mirror of
https://github.com/jart/cosmopolitan.git
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3a8e01a77a
Somehow or another, I previously had missed `BUILD.mk` files. In the process I found a few straggler cases where the modeline was different from the file, including one very involved manual fix where a file had been treated like it was ts=2 and ts=8 on separate occasions. The commit history in the PR shows the gory details; the BUILD.mk was automated, everything else was mostly manual.
1064 lines
29 KiB
C
1064 lines
29 KiB
C
/*-*- mode:c;indent-tabs-mode:nil;c-basic-offset:2;tab-width:8;coding:utf-8 -*-│
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│ vi: set et ft=c ts=2 sts=2 sw=2 fenc=utf-8 :vi │
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╞══════════════════════════════════════════════════════════════════════════════╡
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│ Copyright 2021 Justine Alexandra Roberts Tunney │
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│ │
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│ Permission to use, copy, modify, and/or distribute this software for │
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│ any purpose with or without fee is hereby granted, provided that the │
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│ above copyright notice and this permission notice appear in all copies. │
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│ │
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│ THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL │
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│ WARRANTIES WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED │
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│ WARRANTIES OF MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE │
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||
│ AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL │
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│ DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR │
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│ PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER │
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│ TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR │
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│ PERFORMANCE OF THIS SOFTWARE. │
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╚─────────────────────────────────────────────────────────────────────────────*/
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#include "libc/errno.h"
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#include "libc/math.h"
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#include "libc/tinymath/internal.h"
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#if !(LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024)
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#ifdef __x86_64__
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||
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/**
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* Returns 𝑥^𝑦.
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* @note should take ~56ns
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*/
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long double powl(long double x, long double y) {
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||
long double t, u;
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if (!isunordered(x, y)) {
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if (!isinf(y)) {
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if (!isinf(x)) {
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if (x) {
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if (y) {
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if (x < 0 && y != truncl(y)) {
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#ifndef __NO_MATH_ERRNO__
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errno = EDOM;
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#endif
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return NAN;
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}
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asm("fyl2x" : "=t"(u) : "0"(fabsl(x)), "u"(y) : "st(1)");
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asm("fprem" : "=t"(t) : "0"(u), "u"(1.L));
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asm("f2xm1" : "=t"(t) : "0"(t));
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asm("fscale" : "=t"(t) : "0"(t + 1), "u"(u));
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if (signbit(x)) {
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if (y != truncl(y)) return -NAN;
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if ((int64_t)y & 1) t = -t;
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}
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return t;
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} else {
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return 1;
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}
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} else if (y > 0) {
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if (signbit(x) && y == truncl(y) && ((int64_t)y & 1)) {
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return -0.;
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} else {
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return 0;
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}
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} else if (!y) {
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return 1;
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} else {
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#ifndef __NO_MATH_ERRNO__
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errno = ERANGE;
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#endif
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if (y == truncl(y) && ((int64_t)y & 1)) {
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return copysignl(INFINITY, x);
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} else {
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return INFINITY;
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}
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}
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} else if (signbit(x)) {
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if (!y) return 1;
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x = y < 0 ? 0 : INFINITY;
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if (y == truncl(y) && ((int64_t)y & 1)) x = -x;
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return x;
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} else if (y < 0) {
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return 0;
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} else if (y > 0) {
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return INFINITY;
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} else {
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return 1;
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}
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} else {
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x = fabsl(x);
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if (x < 1) return signbit(y) ? INFINITY : 0;
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if (x > 1) return signbit(y) ? 0 : INFINITY;
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return 1;
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}
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} else if (!y || x == 1) {
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return 1;
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} else {
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return NAN;
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}
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}
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#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
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asm(".ident\t\"\\n\\n\
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OpenBSD libm (ISC License)\\n\
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Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>\"");
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asm(".ident\t\"\\n\\n\
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Musl libc (MIT License)\\n\
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Copyright 2005-2014 Rich Felker, et. al.\"");
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asm(".include \"libc/disclaimer.inc\"");
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// clang-format off
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/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */
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/*
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* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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*
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* Permission to use, copy, modify, and distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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||
* copyright notice and this permission notice appear in all copies.
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||
*
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||
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
|
||
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
|
||
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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||
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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*/
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/* powl.c
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*
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* Power function, long double precision
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*
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*
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* SYNOPSIS:
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*
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* long double x, y, z, powl();
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*
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* z = powl( x, y );
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*
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*
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* DESCRIPTION:
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*
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* Computes x raised to the yth power. Analytically,
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*
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* x**y = exp( y log(x) ).
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*
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* Following Cody and Waite, this program uses a lookup table
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* of 2**-i/32 and pseudo extended precision arithmetic to
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* obtain several extra bits of accuracy in both the logarithm
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* and the exponential.
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*
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*
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* ACCURACY:
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*
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* The relative error of pow(x,y) can be estimated
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* by y dl ln(2), where dl is the absolute error of
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* the internally computed base 2 logarithm. At the ends
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* of the approximation interval the logarithm equal 1/32
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* and its relative error is about 1 lsb = 1.1e-19. Hence
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* the predicted relative error in the result is 2.3e-21 y .
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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*
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* IEEE +-1000 40000 2.8e-18 3.7e-19
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* .001 < x < 1000, with log(x) uniformly distributed.
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* -1000 < y < 1000, y uniformly distributed.
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*
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* IEEE 0,8700 60000 6.5e-18 1.0e-18
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* 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
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*
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*
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* ERROR MESSAGES:
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||
*
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* message condition value returned
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* pow overflow x**y > MAXNUM INFINITY
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* pow underflow x**y < 1/MAXNUM 0.0
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* pow domain x<0 and y noninteger 0.0
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*
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*/
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/* Table size */
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#define NXT 32
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/* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z)
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* on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1
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*/
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static const long double P[] = {
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8.3319510773868690346226E-4L,
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4.9000050881978028599627E-1L,
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1.7500123722550302671919E0L,
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1.4000100839971580279335E0L,
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};
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static const long double Q[] = {
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/* 1.0000000000000000000000E0L,*/
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5.2500282295834889175431E0L,
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8.4000598057587009834666E0L,
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4.2000302519914740834728E0L,
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};
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/* A[i] = 2^(-i/32), rounded to IEEE long double precision.
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* If i is even, A[i] + B[i/2] gives additional accuracy.
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*/
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static const long double A[33] = {
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1.0000000000000000000000E0L,
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9.7857206208770013448287E-1L,
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9.5760328069857364691013E-1L,
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9.3708381705514995065011E-1L,
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9.1700404320467123175367E-1L,
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8.9735453750155359320742E-1L,
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8.7812608018664974155474E-1L,
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8.5930964906123895780165E-1L,
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8.4089641525371454301892E-1L,
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8.2287773907698242225554E-1L,
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8.0524516597462715409607E-1L,
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7.8799042255394324325455E-1L,
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7.7110541270397041179298E-1L,
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7.5458221379671136985669E-1L,
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7.3841307296974965571198E-1L,
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7.2259040348852331001267E-1L,
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7.0710678118654752438189E-1L,
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6.9195494098191597746178E-1L,
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6.7712777346844636413344E-1L,
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6.6261832157987064729696E-1L,
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6.4841977732550483296079E-1L,
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6.3452547859586661129850E-1L,
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6.2092890603674202431705E-1L,
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6.0762367999023443907803E-1L,
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5.9460355750136053334378E-1L,
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5.8186242938878875689693E-1L,
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5.6939431737834582684856E-1L,
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5.5719337129794626814472E-1L,
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5.4525386633262882960438E-1L,
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5.3357020033841180906486E-1L,
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5.2213689121370692017331E-1L,
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5.1094857432705833910408E-1L,
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5.0000000000000000000000E-1L,
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};
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static const long double B[17] = {
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0.0000000000000000000000E0L,
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2.6176170809902549338711E-20L,
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-1.0126791927256478897086E-20L,
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1.3438228172316276937655E-21L,
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1.2207982955417546912101E-20L,
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-6.3084814358060867200133E-21L,
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1.3164426894366316434230E-20L,
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-1.8527916071632873716786E-20L,
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1.8950325588932570796551E-20L,
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1.5564775779538780478155E-20L,
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6.0859793637556860974380E-21L,
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-2.0208749253662532228949E-20L,
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1.4966292219224761844552E-20L,
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3.3540909728056476875639E-21L,
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-8.6987564101742849540743E-22L,
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-1.2327176863327626135542E-20L,
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0.0000000000000000000000E0L,
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};
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/* 2^x = 1 + x P(x),
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* on the interval -1/32 <= x <= 0
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*/
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static const long double R[] = {
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1.5089970579127659901157E-5L,
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1.5402715328927013076125E-4L,
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1.3333556028915671091390E-3L,
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9.6181291046036762031786E-3L,
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5.5504108664798463044015E-2L,
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2.4022650695910062854352E-1L,
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6.9314718055994530931447E-1L,
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};
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#define MEXP (NXT*16384.0L)
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/* The following if denormal numbers are supported, else -MEXP: */
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#define MNEXP (-NXT*(16384.0L+64.0L))
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/* log2(e) - 1 */
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#define LOG2EA 0.44269504088896340735992L
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#define F W
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#define Fa Wa
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#define Fb Wb
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#define G W
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#define Ga Wa
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#define Gb u
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#define H W
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#define Ha Wb
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#define Hb Wb
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static const long double MAXLOGL = 1.1356523406294143949492E4L;
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static const long double MINLOGL = -1.13994985314888605586758E4L;
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static const long double LOGE2L = 6.9314718055994530941723E-1L;
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static const long double huge = 0x1p10000L;
|
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/* XXX Prevent gcc from erroneously constant folding this. */
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static const volatile long double twom10000 = 0x1p-10000L;
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|
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static long double reducl(long double);
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static long double powil(long double, int);
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long double powl(long double x, long double y)
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{
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/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
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int i, nflg, iyflg, yoddint;
|
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long e;
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volatile long double z=0;
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long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
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|
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/* make sure no invalid exception is raised by nan comparision */
|
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if (isnan(x)) {
|
||
if (!isnan(y) && y == 0.0)
|
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return 1.0;
|
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return x;
|
||
}
|
||
if (isnan(y)) {
|
||
if (x == 1.0)
|
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return 1.0;
|
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return y;
|
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}
|
||
if (x == 1.0)
|
||
return 1.0; /* 1**y = 1, even if y is nan */
|
||
if (x == -1.0 && !isfinite(y))
|
||
return 1.0; /* -1**inf = 1 */
|
||
if (y == 0.0)
|
||
return 1.0; /* x**0 = 1, even if x is nan */
|
||
if (y == 1.0)
|
||
return x;
|
||
if (y >= LDBL_MAX) {
|
||
if (x > 1.0 || x < -1.0)
|
||
return INFINITY;
|
||
if (x != 0.0)
|
||
return 0.0;
|
||
}
|
||
if (y <= -LDBL_MAX) {
|
||
if (x > 1.0 || x < -1.0)
|
||
return 0.0;
|
||
if (x != 0.0 || y == -INFINITY)
|
||
return INFINITY;
|
||
}
|
||
if (x >= LDBL_MAX) {
|
||
if (y > 0.0)
|
||
return INFINITY;
|
||
return 0.0;
|
||
}
|
||
|
||
w = floorl(y);
|
||
|
||
/* Set iyflg to 1 if y is an integer. */
|
||
iyflg = 0;
|
||
if (w == y)
|
||
iyflg = 1;
|
||
|
||
/* Test for odd integer y. */
|
||
yoddint = 0;
|
||
if (iyflg) {
|
||
ya = fabsl(y);
|
||
ya = floorl(0.5 * ya);
|
||
yb = 0.5 * fabsl(w);
|
||
if( ya != yb )
|
||
yoddint = 1;
|
||
}
|
||
|
||
if (x <= -LDBL_MAX) {
|
||
if (y > 0.0) {
|
||
if (yoddint)
|
||
return -INFINITY;
|
||
return INFINITY;
|
||
}
|
||
if (y < 0.0) {
|
||
if (yoddint)
|
||
return -0.0;
|
||
return 0.0;
|
||
}
|
||
}
|
||
nflg = 0; /* (x<0)**(odd int) */
|
||
if (x <= 0.0) {
|
||
if (x == 0.0) {
|
||
if (y < 0.0) {
|
||
if (signbit(x) && yoddint)
|
||
/* (-0.0)**(-odd int) = -inf, divbyzero */
|
||
return -1.0/0.0;
|
||
/* (+-0.0)**(negative) = inf, divbyzero */
|
||
return 1.0/0.0;
|
||
}
|
||
if (signbit(x) && yoddint)
|
||
return -0.0;
|
||
return 0.0;
|
||
}
|
||
if (iyflg == 0)
|
||
return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
|
||
/* (x<0)**(integer) */
|
||
if (yoddint)
|
||
nflg = 1; /* negate result */
|
||
x = -x;
|
||
}
|
||
/* (+integer)**(integer) */
|
||
if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
|
||
w = powil(x, (int)y);
|
||
return nflg ? -w : w;
|
||
}
|
||
|
||
/* separate significand from exponent */
|
||
x = frexpl(x, &i);
|
||
e = i;
|
||
|
||
/* find significand in antilog table A[] */
|
||
i = 1;
|
||
if (x <= A[17])
|
||
i = 17;
|
||
if (x <= A[i+8])
|
||
i += 8;
|
||
if (x <= A[i+4])
|
||
i += 4;
|
||
if (x <= A[i+2])
|
||
i += 2;
|
||
if (x >= A[1])
|
||
i = -1;
|
||
i += 1;
|
||
|
||
/* Find (x - A[i])/A[i]
|
||
* in order to compute log(x/A[i]):
|
||
*
|
||
* log(x) = log( a x/a ) = log(a) + log(x/a)
|
||
*
|
||
* log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
|
||
*/
|
||
x -= A[i];
|
||
x -= B[i/2];
|
||
x /= A[i];
|
||
|
||
/* rational approximation for log(1+v):
|
||
*
|
||
* log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
|
||
*/
|
||
z = x*x;
|
||
w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
|
||
w = w - 0.5*z;
|
||
|
||
/* Convert to base 2 logarithm:
|
||
* multiply by log2(e) = 1 + LOG2EA
|
||
*/
|
||
z = LOG2EA * w;
|
||
z += w;
|
||
z += LOG2EA * x;
|
||
z += x;
|
||
|
||
/* Compute exponent term of the base 2 logarithm. */
|
||
w = -i;
|
||
w /= NXT;
|
||
w += e;
|
||
/* Now base 2 log of x is w + z. */
|
||
|
||
/* Multiply base 2 log by y, in extended precision. */
|
||
|
||
/* separate y into large part ya
|
||
* and small part yb less than 1/NXT
|
||
*/
|
||
ya = reducl(y);
|
||
yb = y - ya;
|
||
|
||
/* (w+z)(ya+yb)
|
||
* = w*ya + w*yb + z*y
|
||
*/
|
||
F = z * y + w * yb;
|
||
Fa = reducl(F);
|
||
Fb = F - Fa;
|
||
|
||
G = Fa + w * ya;
|
||
Ga = reducl(G);
|
||
Gb = G - Ga;
|
||
|
||
H = Fb + Gb;
|
||
Ha = reducl(H);
|
||
w = (Ga + Ha) * NXT;
|
||
|
||
/* Test the power of 2 for overflow */
|
||
if (w > MEXP)
|
||
return huge * huge; /* overflow */
|
||
if (w < MNEXP)
|
||
return twom10000 * twom10000; /* underflow */
|
||
|
||
e = w;
|
||
Hb = H - Ha;
|
||
|
||
if (Hb > 0.0) {
|
||
e += 1;
|
||
Hb -= 1.0/NXT; /*0.0625L;*/
|
||
}
|
||
|
||
/* Now the product y * log2(x) = Hb + e/NXT.
|
||
*
|
||
* Compute base 2 exponential of Hb,
|
||
* where -0.0625 <= Hb <= 0.
|
||
*/
|
||
z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */
|
||
|
||
/* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
|
||
* Find lookup table entry for the fractional power of 2.
|
||
*/
|
||
if (e < 0)
|
||
i = 0;
|
||
else
|
||
i = 1;
|
||
i = e/NXT + i;
|
||
e = NXT*i - e;
|
||
w = A[e];
|
||
z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
|
||
z = z + w;
|
||
z = scalbnl(z, i); /* multiply by integer power of 2 */
|
||
|
||
if (nflg)
|
||
z = -z;
|
||
return z;
|
||
}
|
||
|
||
|
||
/* Find a multiple of 1/NXT that is within 1/NXT of x. */
|
||
static long double reducl(long double x)
|
||
{
|
||
long double t;
|
||
|
||
t = x * NXT;
|
||
t = floorl(t);
|
||
t = t / NXT;
|
||
return t;
|
||
}
|
||
|
||
/*
|
||
* Positive real raised to integer power, long double precision
|
||
*
|
||
*
|
||
* SYNOPSIS:
|
||
*
|
||
* long double x, y, powil();
|
||
* int n;
|
||
*
|
||
* y = powil( x, n );
|
||
*
|
||
*
|
||
* DESCRIPTION:
|
||
*
|
||
* Returns argument x>0 raised to the nth power.
|
||
* The routine efficiently decomposes n as a sum of powers of
|
||
* two. The desired power is a product of two-to-the-kth
|
||
* powers of x. Thus to compute the 32767 power of x requires
|
||
* 28 multiplications instead of 32767 multiplications.
|
||
*
|
||
*
|
||
* ACCURACY:
|
||
*
|
||
* Relative error:
|
||
* arithmetic x domain n domain # trials peak rms
|
||
* IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18
|
||
* IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18
|
||
* IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17
|
||
*
|
||
* Returns MAXNUM on overflow, zero on underflow.
|
||
*/
|
||
|
||
static long double powil(long double x, int nn)
|
||
{
|
||
long double ww, y;
|
||
long double s;
|
||
int n, e, sign, lx;
|
||
|
||
if (nn == 0)
|
||
return 1.0;
|
||
|
||
if (nn < 0) {
|
||
sign = -1;
|
||
n = -nn;
|
||
} else {
|
||
sign = 1;
|
||
n = nn;
|
||
}
|
||
|
||
/* Overflow detection */
|
||
|
||
/* Calculate approximate logarithm of answer */
|
||
s = x;
|
||
s = frexpl( s, &lx);
|
||
e = (lx - 1)*n;
|
||
if ((e == 0) || (e > 64) || (e < -64)) {
|
||
s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L);
|
||
s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
|
||
} else {
|
||
s = LOGE2L * e;
|
||
}
|
||
|
||
if (s > MAXLOGL)
|
||
return huge * huge; /* overflow */
|
||
|
||
if (s < MINLOGL)
|
||
return twom10000 * twom10000; /* underflow */
|
||
/* Handle tiny denormal answer, but with less accuracy
|
||
* since roundoff error in 1.0/x will be amplified.
|
||
* The precise demarcation should be the gradual underflow threshold.
|
||
*/
|
||
if (s < -MAXLOGL+2.0) {
|
||
x = 1.0/x;
|
||
sign = -sign;
|
||
}
|
||
|
||
/* First bit of the power */
|
||
if (n & 1)
|
||
y = x;
|
||
else
|
||
y = 1.0;
|
||
|
||
ww = x;
|
||
n >>= 1;
|
||
while (n) {
|
||
ww = ww * ww; /* arg to the 2-to-the-kth power */
|
||
if (n & 1) /* if that bit is set, then include in product */
|
||
y *= ww;
|
||
n >>= 1;
|
||
}
|
||
|
||
if (sign < 0)
|
||
y = 1.0/y;
|
||
return y;
|
||
}
|
||
|
||
#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
|
||
#include "libc/tinymath/freebsd.internal.h"
|
||
|
||
asm(".ident\t\"\\n\\n\
|
||
OpenBSD libm (ISC License)\\n\
|
||
Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>\"");
|
||
|
||
/*-
|
||
* ====================================================
|
||
* 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.
|
||
* ====================================================
|
||
*/
|
||
|
||
/*
|
||
* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
|
||
*
|
||
* Permission to use, copy, modify, and distribute this software for any
|
||
* purpose with or without fee is hereby granted, provided that the above
|
||
* copyright notice and this permission notice appear in all copies.
|
||
*
|
||
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
|
||
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
|
||
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
|
||
* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
|
||
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
|
||
* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
|
||
* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
|
||
*/
|
||
|
||
/* powl(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 113-53 = 60 bit trailing zeros.
|
||
* 2. Perform y*log2(x) = n+y' by simulating multi-precision
|
||
* arithmetic, where |y'|<=0.5.
|
||
* 3. Return x**y = 2**n*exp(y'*log2)
|
||
*
|
||
* Special cases:
|
||
* 1. (anything) ** 0 is 1
|
||
* 2. (anything) ** 1 is itself
|
||
* 3. (anything) ** 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 NAN
|
||
* 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
|
||
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
|
||
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
|
||
* 15. +INF ** (+anything except 0,NAN) is +INF
|
||
* 16. +INF ** (-anything except 0,NAN) is +0
|
||
* 17. -INF ** (anything) = -0 ** (-anything)
|
||
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
||
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
|
||
*
|
||
*/
|
||
|
||
static const long double bp[] = {
|
||
1.0L,
|
||
1.5L,
|
||
};
|
||
|
||
/* log_2(1.5) */
|
||
static const long double dp_h[] = {
|
||
0.0,
|
||
5.8496250072115607565592654282227158546448E-1L
|
||
};
|
||
|
||
/* Low part of log_2(1.5) */
|
||
static const long double dp_l[] = {
|
||
0.0,
|
||
1.0579781240112554492329533686862998106046E-16L
|
||
};
|
||
|
||
static const long double zero = 0.0L,
|
||
one = 1.0L,
|
||
two = 2.0L,
|
||
two113 = 1.0384593717069655257060992658440192E34L,
|
||
huge = 1.0e3000L,
|
||
tiny = 1.0e-3000L;
|
||
|
||
/* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2))
|
||
z = (x-1)/(x+1)
|
||
1 <= x <= 1.25
|
||
Peak relative error 2.3e-37 */
|
||
static const long double LN[] =
|
||
{
|
||
-3.0779177200290054398792536829702930623200E1L,
|
||
6.5135778082209159921251824580292116201640E1L,
|
||
-4.6312921812152436921591152809994014413540E1L,
|
||
1.2510208195629420304615674658258363295208E1L,
|
||
-9.9266909031921425609179910128531667336670E-1L
|
||
};
|
||
static const long double LD[] =
|
||
{
|
||
-5.129862866715009066465422805058933131960E1L,
|
||
1.452015077564081884387441590064272782044E2L,
|
||
-1.524043275549860505277434040464085593165E2L,
|
||
7.236063513651544224319663428634139768808E1L,
|
||
-1.494198912340228235853027849917095580053E1L
|
||
/* 1.0E0 */
|
||
};
|
||
|
||
/* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2)))
|
||
0 <= x <= 0.5
|
||
Peak relative error 5.7e-38 */
|
||
static const long double PN[] =
|
||
{
|
||
5.081801691915377692446852383385968225675E8L,
|
||
9.360895299872484512023336636427675327355E6L,
|
||
4.213701282274196030811629773097579432957E4L,
|
||
5.201006511142748908655720086041570288182E1L,
|
||
9.088368420359444263703202925095675982530E-3L,
|
||
};
|
||
static const long double PD[] =
|
||
{
|
||
3.049081015149226615468111430031590411682E9L,
|
||
1.069833887183886839966085436512368982758E8L,
|
||
8.259257717868875207333991924545445705394E5L,
|
||
1.872583833284143212651746812884298360922E3L,
|
||
/* 1.0E0 */
|
||
};
|
||
|
||
static const long double
|
||
/* ln 2 */
|
||
lg2 = 6.9314718055994530941723212145817656807550E-1L,
|
||
lg2_h = 6.9314718055994528622676398299518041312695E-1L,
|
||
lg2_l = 2.3190468138462996154948554638754786504121E-17L,
|
||
ovt = 8.0085662595372944372e-0017L,
|
||
/* 2/(3*log(2)) */
|
||
cp = 9.6179669392597560490661645400126142495110E-1L,
|
||
cp_h = 9.6179669392597555432899980587535537779331E-1L,
|
||
cp_l = 5.0577616648125906047157785230014751039424E-17L;
|
||
|
||
long double
|
||
powl(long double x, long double y)
|
||
{
|
||
long double z, ax, z_h, z_l, p_h, p_l;
|
||
long double yy1, t1, t2, r, s, t, u, v, w;
|
||
long double s2, s_h, s_l, t_h, t_l;
|
||
int32_t i, j, k, yisint, n;
|
||
uint32_t ix, iy;
|
||
int32_t hx, hy;
|
||
ieee_quad_shape_type o, p, q;
|
||
|
||
p.value = x;
|
||
hx = p.parts32.mswhi;
|
||
ix = hx & 0x7fffffff;
|
||
|
||
q.value = y;
|
||
hy = q.parts32.mswhi;
|
||
iy = hy & 0x7fffffff;
|
||
|
||
|
||
/* y==zero: x**0 = 1 */
|
||
if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
|
||
return one;
|
||
|
||
/* 1.0**y = 1; -1.0**+-Inf = 1 */
|
||
if (x == one)
|
||
return one;
|
||
if (x == -1.0L && iy == 0x7fff0000
|
||
&& (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
|
||
return one;
|
||
|
||
/* +-NaN return x+y */
|
||
if ((ix > 0x7fff0000)
|
||
|| ((ix == 0x7fff0000)
|
||
&& ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0))
|
||
|| (iy > 0x7fff0000)
|
||
|| ((iy == 0x7fff0000)
|
||
&& ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0)))
|
||
return nan_mix(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 >= 0x40700000) /* 2^113 */
|
||
yisint = 2; /* even integer y */
|
||
else if (iy >= 0x3fff0000) /* 1.0 */
|
||
{
|
||
if (floorl (y) == y)
|
||
{
|
||
z = 0.5 * y;
|
||
if (floorl (z) == z)
|
||
yisint = 2;
|
||
else
|
||
yisint = 1;
|
||
}
|
||
}
|
||
}
|
||
|
||
/* special value of y */
|
||
if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0)
|
||
{
|
||
if (iy == 0x7fff0000) /* y is +-inf */
|
||
{
|
||
if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi |
|
||
p.parts32.lswlo) == 0)
|
||
return y - y; /* +-1**inf is NaN */
|
||
else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */
|
||
return (hy >= 0) ? y : zero;
|
||
else /* (|x|<1)**-,+inf = inf,0 */
|
||
return (hy < 0) ? -y : zero;
|
||
}
|
||
if (iy == 0x3fff0000)
|
||
{ /* y is +-1 */
|
||
if (hy < 0)
|
||
return one / x;
|
||
else
|
||
return x;
|
||
}
|
||
if (hy == 0x40000000)
|
||
return x * x; /* y is 2 */
|
||
if (hy == 0x3ffe0000)
|
||
{ /* y is 0.5 */
|
||
if (hx >= 0) /* x >= +0 */
|
||
return sqrtl (x);
|
||
}
|
||
}
|
||
|
||
ax = fabsl (x);
|
||
/* special value of x */
|
||
if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0)
|
||
{
|
||
if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000)
|
||
{
|
||
z = ax; /*x is +-0,+-inf,+-1 */
|
||
if (hy < 0)
|
||
z = one / z; /* z = (1/|x|) */
|
||
if (hx < 0)
|
||
{
|
||
if (((ix - 0x3fff0000) | 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;
|
||
}
|
||
}
|
||
|
||
/* (x<0)**(non-int) is NaN */
|
||
if (((((uint32_t) hx >> 31) - 1) | yisint) == 0)
|
||
return (x - x) / (x - x);
|
||
|
||
/* |y| is huge.
|
||
2^-16495 = 1/2 of smallest representable value.
|
||
If (1 - 1/131072)^y underflows, y > 1.4986e9 */
|
||
if (iy > 0x401d654b)
|
||
{
|
||
/* if (1 - 2^-113)^y underflows, y > 1.1873e38 */
|
||
if (iy > 0x407d654b)
|
||
{
|
||
if (ix <= 0x3ffeffff)
|
||
return (hy < 0) ? huge * huge : tiny * tiny;
|
||
if (ix >= 0x3fff0000)
|
||
return (hy > 0) ? huge * huge : tiny * tiny;
|
||
}
|
||
/* over/underflow if x is not close to one */
|
||
if (ix < 0x3ffeffff)
|
||
return (hy < 0) ? huge * huge : tiny * tiny;
|
||
if (ix > 0x3fff0000)
|
||
return (hy > 0) ? huge * huge : tiny * tiny;
|
||
}
|
||
|
||
n = 0;
|
||
/* take care subnormal number */
|
||
if (ix < 0x00010000)
|
||
{
|
||
ax *= two113;
|
||
n -= 113;
|
||
o.value = ax;
|
||
ix = o.parts32.mswhi;
|
||
}
|
||
n += ((ix) >> 16) - 0x3fff;
|
||
j = ix & 0x0000ffff;
|
||
/* determine interval */
|
||
ix = j | 0x3fff0000; /* normalize ix */
|
||
if (j <= 0x3988)
|
||
k = 0; /* |x|<sqrt(3/2) */
|
||
else if (j < 0xbb67)
|
||
k = 1; /* |x|<sqrt(3) */
|
||
else
|
||
{
|
||
k = 0;
|
||
n += 1;
|
||
ix -= 0x00010000;
|
||
}
|
||
|
||
o.value = ax;
|
||
o.parts32.mswhi = ix;
|
||
ax = o.value;
|
||
|
||
/* compute s = 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 = one / (ax + bp[k]);
|
||
s = u * v;
|
||
s_h = s;
|
||
|
||
o.value = s_h;
|
||
o.parts32.lswlo = 0;
|
||
o.parts32.lswhi &= 0xf8000000;
|
||
s_h = o.value;
|
||
/* t_h=ax+bp[k] High */
|
||
t_h = ax + bp[k];
|
||
o.value = t_h;
|
||
o.parts32.lswlo = 0;
|
||
o.parts32.lswhi &= 0xf8000000;
|
||
t_h = o.value;
|
||
t_l = ax - (t_h - bp[k]);
|
||
s_l = v * ((u - s_h * t_h) - s_h * t_l);
|
||
/* compute log(ax) */
|
||
s2 = s * s;
|
||
u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4])));
|
||
v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2))));
|
||
r = s2 * s2 * u / v;
|
||
r += s_l * (s_h + s);
|
||
s2 = s_h * s_h;
|
||
t_h = 3.0 + s2 + r;
|
||
o.value = t_h;
|
||
o.parts32.lswlo = 0;
|
||
o.parts32.lswhi &= 0xf8000000;
|
||
t_h = o.value;
|
||
t_l = r - ((t_h - 3.0) - s2);
|
||
/* u+v = s*(1+...) */
|
||
u = s_h * t_h;
|
||
v = s_l * t_h + t_l * s;
|
||
/* 2/(3log2)*(s+...) */
|
||
p_h = u + v;
|
||
o.value = p_h;
|
||
o.parts32.lswlo = 0;
|
||
o.parts32.lswhi &= 0xf8000000;
|
||
p_h = o.value;
|
||
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) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
||
t = (long double) n;
|
||
t1 = (((z_h + z_l) + dp_h[k]) + t);
|
||
o.value = t1;
|
||
o.parts32.lswlo = 0;
|
||
o.parts32.lswhi &= 0xf8000000;
|
||
t1 = o.value;
|
||
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
|
||
|
||
/* s (sign of result -ve**odd) = -1 else = 1 */
|
||
s = one;
|
||
if (((((uint32_t) hx >> 31) - 1) | (yisint - 1)) == 0)
|
||
s = -one; /* (-ve)**(odd int) */
|
||
|
||
/* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
|
||
yy1 = y;
|
||
o.value = yy1;
|
||
o.parts32.lswlo = 0;
|
||
o.parts32.lswhi &= 0xf8000000;
|
||
yy1 = o.value;
|
||
p_l = (y - yy1) * t1 + y * t2;
|
||
p_h = yy1 * t1;
|
||
z = p_l + p_h;
|
||
o.value = z;
|
||
j = o.parts32.mswhi;
|
||
if (j >= 0x400d0000) /* z >= 16384 */
|
||
{
|
||
/* if z > 16384 */
|
||
if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi |
|
||
o.parts32.lswlo) != 0)
|
||
return s * huge * huge; /* overflow */
|
||
else
|
||
{
|
||
if (p_l + ovt > z - p_h)
|
||
return s * huge * huge; /* overflow */
|
||
}
|
||
}
|
||
else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */
|
||
{
|
||
/* z < -16495 */
|
||
if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi |
|
||
o.parts32.lswlo)
|
||
!= 0)
|
||
return s * tiny * tiny; /* underflow */
|
||
else
|
||
{
|
||
if (p_l <= z - p_h)
|
||
return s * tiny * tiny; /* underflow */
|
||
}
|
||
}
|
||
/* compute 2**(p_h+p_l) */
|
||
i = j & 0x7fffffff;
|
||
k = (i >> 16) - 0x3fff;
|
||
n = 0;
|
||
if (i > 0x3ffe0000)
|
||
{ /* if |z| > 0.5, set n = [z+0.5] */
|
||
n = floorl (z + 0.5L);
|
||
t = n;
|
||
p_h -= t;
|
||
}
|
||
t = p_l + p_h;
|
||
o.value = t;
|
||
o.parts32.lswlo = 0;
|
||
o.parts32.lswhi &= 0xf8000000;
|
||
t = o.value;
|
||
u = t * lg2_h;
|
||
v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
|
||
z = u + v;
|
||
w = v - (z - u);
|
||
/* exp(z) */
|
||
t = z * z;
|
||
u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4])));
|
||
v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t)));
|
||
t1 = z - t * u / v;
|
||
r = (z * t1) / (t1 - two) - (w + z * w);
|
||
z = one - (r - z);
|
||
o.value = z;
|
||
j = o.parts32.mswhi;
|
||
j += (n << 16);
|
||
if ((j >> 16) <= 0)
|
||
z = scalbnl (z, n); /* subnormal output */
|
||
else
|
||
{
|
||
o.parts32.mswhi = j;
|
||
z = o.value;
|
||
}
|
||
return s * z;
|
||
}
|
||
|
||
#endif /* __x86_64__ */
|
||
|
||
__weak_reference(powl, __powl_finite);
|
||
|
||
#endif /* long double is long */
|