kopia lustrzana https://github.com/micropython/micropython
lib: Add libm_dbl, a double-precision math library, from musl-1.1.16.
Damien George
rodzic
409fc8f9c1
commit
045116551e
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@ -0,0 +1,32 @@
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This directory contains source code for the standard double-precision math
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functions.
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The files lgamma.c, log10.c and tanh.c are too small to have a meaningful
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copyright or license.
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The rest of the files in this directory are copied from the musl library,
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v1.1.16, and, unless otherwise stated in the individual file, have the
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following copyright and MIT license:
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----------------------------------------------------------------------
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Copyright © 2005-2014 Rich Felker, et al.
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Permission is hereby granted, free of charge, to any person obtaining
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a copy of this software and associated documentation files (the
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"Software"), to deal in the Software without restriction, including
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without limitation the rights to use, copy, modify, merge, publish,
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distribute, sublicense, and/or sell copies of the Software, and to
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permit persons to whom the Software is furnished to do so, subject to
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the following conditions:
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The above copyright notice and this permission notice shall be
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included in all copies or substantial portions of the Software.
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
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MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
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IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
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CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
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TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
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SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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----------------------------------------------------------------------
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@ -0,0 +1,71 @@
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/* origin: FreeBSD /usr/src/lib/msun/src/k_cos.c */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/*
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* __cos( x, y )
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* kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
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* Input x is assumed to be bounded by ~pi/4 in magnitude.
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* Input y is the tail of x.
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*
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* Algorithm
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* 1. Since cos(-x) = cos(x), we need only to consider positive x.
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* 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
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* 3. cos(x) is approximated by a polynomial of degree 14 on
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* [0,pi/4]
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* 4 14
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* cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
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* where the remez error is
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*
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* | 2 4 6 8 10 12 14 | -58
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* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
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* | |
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*
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* 4 6 8 10 12 14
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* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
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* cos(x) ~ 1 - x*x/2 + r
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* since cos(x+y) ~ cos(x) - sin(x)*y
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* ~ cos(x) - x*y,
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* a correction term is necessary in cos(x) and hence
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* cos(x+y) = 1 - (x*x/2 - (r - x*y))
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* For better accuracy, rearrange to
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* cos(x+y) ~ w + (tmp + (r-x*y))
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* where w = 1 - x*x/2 and tmp is a tiny correction term
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* (1 - x*x/2 == w + tmp exactly in infinite precision).
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* The exactness of w + tmp in infinite precision depends on w
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* and tmp having the same precision as x. If they have extra
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* precision due to compiler bugs, then the extra precision is
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* only good provided it is retained in all terms of the final
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* expression for cos(). Retention happens in all cases tested
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* under FreeBSD, so don't pessimize things by forcibly clipping
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* any extra precision in w.
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*/
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#include "libm.h"
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static const double
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C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
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C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
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C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
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C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
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C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
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C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
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double __cos(double x, double y)
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{
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double_t hz,z,r,w;
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z = x*x;
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w = z*z;
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r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
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hz = 0.5*z;
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w = 1.0-hz;
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return w + (((1.0-w)-hz) + (z*r-x*y));
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}
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#include "libm.h"
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/* k is such that k*ln2 has minimal relative error and x - kln2 > log(DBL_MIN) */
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static const int k = 2043;
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static const double kln2 = 0x1.62066151add8bp+10;
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/* exp(x)/2 for x >= log(DBL_MAX), slightly better than 0.5*exp(x/2)*exp(x/2) */
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double __expo2(double x)
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{
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double scale;
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/* note that k is odd and scale*scale overflows */
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INSERT_WORDS(scale, (uint32_t)(0x3ff + k/2) << 20, 0);
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/* exp(x - k ln2) * 2**(k-1) */
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return exp(x - kln2) * scale * scale;
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}
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#include <math.h>
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#include <stdint.h>
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int __fpclassifyd(double x)
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{
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union {double f; uint64_t i;} u = {x};
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int e = u.i>>52 & 0x7ff;
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if (!e) return u.i<<1 ? FP_SUBNORMAL : FP_ZERO;
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if (e==0x7ff) return u.i<<12 ? FP_NAN : FP_INFINITE;
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return FP_NORMAL;
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}
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/* origin: FreeBSD /usr/src/lib/msun/src/e_rem_pio2.c */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*
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* Optimized by Bruce D. Evans.
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*/
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/* __rem_pio2(x,y)
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*
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* return the remainder of x rem pi/2 in y[0]+y[1]
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* use __rem_pio2_large() for large x
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*/
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#include "libm.h"
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#if FLT_EVAL_METHOD==0 || FLT_EVAL_METHOD==1
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#define EPS DBL_EPSILON
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#elif FLT_EVAL_METHOD==2
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#define EPS LDBL_EPSILON
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#endif
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/*
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* invpio2: 53 bits of 2/pi
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* pio2_1: first 33 bit of pi/2
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* pio2_1t: pi/2 - pio2_1
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* pio2_2: second 33 bit of pi/2
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* pio2_2t: pi/2 - (pio2_1+pio2_2)
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* pio2_3: third 33 bit of pi/2
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* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
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*/
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static const double
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toint = 1.5/EPS,
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invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
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pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
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pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
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pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
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pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
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pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
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pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
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/* caller must handle the case when reduction is not needed: |x| ~<= pi/4 */
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int __rem_pio2(double x, double *y)
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{
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union {double f; uint64_t i;} u = {x};
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double_t z,w,t,r,fn;
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double tx[3],ty[2];
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uint32_t ix;
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int sign, n, ex, ey, i;
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sign = u.i>>63;
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ix = u.i>>32 & 0x7fffffff;
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if (ix <= 0x400f6a7a) { /* |x| ~<= 5pi/4 */
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if ((ix & 0xfffff) == 0x921fb) /* |x| ~= pi/2 or 2pi/2 */
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goto medium; /* cancellation -- use medium case */
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if (ix <= 0x4002d97c) { /* |x| ~<= 3pi/4 */
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if (!sign) {
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z = x - pio2_1; /* one round good to 85 bits */
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y[0] = z - pio2_1t;
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y[1] = (z-y[0]) - pio2_1t;
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return 1;
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} else {
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z = x + pio2_1;
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y[0] = z + pio2_1t;
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y[1] = (z-y[0]) + pio2_1t;
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return -1;
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}
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} else {
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if (!sign) {
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z = x - 2*pio2_1;
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y[0] = z - 2*pio2_1t;
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y[1] = (z-y[0]) - 2*pio2_1t;
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return 2;
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} else {
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z = x + 2*pio2_1;
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y[0] = z + 2*pio2_1t;
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y[1] = (z-y[0]) + 2*pio2_1t;
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return -2;
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}
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}
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}
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if (ix <= 0x401c463b) { /* |x| ~<= 9pi/4 */
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if (ix <= 0x4015fdbc) { /* |x| ~<= 7pi/4 */
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if (ix == 0x4012d97c) /* |x| ~= 3pi/2 */
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goto medium;
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if (!sign) {
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z = x - 3*pio2_1;
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y[0] = z - 3*pio2_1t;
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y[1] = (z-y[0]) - 3*pio2_1t;
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return 3;
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} else {
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z = x + 3*pio2_1;
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y[0] = z + 3*pio2_1t;
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y[1] = (z-y[0]) + 3*pio2_1t;
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return -3;
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}
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} else {
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if (ix == 0x401921fb) /* |x| ~= 4pi/2 */
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goto medium;
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if (!sign) {
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z = x - 4*pio2_1;
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y[0] = z - 4*pio2_1t;
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y[1] = (z-y[0]) - 4*pio2_1t;
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return 4;
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} else {
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z = x + 4*pio2_1;
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y[0] = z + 4*pio2_1t;
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y[1] = (z-y[0]) + 4*pio2_1t;
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return -4;
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}
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}
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}
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if (ix < 0x413921fb) { /* |x| ~< 2^20*(pi/2), medium size */
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medium:
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/* rint(x/(pi/2)), Assume round-to-nearest. */
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fn = (double_t)x*invpio2 + toint - toint;
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n = (int32_t)fn;
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r = x - fn*pio2_1;
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w = fn*pio2_1t; /* 1st round, good to 85 bits */
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y[0] = r - w;
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u.f = y[0];
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ey = u.i>>52 & 0x7ff;
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ex = ix>>20;
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if (ex - ey > 16) { /* 2nd round, good to 118 bits */
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t = r;
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w = fn*pio2_2;
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r = t - w;
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w = fn*pio2_2t - ((t-r)-w);
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y[0] = r - w;
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u.f = y[0];
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ey = u.i>>52 & 0x7ff;
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if (ex - ey > 49) { /* 3rd round, good to 151 bits, covers all cases */
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t = r;
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w = fn*pio2_3;
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r = t - w;
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w = fn*pio2_3t - ((t-r)-w);
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y[0] = r - w;
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}
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}
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y[1] = (r - y[0]) - w;
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return n;
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}
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/*
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* all other (large) arguments
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*/
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if (ix >= 0x7ff00000) { /* x is inf or NaN */
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y[0] = y[1] = x - x;
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return 0;
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}
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/* set z = scalbn(|x|,-ilogb(x)+23) */
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u.f = x;
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u.i &= (uint64_t)-1>>12;
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u.i |= (uint64_t)(0x3ff + 23)<<52;
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z = u.f;
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for (i=0; i < 2; i++) {
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tx[i] = (double)(int32_t)z;
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z = (z-tx[i])*0x1p24;
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}
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tx[i] = z;
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/* skip zero terms, first term is non-zero */
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while (tx[i] == 0.0)
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i--;
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n = __rem_pio2_large(tx,ty,(int)(ix>>20)-(0x3ff+23),i+1,1);
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if (sign) {
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y[0] = -ty[0];
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y[1] = -ty[1];
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return -n;
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}
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y[0] = ty[0];
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y[1] = ty[1];
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return n;
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}
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@ -0,0 +1,442 @@
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/* origin: FreeBSD /usr/src/lib/msun/src/k_rem_pio2.c */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
|
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* software is freely granted, provided that this notice
|
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* is preserved.
|
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* ====================================================
|
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*/
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/*
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* __rem_pio2_large(x,y,e0,nx,prec)
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* double x[],y[]; int e0,nx,prec;
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*
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* __rem_pio2_large return the last three digits of N with
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* y = x - N*pi/2
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* so that |y| < pi/2.
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*
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* The method is to compute the integer (mod 8) and fraction parts of
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* (2/pi)*x without doing the full multiplication. In general we
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* skip the part of the product that are known to be a huge integer (
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* more accurately, = 0 mod 8 ). Thus the number of operations are
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* independent of the exponent of the input.
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*
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* (2/pi) is represented by an array of 24-bit integers in ipio2[].
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*
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* Input parameters:
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* x[] The input value (must be positive) is broken into nx
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* pieces of 24-bit integers in double precision format.
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* x[i] will be the i-th 24 bit of x. The scaled exponent
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* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
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* match x's up to 24 bits.
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*
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* Example of breaking a double positive z into x[0]+x[1]+x[2]:
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* e0 = ilogb(z)-23
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* z = scalbn(z,-e0)
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* for i = 0,1,2
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* x[i] = floor(z)
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* z = (z-x[i])*2**24
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*
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*
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* y[] ouput result in an array of double precision numbers.
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* The dimension of y[] is:
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* 24-bit precision 1
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* 53-bit precision 2
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* 64-bit precision 2
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* 113-bit precision 3
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* The actual value is the sum of them. Thus for 113-bit
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* precison, one may have to do something like:
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*
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* long double t,w,r_head, r_tail;
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* t = (long double)y[2] + (long double)y[1];
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* w = (long double)y[0];
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* r_head = t+w;
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* r_tail = w - (r_head - t);
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*
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* e0 The exponent of x[0]. Must be <= 16360 or you need to
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* expand the ipio2 table.
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*
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* nx dimension of x[]
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*
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* prec an integer indicating the precision:
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* 0 24 bits (single)
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* 1 53 bits (double)
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* 2 64 bits (extended)
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* 3 113 bits (quad)
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*
|
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* External function:
|
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* double scalbn(), floor();
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*
|
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*
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* Here is the description of some local variables:
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*
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* jk jk+1 is the initial number of terms of ipio2[] needed
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* in the computation. The minimum and recommended value
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* for jk is 3,4,4,6 for single, double, extended, and quad.
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* jk+1 must be 2 larger than you might expect so that our
|
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* recomputation test works. (Up to 24 bits in the integer
|
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* part (the 24 bits of it that we compute) and 23 bits in
|
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* the fraction part may be lost to cancelation before we
|
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* recompute.)
|
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*
|
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* jz local integer variable indicating the number of
|
||||
* terms of ipio2[] used.
|
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*
|
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* jx nx - 1
|
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*
|
||||
* 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
|
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* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,12 @@
|
|||
#include "libm.h"
|
||||
|
||||
int __signbitd(double x)
|
||||
{
|
||||
union {
|
||||
double d;
|
||||
uint64_t i;
|
||||
} y = { x };
|
||||
return y.i>>63;
|
||||
}
|
||||
|
||||
|
||||
|
|
@ -0,0 +1,64 @@
|
|||
/* 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);
|
||||
}
|
||||
|
|
@ -0,0 +1,110 @@
|
|||
/* 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);
|
||||
}
|
||||
|
|
@ -0,0 +1,101 @@
|
|||
/* 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);
|
||||
}
|
||||
|
|
@ -0,0 +1,24 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,107 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,28 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,116 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,107 @@
|
|||
/* 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(-,-) */
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,29 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,31 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,77 @@
|
|||
/* 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);
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,40 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,273 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,134 @@
|
|||
/* 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);
|
||||
}
|
||||
|
|
@ -0,0 +1,201 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,31 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,68 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,23 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,6 @@
|
|||
#include <math.h>
|
||||
|
||||
double ldexp(double x, int n)
|
||||
{
|
||||
return scalbn(x, n);
|
||||
}
|
||||
|
|
@ -0,0 +1,8 @@
|
|||
#include <math.h>
|
||||
|
||||
double __lgamma_r(double, int*);
|
||||
|
||||
double lgamma(double x) {
|
||||
int sign;
|
||||
return __lgamma_r(x, &sign);
|
||||
}
|
||||
|
|
@ -0,0 +1,96 @@
|
|||
// 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);
|
||||
|
|
@ -0,0 +1,118 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,7 @@
|
|||
#include <math.h>
|
||||
|
||||
static const double _M_LN10 = 2.302585092994046;
|
||||
|
||||
double log10(double x) {
|
||||
return log(x) / (double)_M_LN10;
|
||||
}
|
||||
|
|
@ -0,0 +1,122 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,34 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,20 @@
|
|||
//#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,328 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,28 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,33 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,78 @@
|
|||
/* 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]);
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,39 @@
|
|||
#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;
|
||||
}
|
||||
|
|
@ -0,0 +1,185 @@
|
|||
/* 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;
|
||||
}
|
||||
|
|
@ -0,0 +1,70 @@
|
|||
/* 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);
|
||||
}
|
||||
|
|
@ -0,0 +1,5 @@
|
|||
#include <math.h>
|
||||
|
||||
double tanh(double x) {
|
||||
return sinh(x) / cosh(x);
|
||||
}
|
||||
|
|
@ -0,0 +1,222 @@
|
|||
/*
|
||||
"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
|
||||
|
|
@ -0,0 +1,19 @@
|
|||
#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;
|
||||
}
|
||||
Ładowanie…
Reference in New Issue