = 0;
n += 1;
ix -= 0x00100000;
}
SET_HIGH_WORD(ax, ix);
/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax - bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
v = 1.0/(ax+bp[k]);
ss = u*v;
s_h = ss;
SET_LOW_WORD(s_h, 0);
/* t_h=ax+bp[k] High */
t_h = 0.0;
SET_HIGH_WORD(t_h, ((ix>>1)|0x20000000) + 0x00080000 + (k<<18));
t_l = ax - (t_h-bp[k]);
s_l = v*((u-s_h*t_h)-s_h*t_l);
/* compute log(ax) */
s2 = ss*ss;
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
r += s_l*(s_h+ss);
s2 = s_h*s_h;
t_h = 3.0 + s2 + r;
SET_LOW_WORD(t_h, 0);
t_l = r - ((t_h-3.0)-s2);
/* u+v = ss*(1+...) */
u = s_h*t_h;
v = s_l*t_h + t_l*ss;
/* 2/(3log2)*(ss+...) */
p_h = u + v;
SET_LOW_WORD(p_h, 0);
p_l = v - (p_h-u);
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
z_l = cp_l*p_h+p_l*cp + dp_l[k];
/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = (double)n;
t1 = ((z_h + z_l) + dp_h[k]) + t;
SET_LOW_WORD(t1, 0);
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
}
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
y1 = y;
SET_LOW_WORD(y1, 0);
p_l = (y-y1)*t1 + y*t2;
p_h = y1*t1;
z = p_l + p_h;
EXTRACT_WORDS(j, i, z);
if (j >= 0x40900000) { /* z >= 1024 */
if (((j-0x40900000)|i) != 0) /* if z > 1024 */
return s*huge*huge; /* overflow */
if (p_l + ovt > z - p_h)
return s*huge*huge; /* overflow */
} else if ((j&0x7fffffff) >= 0x4090cc00) { /* z <= -1075 */ // FIXME: instead of abs(j) use unsigned j
if (((j-0xc090cc00)|i) != 0) /* z < -1075 */
return s*tiny*tiny; /* underflow */
if (p_l <= z - p_h)
return s*tiny*tiny; /* underflow */
}
/*
* compute 2**(p_h+p_l)
*/
i = j & 0x7fffffff;
k = (i>>20) - 0x3ff;
n = 0;
if (i > 0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
n = j + (0x00100000>>(k+1));
k = ((n&0x7fffffff)>>20) - 0x3ff; /* new k for n */
t = 0.0;
SET_HIGH_WORD(t, n & ~(0x000fffff>>k));
n = ((n&0x000fffff)|0x00100000)>>(20-k);
if (j < 0)
n = -n;
p_h -= t;
}
t = p_l + p_h;
SET_LOW_WORD(t, 0);
u = t*lg2_h;
v = (p_l-(t-p_h))*lg2 + t*lg2_l;
z = u + v;
w = v - (z-u);
t = z*z;
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
r = (z*t1)/(t1-2.0) - (w + z*w);
z = 1.0 - (r-z);
GET_HIGH_WORD(j, z);
j += n<<20;
if ((j>>20) <= 0) /* subnormal output */
z = scalbn(z,n);
else
SET_HIGH_WORD(z, j);
return s*z;
}

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#include <float.h>
#include <math.h>
#include <stdint.h>
#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
#define EPS DBL_EPSILON
#elif FLT_EVAL_METHOD==2
#define EPS LDBL_EPSILON
#endif
static const double_t toint = 1/EPS;
double rint(double x)
{
union {double f; uint64_t i;} u = {x};
int e = u.i>>52 & 0x7ff;
int s = u.i>>63;
double_t y;
if (e >= 0x3ff+52)
return x;
if (s)
y = x - toint + toint;
else
y = x + toint - toint;
if (y == 0)
return s ? -0.0 : 0;
return y;
}

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#include <math.h>
#include <stdint.h>
double scalbn(double x, int n)
{
union {double f; uint64_t i;} u;
double_t y = x;
if (n > 1023) {
y *= 0x1p1023;
n -= 1023;
if (n > 1023) {
y *= 0x1p1023;
n -= 1023;
if (n > 1023)
n = 1023;
}
} else if (n < -1022) {
/* make sure final n < -53 to avoid double
rounding in the subnormal range */
y *= 0x1p-1022 * 0x1p53;
n += 1022 - 53;
if (n < -1022) {
y *= 0x1p-1022 * 0x1p53;
n += 1022 - 53;
if (n < -1022)
n = -1022;
}
}
u.i = (uint64_t)(0x3ff+n)<<52;
x = y * u.f;
return x;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/s_sin.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* sin(x)
* Return sine function of x.
*
* kernel function:
* __sin ... sine function on [-pi/4,pi/4]
* __cos ... cose function on [-pi/4,pi/4]
* __rem_pio2 ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
#include "libm.h"
double sin(double x)
{
double y[2];
uint32_t ix;
unsigned n;
/* High word of x. */
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
/* |x| ~< pi/4 */
if (ix <= 0x3fe921fb) {
if (ix < 0x3e500000) { /* |x| < 2**-26 */
/* raise inexact if x != 0 and underflow if subnormal*/
FORCE_EVAL(ix < 0x00100000 ? x/0x1p120f : x+0x1p120f);
return x;
}
return __sin(x, 0.0, 0);
}
/* sin(Inf or NaN) is NaN */
if (ix >= 0x7ff00000)
return x - x;
/* argument reduction needed */
n = __rem_pio2(x, y);
switch (n&3) {
case 0: return __sin(y[0], y[1], 1);
case 1: return __cos(y[0], y[1]);
case 2: return -__sin(y[0], y[1], 1);
default:
return -__cos(y[0], y[1]);
}
}

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#include "libm.h"
/* sinh(x) = (exp(x) - 1/exp(x))/2
* = (exp(x)-1 + (exp(x)-1)/exp(x))/2
* = x + x^3/6 + o(x^5)
*/
double sinh(double x)
{
union {double f; uint64_t i;} u = {.f = x};
uint32_t w;
double t, h, absx;
h = 0.5;
if (u.i >> 63)
h = -h;
/* |x| */
u.i &= (uint64_t)-1/2;
absx = u.f;
w = u.i >> 32;
/* |x| < log(DBL_MAX) */
if (w < 0x40862e42) {
t = expm1(absx);
if (w < 0x3ff00000) {
if (w < 0x3ff00000 - (26<<20))
/* note: inexact and underflow are raised by expm1 */
/* note: this branch avoids spurious underflow */
return x;
return h*(2*t - t*t/(t+1));
}
/* note: |x|>log(0x1p26)+eps could be just h*exp(x) */
return h*(t + t/(t+1));
}
/* |x| > log(DBL_MAX) or nan */
/* note: the result is stored to handle overflow */
t = 2*h*__expo2(absx);
return t;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/e_sqrt.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* sqrt(x)
* Return correctly rounded sqrt.
* ------------------------------------------
* | Use the hardware sqrt if you have one |
* ------------------------------------------
* Method:
* Bit by bit method using integer arithmetic. (Slow, but portable)
* 1. Normalization
* Scale x to y in [1,4) with even powers of 2:
* find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
* sqrt(x) = 2^k * sqrt(y)
* 2. Bit by bit computation
* Let q = sqrt(y) truncated to i bit after binary point (q = 1),
* i 0
* i+1 2
* s = 2*q , and y = 2 * ( y - q ). (1)
* i i i i
*
* To compute q from q , one checks whether
* i+1 i
*
* -(i+1) 2
* (q + 2 ) <= y. (2)
* i
* -(i+1)
* If (2) is false, then q = q ; otherwise q = q + 2 .
* i+1 i i+1 i
*
* With some algebric manipulation, it is not difficult to see
* that (2) is equivalent to
* -(i+1)
* s + 2 <= y (3)
* i i
*
* The advantage of (3) is that s and y can be computed by
* i i
* the following recurrence formula:
* if (3) is false
*
* s = s , y = y ; (4)
* i+1 i i+1 i
*
* otherwise,
* -i -(i+1)
* s = s + 2 , y = y - s - 2 (5)
* i+1 i i+1 i i
*
* One may easily use induction to prove (4) and (5).
* Note. Since the left hand side of (3) contain only i+2 bits,
* it does not necessary to do a full (53-bit) comparison
* in (3).
* 3. Final rounding
* After generating the 53 bits result, we compute one more bit.
* Together with the remainder, we can decide whether the
* result is exact, bigger than 1/2ulp, or less than 1/2ulp
* (it will never equal to 1/2ulp).
* The rounding mode can be detected by checking whether
* huge + tiny is equal to huge, and whether huge - tiny is
* equal to huge for some floating point number "huge" and "tiny".
*
* Special cases:
* sqrt(+-0) = +-0 ... exact
* sqrt(inf) = inf
* sqrt(-ve) = NaN ... with invalid signal
* sqrt(NaN) = NaN ... with invalid signal for signaling NaN
*/
#include "libm.h"
static const double tiny = 1.0e-300;
double sqrt(double x)
{
double z;
int32_t sign = (int)0x80000000;
int32_t ix0,s0,q,m,t,i;
uint32_t r,t1,s1,ix1,q1;
EXTRACT_WORDS(ix0, ix1, x);
/* take care of Inf and NaN */
if ((ix0&0x7ff00000) == 0x7ff00000) {
return x*x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf, sqrt(-inf)=sNaN */
}
/* take care of zero */
if (ix0 <= 0) {
if (((ix0&~sign)|ix1) == 0)
return x; /* sqrt(+-0) = +-0 */
if (ix0 < 0)
return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
}
/* normalize x */
m = ix0>>20;
if (m == 0) { /* subnormal x */
while (ix0 == 0) {
m -= 21;
ix0 |= (ix1>>11);
ix1 <<= 21;
}
for (i=0; (ix0&0x00100000) == 0; i++)
ix0<<=1;
m -= i - 1;
ix0 |= ix1>>(32-i);
ix1 <<= i;
}
m -= 1023; /* unbias exponent */
ix0 = (ix0&0x000fffff)|0x00100000;
if (m & 1) { /* odd m, double x to make it even */
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
}
m >>= 1; /* m = [m/2] */
/* generate sqrt(x) bit by bit */
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
r = 0x00200000; /* r = moving bit from right to left */
while (r != 0) {
t = s0 + r;
if (t <= ix0) {
s0 = t + r;
ix0 -= t;
q += r;
}
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
r >>= 1;
}
r = sign;
while (r != 0) {
t1 = s1 + r;
t = s0;
if (t < ix0 || (t == ix0 && t1 <= ix1)) {
s1 = t1 + r;
if ((t1&sign) == sign && (s1&sign) == 0)
s0++;
ix0 -= t;
if (ix1 < t1)
ix0--;
ix1 -= t1;
q1 += r;
}
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
r >>= 1;
}
/* use floating add to find out rounding direction */
if ((ix0|ix1) != 0) {
z = 1.0 - tiny; /* raise inexact flag */
if (z >= 1.0) {
z = 1.0 + tiny;
if (q1 == (uint32_t)0xffffffff) {
q1 = 0;
q++;
} else if (z > 1.0) {
if (q1 == (uint32_t)0xfffffffe)
q++;
q1 += 2;
} else
q1 += q1 & 1;
}
}
ix0 = (q>>1) + 0x3fe00000;
ix1 = q1>>1;
if (q&1)
ix1 |= sign;
ix0 += m << 20;
INSERT_WORDS(z, ix0, ix1);
return z;
}

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/* origin: FreeBSD /usr/src/lib/msun/src/s_tan.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* tan(x)
* Return tangent function of x.
*
* kernel function:
* __tan ... tangent function on [-pi/4,pi/4]
* __rem_pio2 ... argument reduction routine
*
* Method.
* Let S,C and T denote the sin, cos and tan respectively on
* [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
* in [-pi/4 , +pi/4], and let n = k mod 4.
* We have
*
* n sin(x) cos(x) tan(x)
* ----------------------------------------------------------
* 0 S C T
* 1 C -S -1/T
* 2 -S -C T
* 3 -C S -1/T
* ----------------------------------------------------------
*
* Special cases:
* Let trig be any of sin, cos, or tan.
* trig(+-INF) is NaN, with signals;
* trig(NaN) is that NaN;
*
* Accuracy:
* TRIG(x) returns trig(x) nearly rounded
*/
#include "libm.h"
double tan(double x)
{
double y[2];
uint32_t ix;
unsigned n;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
/* |x| ~< pi/4 */
if (ix <= 0x3fe921fb) {
if (ix < 0x3e400000) { /* |x| < 2**-27 */
/* raise inexact if x!=0 and underflow if subnormal */
FORCE_EVAL(ix < 0x00100000 ? x/0x1p120f : x+0x1p120f);
return x;
}
return __tan(x, 0.0, 0);
}
/* tan(Inf or NaN) is NaN */
if (ix >= 0x7ff00000)
return x - x;
/* argument reduction */
n = __rem_pio2(x, y);
return __tan(y[0], y[1], n&1);
}

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#include <math.h>
double tanh(double x) {
return sinh(x) / cosh(x);
}

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/*
"A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964)
"Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001)
"An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004)
approximation method:
(x - 0.5) S(x)
Gamma(x) = (x + g - 0.5) * ----------------
exp(x + g - 0.5)
with
a1 a2 a3 aN
S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ]
x + 1 x + 2 x + 3 x + N
with a0, a1, a2, a3,.. aN constants which depend on g.
for x < 0 the following reflection formula is used:
Gamma(x)*Gamma(-x) = -pi/(x sin(pi x))
most ideas and constants are from boost and python
*/
#include "libm.h"
static const double pi = 3.141592653589793238462643383279502884;
/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
static double sinpi(double x)
{
int n;
/* argument reduction: x = |x| mod 2 */
/* spurious inexact when x is odd int */
x = x * 0.5;
x = 2 * (x - floor(x));
/* reduce x into [-.25,.25] */
n = 4 * x;
n = (n+1)/2;
x -= n * 0.5;
x *= pi;
switch (n) {
default: /* case 4 */
case 0:
return __sin(x, 0, 0);
case 1:
return __cos(x, 0);
case 2:
return __sin(-x, 0, 0);
case 3:
return -__cos(x, 0);
}
}
#define N 12
//static const double g = 6.024680040776729583740234375;
static const double gmhalf = 5.524680040776729583740234375;
static const double Snum[N+1] = {
23531376880.410759688572007674451636754734846804940,
42919803642.649098768957899047001988850926355848959,
35711959237.355668049440185451547166705960488635843,
17921034426.037209699919755754458931112671403265390,
6039542586.3520280050642916443072979210699388420708,
1439720407.3117216736632230727949123939715485786772,
248874557.86205415651146038641322942321632125127801,
31426415.585400194380614231628318205362874684987640,
2876370.6289353724412254090516208496135991145378768,
186056.26539522349504029498971604569928220784236328,
8071.6720023658162106380029022722506138218516325024,
210.82427775157934587250973392071336271166969580291,
2.5066282746310002701649081771338373386264310793408,
};
static const double Sden[N+1] = {
0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535,
2637558, 357423, 32670, 1925, 66, 1,
};
/* n! for small integer n */
static const double fact[] = {
1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0,
479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0,
355687428096000.0, 6402373705728000.0, 121645100408832000.0,
2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0,
};
/* S(x) rational function for positive x */
static double S(double x)
{
double_t num = 0, den = 0;
int i;
/* to avoid overflow handle large x differently */
if (x < 8)
for (i = N; i >= 0; i--) {
num = num * x + Snum[i];
den = den * x + Sden[i];
}
else
for (i = 0; i <= N; i++) {
num = num / x + Snum[i];
den = den / x + Sden[i];
}
return num/den;
}
double tgamma(double x)
{
union {double f; uint64_t i;} u = {x};
double absx, y;
double_t dy, z, r;
uint32_t ix = u.i>>32 & 0x7fffffff;
int sign = u.i>>63;
/* special cases */
if (ix >= 0x7ff00000)
/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
return x + INFINITY;
if (ix < (0x3ff-54)<<20)
/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
return 1/x;
/* integer arguments */
/* raise inexact when non-integer */
if (x == floor(x)) {
if (sign)
return 0/0.0;
if (x <= sizeof fact/sizeof *fact)
return fact[(int)x - 1];
}
/* x >= 172: tgamma(x)=inf with overflow */
/* x =< -184: tgamma(x)=+-0 with underflow */
if (ix >= 0x40670000) { /* |x| >= 184 */
if (sign) {
FORCE_EVAL((float)(0x1p-126/x));
if (floor(x) * 0.5 == floor(x * 0.5))
return 0;
return -0.0;
}
x *= 0x1p1023;
return x;
}
absx = sign ? -x : x;
/* handle the error of x + g - 0.5 */
y = absx + gmhalf;
if (absx > gmhalf) {
dy = y - absx;
dy -= gmhalf;
} else {
dy = y - gmhalf;
dy -= absx;
}
z = absx - 0.5;
r = S(absx) * exp(-y);
if (x < 0) {
/* reflection formula for negative x */
/* sinpi(absx) is not 0, integers are already handled */
r = -pi / (sinpi(absx) * absx * r);
dy = -dy;
z = -z;
}
r += dy * (gmhalf+0.5) * r / y;
z = pow(y, 0.5*z);
y = r * z * z;
return y;
}
#if 1
double __lgamma_r(double x, int *sign)
{
double r, absx;
*sign = 1;
/* special cases */
if (!isfinite(x))
/* lgamma(nan)=nan, lgamma(+-inf)=inf */
return x*x;
/* integer arguments */
if (x == floor(x) && x <= 2) {
/* n <= 0: lgamma(n)=inf with divbyzero */
/* n == 1,2: lgamma(n)=0 */
if (x <= 0)
return 1/0.0;
return 0;
}
absx = fabs(x);
/* lgamma(x) ~ -log(|x|) for tiny |x| */
if (absx < 0x1p-54) {
*sign = 1 - 2*!!signbit(x);
return -log(absx);
}
/* use tgamma for smaller |x| */
if (absx < 128) {
x = tgamma(x);
*sign = 1 - 2*!!signbit(x);
return log(fabs(x));
}
/* second term (log(S)-g) could be more precise here.. */
/* or with stirling: (|x|-0.5)*(log(|x|)-1) + poly(1/|x|) */
r = (absx-0.5)*(log(absx+gmhalf)-1) + (log(S(absx)) - (gmhalf+0.5));
if (x < 0) {
/* reflection formula for negative x */
x = sinpi(absx);
*sign = 2*!!signbit(x) - 1;
r = log(pi/(fabs(x)*absx)) - r;
}
return r;
}
//weak_alias(__lgamma_r, lgamma_r);
#endif

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#include "libm.h"
double trunc(double x)
{
union {double f; uint64_t i;} u = {x};
int e = (int)(u.i >> 52 & 0x7ff) - 0x3ff + 12;
uint64_t m;
if (e >= 52 + 12)
return x;
if (e < 12)
e = 1;
m = -1ULL >> e;
if ((u.i & m) == 0)
return x;
FORCE_EVAL(x + 0x1p120f);
u.i &= ~m;
return u.f;
}