327 lines
10 KiB
C
327 lines
10 KiB
C
/*-
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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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