260ba0bff1
Mainly focus on files that use BSD 2-Clause license, however the tool I was using mis-identified many licenses so this was mostly a manual - error prone - task. The Software Package Data Exchange (SPDX) group provides a specification to make it easier for automated tools to detect and summarize well known opensource licenses. We are gradually adopting the specification, noting that the tags are considered only advisory and do not, in any way, superceed or replace the license texts.
329 lines
10 KiB
C
329 lines
10 KiB
C
/*-
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* SPDX-License-Identifier: BSD-2-Clause-FreeBSD
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*
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* Copyright (c) 2009-2013 Steven G. Kargl
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* All rights reserved.
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*
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* 1. Redistributions of source code must retain the above copyright
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* notice unmodified, this list of conditions, and the following
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* disclaimer.
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* 2. Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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*
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* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
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* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
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* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
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* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
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* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
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* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* Optimized by Bruce D. Evans.
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*/
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/*
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* ld128 version of s_expl.c. See ../ld80/s_expl.c for most comments.
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*/
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#include <float.h>
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#include "fpmath.h"
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#include "math.h"
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#include "math_private.h"
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#include "k_expl.h"
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/* XXX Prevent compilers from erroneously constant folding these: */
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static const volatile long double
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huge = 0x1p10000L,
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tiny = 0x1p-10000L;
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static const long double
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twom10000 = 0x1p-10000L;
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static const long double
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/* log(2**16384 - 0.5) rounded towards zero: */
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/* log(2**16384 - 0.5 + 1) rounded towards zero for expm1l() is the same: */
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o_threshold = 11356.523406294143949491931077970763428L,
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/* log(2**(-16381-64-1)) rounded towards zero: */
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u_threshold = -11433.462743336297878837243843452621503L;
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long double
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expl(long double x)
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{
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union IEEEl2bits u;
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long double hi, lo, t, twopk;
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int k;
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uint16_t hx, ix;
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DOPRINT_START(&x);
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/* Filter out exceptional cases. */
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u.e = x;
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hx = u.xbits.expsign;
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ix = hx & 0x7fff;
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if (ix >= BIAS + 13) { /* |x| >= 8192 or x is NaN */
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if (ix == BIAS + LDBL_MAX_EXP) {
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if (hx & 0x8000) /* x is -Inf or -NaN */
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RETURNP(-1 / x);
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RETURNP(x + x); /* x is +Inf or +NaN */
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}
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if (x > o_threshold)
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RETURNP(huge * huge);
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if (x < u_threshold)
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RETURNP(tiny * tiny);
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} else if (ix < BIAS - 114) { /* |x| < 0x1p-114 */
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RETURN2P(1, x); /* 1 with inexact iff x != 0 */
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}
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ENTERI();
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twopk = 1;
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__k_expl(x, &hi, &lo, &k);
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t = SUM2P(hi, lo);
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/* Scale by 2**k. */
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/* XXX sparc64 multiplication is so slow that scalbnl() is faster. */
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if (k >= LDBL_MIN_EXP) {
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if (k == LDBL_MAX_EXP)
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RETURNI(t * 2 * 0x1p16383L);
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SET_LDBL_EXPSIGN(twopk, BIAS + k);
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RETURNI(t * twopk);
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} else {
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SET_LDBL_EXPSIGN(twopk, BIAS + k + 10000);
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RETURNI(t * twopk * twom10000);
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}
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}
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/*
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* Our T1 and T2 are chosen to be approximately the points where method
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* A and method B have the same accuracy. Tang's T1 and T2 are the
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* points where method A's accuracy changes by a full bit. For Tang,
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* this drop in accuracy makes method A immediately less accurate than
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* method B, but our larger INTERVALS makes method A 2 bits more
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* accurate so it remains the most accurate method significantly
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* closer to the origin despite losing the full bit in our extended
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* range for it.
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*
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* Split the interval [T1, T2] into two intervals [T1, T3] and [T3, T2].
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* Setting T3 to 0 would require the |x| < 0x1p-113 condition to appear
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* in both subintervals, so set T3 = 2**-5, which places the condition
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* into the [T1, T3] interval.
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*
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* XXX we now do this more to (partially) balance the number of terms
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* in the C and D polys than to avoid checking the condition in both
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* intervals.
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*
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* XXX these micro-optimizations are excessive.
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*/
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static const double
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T1 = -0.1659, /* ~-30.625/128 * log(2) */
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T2 = 0.1659, /* ~30.625/128 * log(2) */
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T3 = 0.03125;
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/*
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* Domain [-0.1659, 0.03125], range ~[2.9134e-44, 1.8404e-37]:
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* |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-122.03
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*
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* XXX none of the long double C or D coeffs except C10 is correctly printed.
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* If you re-print their values in %.35Le format, the result is always
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* different. For example, the last 2 digits in C3 should be 59, not 67.
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* 67 is apparently from rounding an extra-precision value to 36 decimal
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* places.
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*/
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static const long double
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C3 = 1.66666666666666666666666666666666667e-1L,
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C4 = 4.16666666666666666666666666666666645e-2L,
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C5 = 8.33333333333333333333333333333371638e-3L,
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C6 = 1.38888888888888888888888888891188658e-3L,
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C7 = 1.98412698412698412698412697235950394e-4L,
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C8 = 2.48015873015873015873015112487849040e-5L,
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C9 = 2.75573192239858906525606685484412005e-6L,
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C10 = 2.75573192239858906612966093057020362e-7L,
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C11 = 2.50521083854417203619031960151253944e-8L,
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C12 = 2.08767569878679576457272282566520649e-9L,
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C13 = 1.60590438367252471783548748824255707e-10L;
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/*
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* XXX this has 1 more coeff than needed.
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* XXX can start the double coeffs but not the double mults at C10.
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* With my coeffs (C10-C17 double; s = best_s):
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* Domain [-0.1659, 0.03125], range ~[-1.1976e-37, 1.1976e-37]:
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* |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
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*/
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static const double
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C14 = 1.1470745580491932e-11, /* 0x1.93974a81dae30p-37 */
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C15 = 7.6471620181090468e-13, /* 0x1.ae7f3820adab1p-41 */
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C16 = 4.7793721460260450e-14, /* 0x1.ae7cd18a18eacp-45 */
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C17 = 2.8074757356658877e-15, /* 0x1.949992a1937d9p-49 */
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C18 = 1.4760610323699476e-16; /* 0x1.545b43aabfbcdp-53 */
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/*
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* Domain [0.03125, 0.1659], range ~[-2.7676e-37, -1.0367e-38]:
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* |(exp(x)-1-x-x**2/2)/x - p(x)| < 2**-121.44
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*/
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static const long double
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D3 = 1.66666666666666666666666666666682245e-1L,
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D4 = 4.16666666666666666666666666634228324e-2L,
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D5 = 8.33333333333333333333333364022244481e-3L,
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D6 = 1.38888888888888888888887138722762072e-3L,
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D7 = 1.98412698412698412699085805424661471e-4L,
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D8 = 2.48015873015873015687993712101479612e-5L,
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D9 = 2.75573192239858944101036288338208042e-6L,
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D10 = 2.75573192239853161148064676533754048e-7L,
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D11 = 2.50521083855084570046480450935267433e-8L,
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D12 = 2.08767569819738524488686318024854942e-9L,
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D13 = 1.60590442297008495301927448122499313e-10L;
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/*
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* XXX this has 1 more coeff than needed.
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* XXX can start the double coeffs but not the double mults at D11.
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* With my coeffs (D11-D16 double):
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* Domain [0.03125, 0.1659], range ~[-1.1980e-37, 1.1980e-37]:
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* |(exp(x)-1-x-x**2/2)/x - p(x)| ~< 2**-122.65
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*/
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static const double
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D14 = 1.1470726176204336e-11, /* 0x1.93971dc395d9ep-37 */
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D15 = 7.6478532249581686e-13, /* 0x1.ae892e3D16fcep-41 */
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D16 = 4.7628892832607741e-14, /* 0x1.ad00Dfe41feccp-45 */
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D17 = 3.0524857220358650e-15; /* 0x1.D7e8d886Df921p-49 */
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long double
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expm1l(long double x)
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{
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union IEEEl2bits u, v;
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long double hx2_hi, hx2_lo, q, r, r1, t, twomk, twopk, x_hi;
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long double x_lo, x2;
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double dr, dx, fn, r2;
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int k, n, n2;
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uint16_t hx, ix;
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DOPRINT_START(&x);
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/* Filter out exceptional cases. */
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u.e = x;
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hx = u.xbits.expsign;
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ix = hx & 0x7fff;
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if (ix >= BIAS + 7) { /* |x| >= 128 or x is NaN */
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if (ix == BIAS + LDBL_MAX_EXP) {
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if (hx & 0x8000) /* x is -Inf or -NaN */
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RETURNP(-1 / x - 1);
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RETURNP(x + x); /* x is +Inf or +NaN */
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}
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if (x > o_threshold)
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RETURNP(huge * huge);
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/*
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* expm1l() never underflows, but it must avoid
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* unrepresentable large negative exponents. We used a
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* much smaller threshold for large |x| above than in
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* expl() so as to handle not so large negative exponents
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* in the same way as large ones here.
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*/
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if (hx & 0x8000) /* x <= -128 */
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RETURN2P(tiny, -1); /* good for x < -114ln2 - eps */
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}
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ENTERI();
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if (T1 < x && x < T2) {
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x2 = x * x;
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dx = x;
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if (x < T3) {
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if (ix < BIAS - 113) { /* |x| < 0x1p-113 */
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/* x (rounded) with inexact if x != 0: */
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RETURNPI(x == 0 ? x :
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(0x1p200 * x + fabsl(x)) * 0x1p-200);
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}
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q = x * x2 * C3 + x2 * x2 * (C4 + x * (C5 + x * (C6 +
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x * (C7 + x * (C8 + x * (C9 + x * (C10 +
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x * (C11 + x * (C12 + x * (C13 +
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dx * (C14 + dx * (C15 + dx * (C16 +
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dx * (C17 + dx * C18))))))))))))));
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} else {
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q = x * x2 * D3 + x2 * x2 * (D4 + x * (D5 + x * (D6 +
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x * (D7 + x * (D8 + x * (D9 + x * (D10 +
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x * (D11 + x * (D12 + x * (D13 +
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dx * (D14 + dx * (D15 + dx * (D16 +
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dx * D17)))))))))))));
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}
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x_hi = (float)x;
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x_lo = x - x_hi;
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hx2_hi = x_hi * x_hi / 2;
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hx2_lo = x_lo * (x + x_hi) / 2;
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if (ix >= BIAS - 7)
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RETURN2PI(hx2_hi + x_hi, hx2_lo + x_lo + q);
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else
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RETURN2PI(x, hx2_lo + q + hx2_hi);
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}
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/* Reduce x to (k*ln2 + endpoint[n2] + r1 + r2). */
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/* Use a specialized rint() to get fn. Assume round-to-nearest. */
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fn = (double)x * INV_L + 0x1.8p52 - 0x1.8p52;
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#if defined(HAVE_EFFICIENT_IRINT)
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n = irint(fn);
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#else
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n = (int)fn;
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#endif
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n2 = (unsigned)n % INTERVALS;
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k = n >> LOG2_INTERVALS;
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r1 = x - fn * L1;
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r2 = fn * -L2;
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r = r1 + r2;
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/* Prepare scale factor. */
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v.e = 1;
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v.xbits.expsign = BIAS + k;
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twopk = v.e;
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/*
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* Evaluate lower terms of
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* expl(endpoint[n2] + r1 + r2) = tbl[n2] * expl(r1 + r2).
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*/
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dr = r;
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q = r2 + r * r * (A2 + r * (A3 + r * (A4 + r * (A5 + r * (A6 +
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dr * (A7 + dr * (A8 + dr * (A9 + dr * A10))))))));
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t = tbl[n2].lo + tbl[n2].hi;
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if (k == 0) {
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t = SUM2P(tbl[n2].hi - 1, tbl[n2].lo * (r1 + 1) + t * q +
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tbl[n2].hi * r1);
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RETURNI(t);
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}
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if (k == -1) {
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t = SUM2P(tbl[n2].hi - 2, tbl[n2].lo * (r1 + 1) + t * q +
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tbl[n2].hi * r1);
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RETURNI(t / 2);
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}
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if (k < -7) {
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t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
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RETURNI(t * twopk - 1);
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}
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if (k > 2 * LDBL_MANT_DIG - 1) {
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t = SUM2P(tbl[n2].hi, tbl[n2].lo + t * (q + r1));
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if (k == LDBL_MAX_EXP)
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RETURNI(t * 2 * 0x1p16383L - 1);
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RETURNI(t * twopk - 1);
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}
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v.xbits.expsign = BIAS - k;
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twomk = v.e;
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if (k > LDBL_MANT_DIG - 1)
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t = SUM2P(tbl[n2].hi, tbl[n2].lo - twomk + t * (q + r1));
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else
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t = SUM2P(tbl[n2].hi - twomk, tbl[n2].lo + t * (q + r1));
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RETURNI(t * twopk);
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}
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