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650ef4bd79
http://sourceware.org/ml/libc-alpha/2013-08/msg00084.html Another batch of ieee854 macros and union replacement. These four files also have bugs fixed with this patch. The fact that the two doubles in an IBM long double may have different signs means that negation and absolute value operations can't just twiddle one sign bit as you can with ieee864 style extended double. fmodl, remainderl, erfl and erfcl all had errors of this type. erfl also returned +1 for large magnitude negative input where it should return -1. The hypotl error is innocuous since the value adjusted twice is only used as a flag. The e_hypotl.c tests for large "a" and small "b" are mutually exclusive because we've already exited when x/y > 2**120. That allows some further small simplifications. [BZ #15734], [BZ #15735] * sysdeps/ieee754/ldbl-128ibm/e_fmodl.c (__ieee754_fmodl): Rewrite all uses of ieee875 long double macros and unions. Simplify test for 0.0L. Correct |x|<|y| and |x|=|y| test. Use ldbl_extract_mantissa value for ix,iy exponents. Properly normalize after ldbl_extract_mantissa, and don't add hidden bit already handled. Don't treat low word of ieee854 mantissa like low word of IBM long double and mask off bit when testing for zero. * sysdeps/ieee754/ldbl-128ibm/e_hypotl.c (__ieee754_hypotl): Rewrite all uses of ieee875 long double macros and unions. Simplify tests for 0.0L and inf. Correct double adjustment of k. Delete dead code adjusting ha,hb. Simplify code setting kld. Delete two600 and two1022, instead use their values. Recognise that tests for large "a" and small "b" are mutually exclusive. Rename vars. Comment. * sysdeps/ieee754/ldbl-128ibm/e_remainderl.c (__ieee754_remainderl): Rewrite all uses of ieee875 long double macros and unions. Simplify test for 0.0L and nan. Correct negation. * sysdeps/ieee754/ldbl-128ibm/s_erfl.c (__erfl): Rewrite all uses of ieee875 long double macros and unions. Correct output for large magnitude x. Correct absolute value calculation. (__erfcl): Likewise. * math/libm-test.inc: Add tests for errors discovered in IBM long double versions of fmodl, remainderl, erfl and erfcl.
957 lines
30 KiB
C
957 lines
30 KiB
C
/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* Modifications and expansions for 128-bit long double are
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Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
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and are incorporated herein by permission of the author. The author
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reserves the right to distribute this material elsewhere under different
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copying permissions. These modifications are distributed here under
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the following terms:
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This library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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This library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with this library; if not, see
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<http://www.gnu.org/licenses/>. */
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/* double erf(double x)
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* double erfc(double x)
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* x
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* 2 |\
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* erf(x) = --------- | exp(-t*t)dt
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* sqrt(pi) \|
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* 0
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*
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* erfc(x) = 1-erf(x)
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* Note that
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* erf(-x) = -erf(x)
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* erfc(-x) = 2 - erfc(x)
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*
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* Method:
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* 1. erf(x) = x + x*R(x^2) for |x| in [0, 7/8]
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* Remark. The formula is derived by noting
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* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
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* and that
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* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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* is close to one.
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*
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* 1a. erf(x) = 1 - erfc(x), for |x| > 1.0
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* erfc(x) = 1 - erf(x) if |x| < 1/4
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*
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* 2. For |x| in [7/8, 1], let s = |x| - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(s + c) = sign(x) * (c + P1(s)/Q1(s))
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* Remark: here we use the taylor series expansion at x=1.
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* erf(1+s) = erf(1) + s*Poly(s)
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* = 0.845.. + P1(s)/Q1(s)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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*
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* 3. For x in [1/4, 5/4],
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* erfc(s + const) = erfc(const) + s P1(s)/Q1(s)
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* for const = 1/4, 3/8, ..., 9/8
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* and 0 <= s <= 1/8 .
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*
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* 4. For x in [5/4, 107],
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* erfc(x) = (1/x)*exp(-x*x-0.5625 + R(z))
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* z=1/x^2
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* The interval is partitioned into several segments
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* of width 1/8 in 1/x.
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* erf(x) = 1.0 - erfc(x) if x < 25.6283 else
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* erf(x) = sign(x)*(1.0 - tiny)
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*
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* Note1:
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* To compute exp(-x*x-0.5625+R/S), let s be a single
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* precision number and s := x; then
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* -x*x = -s*s + (s-x)*(s+x)
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* exp(-x*x-0.5626+R/S) =
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* exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
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* Note2:
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* Here 4 and 5 make use of the asymptotic series
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* exp(-x*x)
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* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
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* x*sqrt(pi)
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*
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* Note3:
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* For x higher than 25.6283, erf(x) underflows.
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*
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* 5. For inf > x >= 107
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* erf(x) = sign(x) *(1 - tiny) (raise inexact)
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* erfc(x) = tiny*tiny (raise underflow) if x > 0
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* = 2 - tiny if x<0
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*
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* 7. Special case:
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* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
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* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
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* erfc/erf(NaN) is NaN
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*/
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#include <math.h>
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#include <math_private.h>
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#include <math_ldbl_opt.h>
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/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
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static long double
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neval (long double x, const long double *p, int n)
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{
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long double y;
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p += n;
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y = *p--;
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do
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{
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y = y * x + *p--;
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}
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while (--n > 0);
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return y;
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}
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/* Evaluate x^n+1 + P[n] x^(n) + P[n-1] x^(n-1) + ... + P[0] */
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static long double
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deval (long double x, const long double *p, int n)
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{
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long double y;
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p += n;
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y = x + *p--;
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do
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{
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y = y * x + *p--;
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}
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while (--n > 0);
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return y;
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}
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static const long double
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tiny = 1e-300L,
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half = 0.5L,
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one = 1.0L,
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two = 2.0L,
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/* 2/sqrt(pi) - 1 */
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efx = 1.2837916709551257389615890312154517168810E-1L,
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/* 8 * (2/sqrt(pi) - 1) */
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efx8 = 1.0270333367641005911692712249723613735048E0L;
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/* erf(x) = x + x R(x^2)
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0 <= x <= 7/8
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Peak relative error 1.8e-35 */
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#define NTN1 8
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static const long double TN1[NTN1 + 1] =
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{
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-3.858252324254637124543172907442106422373E10L,
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9.580319248590464682316366876952214879858E10L,
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1.302170519734879977595901236693040544854E10L,
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2.922956950426397417800321486727032845006E9L,
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1.764317520783319397868923218385468729799E8L,
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1.573436014601118630105796794840834145120E7L,
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4.028077380105721388745632295157816229289E5L,
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1.644056806467289066852135096352853491530E4L,
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3.390868480059991640235675479463287886081E1L
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};
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#define NTD1 8
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static const long double TD1[NTD1 + 1] =
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{
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-3.005357030696532927149885530689529032152E11L,
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-1.342602283126282827411658673839982164042E11L,
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-2.777153893355340961288511024443668743399E10L,
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-3.483826391033531996955620074072768276974E9L,
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-2.906321047071299585682722511260895227921E8L,
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-1.653347985722154162439387878512427542691E7L,
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-6.245520581562848778466500301865173123136E5L,
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-1.402124304177498828590239373389110545142E4L,
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-1.209368072473510674493129989468348633579E2L
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/* 1.0E0 */
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};
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/* erf(z+1) = erf_const + P(z)/Q(z)
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-.125 <= z <= 0
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Peak relative error 7.3e-36 */
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static const long double erf_const = 0.845062911510467529296875L;
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#define NTN2 8
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static const long double TN2[NTN2 + 1] =
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{
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-4.088889697077485301010486931817357000235E1L,
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7.157046430681808553842307502826960051036E3L,
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-2.191561912574409865550015485451373731780E3L,
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2.180174916555316874988981177654057337219E3L,
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2.848578658049670668231333682379720943455E2L,
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1.630362490952512836762810462174798925274E2L,
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6.317712353961866974143739396865293596895E0L,
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2.450441034183492434655586496522857578066E1L,
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5.127662277706787664956025545897050896203E-1L
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};
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#define NTD2 8
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static const long double TD2[NTD2 + 1] =
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{
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1.731026445926834008273768924015161048885E4L,
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1.209682239007990370796112604286048173750E4L,
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1.160950290217993641320602282462976163857E4L,
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5.394294645127126577825507169061355698157E3L,
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2.791239340533632669442158497532521776093E3L,
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8.989365571337319032943005387378993827684E2L,
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2.974016493766349409725385710897298069677E2L,
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6.148192754590376378740261072533527271947E1L,
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1.178502892490738445655468927408440847480E1L
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/* 1.0E0 */
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};
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/* erfc(x + 0.25) = erfc(0.25) + x R(x)
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0 <= x < 0.125
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Peak relative error 1.4e-35 */
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#define NRNr13 8
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static const long double RNr13[NRNr13 + 1] =
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{
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-2.353707097641280550282633036456457014829E3L,
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3.871159656228743599994116143079870279866E2L,
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-3.888105134258266192210485617504098426679E2L,
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-2.129998539120061668038806696199343094971E1L,
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-8.125462263594034672468446317145384108734E1L,
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8.151549093983505810118308635926270319660E0L,
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-5.033362032729207310462422357772568553670E0L,
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-4.253956621135136090295893547735851168471E-2L,
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-8.098602878463854789780108161581050357814E-2L
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};
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#define NRDr13 7
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static const long double RDr13[NRDr13 + 1] =
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{
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2.220448796306693503549505450626652881752E3L,
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1.899133258779578688791041599040951431383E2L,
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1.061906712284961110196427571557149268454E3L,
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7.497086072306967965180978101974566760042E1L,
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2.146796115662672795876463568170441327274E2L,
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1.120156008362573736664338015952284925592E1L,
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2.211014952075052616409845051695042741074E1L,
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6.469655675326150785692908453094054988938E-1L
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/* 1.0E0 */
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};
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/* erfc(0.25) = C13a + C13b to extra precision. */
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static const long double C13a = 0.723663330078125L;
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static const long double C13b = 1.0279753638067014931732235184287934646022E-5L;
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/* erfc(x + 0.375) = erfc(0.375) + x R(x)
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0 <= x < 0.125
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Peak relative error 1.2e-35 */
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#define NRNr14 8
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static const long double RNr14[NRNr14 + 1] =
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{
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-2.446164016404426277577283038988918202456E3L,
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6.718753324496563913392217011618096698140E2L,
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-4.581631138049836157425391886957389240794E2L,
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-2.382844088987092233033215402335026078208E1L,
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-7.119237852400600507927038680970936336458E1L,
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1.313609646108420136332418282286454287146E1L,
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-6.188608702082264389155862490056401365834E0L,
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-2.787116601106678287277373011101132659279E-2L,
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-2.230395570574153963203348263549700967918E-2L
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};
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#define NRDr14 7
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static const long double RDr14[NRDr14 + 1] =
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{
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2.495187439241869732696223349840963702875E3L,
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2.503549449872925580011284635695738412162E2L,
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1.159033560988895481698051531263861842461E3L,
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9.493751466542304491261487998684383688622E1L,
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2.276214929562354328261422263078480321204E2L,
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1.367697521219069280358984081407807931847E1L,
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2.276988395995528495055594829206582732682E1L,
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7.647745753648996559837591812375456641163E-1L
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/* 1.0E0 */
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};
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/* erfc(0.375) = C14a + C14b to extra precision. */
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static const long double C14a = 0.5958709716796875L;
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static const long double C14b = 1.2118885490201676174914080878232469565953E-5L;
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/* erfc(x + 0.5) = erfc(0.5) + x R(x)
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0 <= x < 0.125
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Peak relative error 4.7e-36 */
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#define NRNr15 8
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static const long double RNr15[NRNr15 + 1] =
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{
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-2.624212418011181487924855581955853461925E3L,
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8.473828904647825181073831556439301342756E2L,
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-5.286207458628380765099405359607331669027E2L,
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-3.895781234155315729088407259045269652318E1L,
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-6.200857908065163618041240848728398496256E1L,
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1.469324610346924001393137895116129204737E1L,
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-6.961356525370658572800674953305625578903E0L,
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5.145724386641163809595512876629030548495E-3L,
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1.990253655948179713415957791776180406812E-2L
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};
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#define NRDr15 7
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static const long double RDr15[NRDr15 + 1] =
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{
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2.986190760847974943034021764693341524962E3L,
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5.288262758961073066335410218650047725985E2L,
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1.363649178071006978355113026427856008978E3L,
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1.921707975649915894241864988942255320833E2L,
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2.588651100651029023069013885900085533226E2L,
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2.628752920321455606558942309396855629459E1L,
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2.455649035885114308978333741080991380610E1L,
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1.378826653595128464383127836412100939126E0L
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/* 1.0E0 */
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};
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/* erfc(0.5) = C15a + C15b to extra precision. */
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static const long double C15a = 0.4794921875L;
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static const long double C15b = 7.9346869534623172533461080354712635484242E-6L;
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/* erfc(x + 0.625) = erfc(0.625) + x R(x)
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0 <= x < 0.125
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Peak relative error 5.1e-36 */
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#define NRNr16 8
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static const long double RNr16[NRNr16 + 1] =
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{
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-2.347887943200680563784690094002722906820E3L,
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8.008590660692105004780722726421020136482E2L,
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-5.257363310384119728760181252132311447963E2L,
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-4.471737717857801230450290232600243795637E1L,
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-4.849540386452573306708795324759300320304E1L,
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1.140885264677134679275986782978655952843E1L,
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-6.731591085460269447926746876983786152300E0L,
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1.370831653033047440345050025876085121231E-1L,
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2.022958279982138755020825717073966576670E-2L,
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};
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#define NRDr16 7
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static const long double RDr16[NRDr16 + 1] =
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{
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3.075166170024837215399323264868308087281E3L,
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8.730468942160798031608053127270430036627E2L,
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1.458472799166340479742581949088453244767E3L,
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3.230423687568019709453130785873540386217E2L,
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2.804009872719893612081109617983169474655E2L,
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4.465334221323222943418085830026979293091E1L,
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2.612723259683205928103787842214809134746E1L,
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2.341526751185244109722204018543276124997E0L,
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/* 1.0E0 */
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};
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/* erfc(0.625) = C16a + C16b to extra precision. */
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static const long double C16a = 0.3767547607421875L;
|
|
static const long double C16b = 4.3570693945275513594941232097252997287766E-6L;
|
|
|
|
/* erfc(x + 0.75) = erfc(0.75) + x R(x)
|
|
0 <= x < 0.125
|
|
Peak relative error 1.7e-35 */
|
|
#define NRNr17 8
|
|
static const long double RNr17[NRNr17 + 1] =
|
|
{
|
|
-1.767068734220277728233364375724380366826E3L,
|
|
6.693746645665242832426891888805363898707E2L,
|
|
-4.746224241837275958126060307406616817753E2L,
|
|
-2.274160637728782675145666064841883803196E1L,
|
|
-3.541232266140939050094370552538987982637E1L,
|
|
6.988950514747052676394491563585179503865E0L,
|
|
-5.807687216836540830881352383529281215100E0L,
|
|
3.631915988567346438830283503729569443642E-1L,
|
|
-1.488945487149634820537348176770282391202E-2L
|
|
};
|
|
#define NRDr17 7
|
|
static const long double RDr17[NRDr17 + 1] =
|
|
{
|
|
2.748457523498150741964464942246913394647E3L,
|
|
1.020213390713477686776037331757871252652E3L,
|
|
1.388857635935432621972601695296561952738E3L,
|
|
3.903363681143817750895999579637315491087E2L,
|
|
2.784568344378139499217928969529219886578E2L,
|
|
5.555800830216764702779238020065345401144E1L,
|
|
2.646215470959050279430447295801291168941E1L,
|
|
2.984905282103517497081766758550112011265E0L,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(0.75) = C17a + C17b to extra precision. */
|
|
static const long double C17a = 0.2888336181640625L;
|
|
static const long double C17b = 1.0748182422368401062165408589222625794046E-5L;
|
|
|
|
|
|
/* erfc(x + 0.875) = erfc(0.875) + x R(x)
|
|
0 <= x < 0.125
|
|
Peak relative error 2.2e-35 */
|
|
#define NRNr18 8
|
|
static const long double RNr18[NRNr18 + 1] =
|
|
{
|
|
-1.342044899087593397419622771847219619588E3L,
|
|
6.127221294229172997509252330961641850598E2L,
|
|
-4.519821356522291185621206350470820610727E2L,
|
|
1.223275177825128732497510264197915160235E1L,
|
|
-2.730789571382971355625020710543532867692E1L,
|
|
4.045181204921538886880171727755445395862E0L,
|
|
-4.925146477876592723401384464691452700539E0L,
|
|
5.933878036611279244654299924101068088582E-1L,
|
|
-5.557645435858916025452563379795159124753E-2L
|
|
};
|
|
#define NRDr18 7
|
|
static const long double RDr18[NRDr18 + 1] =
|
|
{
|
|
2.557518000661700588758505116291983092951E3L,
|
|
1.070171433382888994954602511991940418588E3L,
|
|
1.344842834423493081054489613250688918709E3L,
|
|
4.161144478449381901208660598266288188426E2L,
|
|
2.763670252219855198052378138756906980422E2L,
|
|
5.998153487868943708236273854747564557632E1L,
|
|
2.657695108438628847733050476209037025318E1L,
|
|
3.252140524394421868923289114410336976512E0L,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(0.875) = C18a + C18b to extra precision. */
|
|
static const long double C18a = 0.215911865234375L;
|
|
static const long double C18b = 1.3073705765341685464282101150637224028267E-5L;
|
|
|
|
/* erfc(x + 1.0) = erfc(1.0) + x R(x)
|
|
0 <= x < 0.125
|
|
Peak relative error 1.6e-35 */
|
|
#define NRNr19 8
|
|
static const long double RNr19[NRNr19 + 1] =
|
|
{
|
|
-1.139180936454157193495882956565663294826E3L,
|
|
6.134903129086899737514712477207945973616E2L,
|
|
-4.628909024715329562325555164720732868263E2L,
|
|
4.165702387210732352564932347500364010833E1L,
|
|
-2.286979913515229747204101330405771801610E1L,
|
|
1.870695256449872743066783202326943667722E0L,
|
|
-4.177486601273105752879868187237000032364E0L,
|
|
7.533980372789646140112424811291782526263E-1L,
|
|
-8.629945436917752003058064731308767664446E-2L
|
|
};
|
|
#define NRDr19 7
|
|
static const long double RDr19[NRDr19 + 1] =
|
|
{
|
|
2.744303447981132701432716278363418643778E3L,
|
|
1.266396359526187065222528050591302171471E3L,
|
|
1.466739461422073351497972255511919814273E3L,
|
|
4.868710570759693955597496520298058147162E2L,
|
|
2.993694301559756046478189634131722579643E2L,
|
|
6.868976819510254139741559102693828237440E1L,
|
|
2.801505816247677193480190483913753613630E1L,
|
|
3.604439909194350263552750347742663954481E0L,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(1.0) = C19a + C19b to extra precision. */
|
|
static const long double C19a = 0.15728759765625L;
|
|
static const long double C19b = 1.1609394035130658779364917390740703933002E-5L;
|
|
|
|
/* erfc(x + 1.125) = erfc(1.125) + x R(x)
|
|
0 <= x < 0.125
|
|
Peak relative error 3.6e-36 */
|
|
#define NRNr20 8
|
|
static const long double RNr20[NRNr20 + 1] =
|
|
{
|
|
-9.652706916457973956366721379612508047640E2L,
|
|
5.577066396050932776683469951773643880634E2L,
|
|
-4.406335508848496713572223098693575485978E2L,
|
|
5.202893466490242733570232680736966655434E1L,
|
|
-1.931311847665757913322495948705563937159E1L,
|
|
-9.364318268748287664267341457164918090611E-2L,
|
|
-3.306390351286352764891355375882586201069E0L,
|
|
7.573806045289044647727613003096916516475E-1L,
|
|
-9.611744011489092894027478899545635991213E-2L
|
|
};
|
|
#define NRDr20 7
|
|
static const long double RDr20[NRDr20 + 1] =
|
|
{
|
|
3.032829629520142564106649167182428189014E3L,
|
|
1.659648470721967719961167083684972196891E3L,
|
|
1.703545128657284619402511356932569292535E3L,
|
|
6.393465677731598872500200253155257708763E2L,
|
|
3.489131397281030947405287112726059221934E2L,
|
|
8.848641738570783406484348434387611713070E1L,
|
|
3.132269062552392974833215844236160958502E1L,
|
|
4.430131663290563523933419966185230513168E0L
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(1.125) = C20a + C20b to extra precision. */
|
|
static const long double C20a = 0.111602783203125L;
|
|
static const long double C20b = 8.9850951672359304215530728365232161564636E-6L;
|
|
|
|
/* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2))
|
|
7/8 <= 1/x < 1
|
|
Peak relative error 1.4e-35 */
|
|
#define NRNr8 9
|
|
static const long double RNr8[NRNr8 + 1] =
|
|
{
|
|
3.587451489255356250759834295199296936784E1L,
|
|
5.406249749087340431871378009874875889602E2L,
|
|
2.931301290625250886238822286506381194157E3L,
|
|
7.359254185241795584113047248898753470923E3L,
|
|
9.201031849810636104112101947312492532314E3L,
|
|
5.749697096193191467751650366613289284777E3L,
|
|
1.710415234419860825710780802678697889231E3L,
|
|
2.150753982543378580859546706243022719599E2L,
|
|
8.740953582272147335100537849981160931197E0L,
|
|
4.876422978828717219629814794707963640913E-2L
|
|
};
|
|
#define NRDr8 8
|
|
static const long double RDr8[NRDr8 + 1] =
|
|
{
|
|
6.358593134096908350929496535931630140282E1L,
|
|
9.900253816552450073757174323424051765523E2L,
|
|
5.642928777856801020545245437089490805186E3L,
|
|
1.524195375199570868195152698617273739609E4L,
|
|
2.113829644500006749947332935305800887345E4L,
|
|
1.526438562626465706267943737310282977138E4L,
|
|
5.561370922149241457131421914140039411782E3L,
|
|
9.394035530179705051609070428036834496942E2L,
|
|
6.147019596150394577984175188032707343615E1L
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
/* erfc(1/x) = 1/x exp (-1/x^2 - 0.5625 + R(1/x^2))
|
|
0.75 <= 1/x <= 0.875
|
|
Peak relative error 2.0e-36 */
|
|
#define NRNr7 9
|
|
static const long double RNr7[NRNr7 + 1] =
|
|
{
|
|
1.686222193385987690785945787708644476545E1L,
|
|
1.178224543567604215602418571310612066594E3L,
|
|
1.764550584290149466653899886088166091093E4L,
|
|
1.073758321890334822002849369898232811561E5L,
|
|
3.132840749205943137619839114451290324371E5L,
|
|
4.607864939974100224615527007793867585915E5L,
|
|
3.389781820105852303125270837910972384510E5L,
|
|
1.174042187110565202875011358512564753399E5L,
|
|
1.660013606011167144046604892622504338313E4L,
|
|
6.700393957480661937695573729183733234400E2L
|
|
};
|
|
#define NRDr7 9
|
|
static const long double RDr7[NRDr7 + 1] =
|
|
{
|
|
-1.709305024718358874701575813642933561169E3L,
|
|
-3.280033887481333199580464617020514788369E4L,
|
|
-2.345284228022521885093072363418750835214E5L,
|
|
-8.086758123097763971926711729242327554917E5L,
|
|
-1.456900414510108718402423999575992450138E6L,
|
|
-1.391654264881255068392389037292702041855E6L,
|
|
-6.842360801869939983674527468509852583855E5L,
|
|
-1.597430214446573566179675395199807533371E5L,
|
|
-1.488876130609876681421645314851760773480E4L,
|
|
-3.511762950935060301403599443436465645703E2L
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
|
|
5/8 <= 1/x < 3/4
|
|
Peak relative error 1.9e-35 */
|
|
#define NRNr6 9
|
|
static const long double RNr6[NRNr6 + 1] =
|
|
{
|
|
1.642076876176834390623842732352935761108E0L,
|
|
1.207150003611117689000664385596211076662E2L,
|
|
2.119260779316389904742873816462800103939E3L,
|
|
1.562942227734663441801452930916044224174E4L,
|
|
5.656779189549710079988084081145693580479E4L,
|
|
1.052166241021481691922831746350942786299E5L,
|
|
9.949798524786000595621602790068349165758E4L,
|
|
4.491790734080265043407035220188849562856E4L,
|
|
8.377074098301530326270432059434791287601E3L,
|
|
4.506934806567986810091824791963991057083E2L
|
|
};
|
|
#define NRDr6 9
|
|
static const long double RDr6[NRDr6 + 1] =
|
|
{
|
|
-1.664557643928263091879301304019826629067E2L,
|
|
-3.800035902507656624590531122291160668452E3L,
|
|
-3.277028191591734928360050685359277076056E4L,
|
|
-1.381359471502885446400589109566587443987E5L,
|
|
-3.082204287382581873532528989283748656546E5L,
|
|
-3.691071488256738343008271448234631037095E5L,
|
|
-2.300482443038349815750714219117566715043E5L,
|
|
-6.873955300927636236692803579555752171530E4L,
|
|
-8.262158817978334142081581542749986845399E3L,
|
|
-2.517122254384430859629423488157361983661E2L
|
|
/* 1.00 */
|
|
};
|
|
|
|
/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
|
|
1/2 <= 1/x < 5/8
|
|
Peak relative error 4.6e-36 */
|
|
#define NRNr5 10
|
|
static const long double RNr5[NRNr5 + 1] =
|
|
{
|
|
-3.332258927455285458355550878136506961608E-3L,
|
|
-2.697100758900280402659586595884478660721E-1L,
|
|
-6.083328551139621521416618424949137195536E0L,
|
|
-6.119863528983308012970821226810162441263E1L,
|
|
-3.176535282475593173248810678636522589861E2L,
|
|
-8.933395175080560925809992467187963260693E2L,
|
|
-1.360019508488475978060917477620199499560E3L,
|
|
-1.075075579828188621541398761300910213280E3L,
|
|
-4.017346561586014822824459436695197089916E2L,
|
|
-5.857581368145266249509589726077645791341E1L,
|
|
-2.077715925587834606379119585995758954399E0L
|
|
};
|
|
#define NRDr5 9
|
|
static const long double RDr5[NRDr5 + 1] =
|
|
{
|
|
3.377879570417399341550710467744693125385E-1L,
|
|
1.021963322742390735430008860602594456187E1L,
|
|
1.200847646592942095192766255154827011939E2L,
|
|
7.118915528142927104078182863387116942836E2L,
|
|
2.318159380062066469386544552429625026238E3L,
|
|
4.238729853534009221025582008928765281620E3L,
|
|
4.279114907284825886266493994833515580782E3L,
|
|
2.257277186663261531053293222591851737504E3L,
|
|
5.570475501285054293371908382916063822957E2L,
|
|
5.142189243856288981145786492585432443560E1L
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
|
|
3/8 <= 1/x < 1/2
|
|
Peak relative error 2.0e-36 */
|
|
#define NRNr4 10
|
|
static const long double RNr4[NRNr4 + 1] =
|
|
{
|
|
3.258530712024527835089319075288494524465E-3L,
|
|
2.987056016877277929720231688689431056567E-1L,
|
|
8.738729089340199750734409156830371528862E0L,
|
|
1.207211160148647782396337792426311125923E2L,
|
|
8.997558632489032902250523945248208224445E2L,
|
|
3.798025197699757225978410230530640879762E3L,
|
|
9.113203668683080975637043118209210146846E3L,
|
|
1.203285891339933238608683715194034900149E4L,
|
|
8.100647057919140328536743641735339740855E3L,
|
|
2.383888249907144945837976899822927411769E3L,
|
|
2.127493573166454249221983582495245662319E2L
|
|
};
|
|
#define NRDr4 10
|
|
static const long double RDr4[NRDr4 + 1] =
|
|
{
|
|
-3.303141981514540274165450687270180479586E-1L,
|
|
-1.353768629363605300707949368917687066724E1L,
|
|
-2.206127630303621521950193783894598987033E2L,
|
|
-1.861800338758066696514480386180875607204E3L,
|
|
-8.889048775872605708249140016201753255599E3L,
|
|
-2.465888106627948210478692168261494857089E4L,
|
|
-3.934642211710774494879042116768390014289E4L,
|
|
-3.455077258242252974937480623730228841003E4L,
|
|
-1.524083977439690284820586063729912653196E4L,
|
|
-2.810541887397984804237552337349093953857E3L,
|
|
-1.343929553541159933824901621702567066156E2L
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
|
|
1/4 <= 1/x < 3/8
|
|
Peak relative error 8.4e-37 */
|
|
#define NRNr3 11
|
|
static const long double RNr3[NRNr3 + 1] =
|
|
{
|
|
-1.952401126551202208698629992497306292987E-6L,
|
|
-2.130881743066372952515162564941682716125E-4L,
|
|
-8.376493958090190943737529486107282224387E-3L,
|
|
-1.650592646560987700661598877522831234791E-1L,
|
|
-1.839290818933317338111364667708678163199E0L,
|
|
-1.216278715570882422410442318517814388470E1L,
|
|
-4.818759344462360427612133632533779091386E1L,
|
|
-1.120994661297476876804405329172164436784E2L,
|
|
-1.452850765662319264191141091859300126931E2L,
|
|
-9.485207851128957108648038238656777241333E1L,
|
|
-2.563663855025796641216191848818620020073E1L,
|
|
-1.787995944187565676837847610706317833247E0L
|
|
};
|
|
#define NRDr3 10
|
|
static const long double RDr3[NRDr3 + 1] =
|
|
{
|
|
1.979130686770349481460559711878399476903E-4L,
|
|
1.156941716128488266238105813374635099057E-2L,
|
|
2.752657634309886336431266395637285974292E-1L,
|
|
3.482245457248318787349778336603569327521E0L,
|
|
2.569347069372696358578399521203959253162E1L,
|
|
1.142279000180457419740314694631879921561E2L,
|
|
3.056503977190564294341422623108332700840E2L,
|
|
4.780844020923794821656358157128719184422E2L,
|
|
4.105972727212554277496256802312730410518E2L,
|
|
1.724072188063746970865027817017067646246E2L,
|
|
2.815939183464818198705278118326590370435E1L
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
|
|
1/8 <= 1/x < 1/4
|
|
Peak relative error 1.5e-36 */
|
|
#define NRNr2 11
|
|
static const long double RNr2[NRNr2 + 1] =
|
|
{
|
|
-2.638914383420287212401687401284326363787E-8L,
|
|
-3.479198370260633977258201271399116766619E-6L,
|
|
-1.783985295335697686382487087502222519983E-4L,
|
|
-4.777876933122576014266349277217559356276E-3L,
|
|
-7.450634738987325004070761301045014986520E-2L,
|
|
-7.068318854874733315971973707247467326619E-1L,
|
|
-4.113919921935944795764071670806867038732E0L,
|
|
-1.440447573226906222417767283691888875082E1L,
|
|
-2.883484031530718428417168042141288943905E1L,
|
|
-2.990886974328476387277797361464279931446E1L,
|
|
-1.325283914915104866248279787536128997331E1L,
|
|
-1.572436106228070195510230310658206154374E0L
|
|
};
|
|
#define NRDr2 10
|
|
static const long double RDr2[NRDr2 + 1] =
|
|
{
|
|
2.675042728136731923554119302571867799673E-6L,
|
|
2.170997868451812708585443282998329996268E-4L,
|
|
7.249969752687540289422684951196241427445E-3L,
|
|
1.302040375859768674620410563307838448508E-1L,
|
|
1.380202483082910888897654537144485285549E0L,
|
|
8.926594113174165352623847870299170069350E0L,
|
|
3.521089584782616472372909095331572607185E1L,
|
|
8.233547427533181375185259050330809105570E1L,
|
|
1.072971579885803033079469639073292840135E2L,
|
|
6.943803113337964469736022094105143158033E1L,
|
|
1.775695341031607738233608307835017282662E1L
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
/* erfc(1/x) = 1/x exp(-1/x^2 - 0.5625 + R(1/x^2))
|
|
1/128 <= 1/x < 1/8
|
|
Peak relative error 2.2e-36 */
|
|
#define NRNr1 9
|
|
static const long double RNr1[NRNr1 + 1] =
|
|
{
|
|
-4.250780883202361946697751475473042685782E-8L,
|
|
-5.375777053288612282487696975623206383019E-6L,
|
|
-2.573645949220896816208565944117382460452E-4L,
|
|
-6.199032928113542080263152610799113086319E-3L,
|
|
-8.262721198693404060380104048479916247786E-2L,
|
|
-6.242615227257324746371284637695778043982E-1L,
|
|
-2.609874739199595400225113299437099626386E0L,
|
|
-5.581967563336676737146358534602770006970E0L,
|
|
-5.124398923356022609707490956634280573882E0L,
|
|
-1.290865243944292370661544030414667556649E0L
|
|
};
|
|
#define NRDr1 8
|
|
static const long double RDr1[NRDr1 + 1] =
|
|
{
|
|
4.308976661749509034845251315983612976224E-6L,
|
|
3.265390126432780184125233455960049294580E-4L,
|
|
9.811328839187040701901866531796570418691E-3L,
|
|
1.511222515036021033410078631914783519649E-1L,
|
|
1.289264341917429958858379585970225092274E0L,
|
|
6.147640356182230769548007536914983522270E0L,
|
|
1.573966871337739784518246317003956180750E1L,
|
|
1.955534123435095067199574045529218238263E1L,
|
|
9.472613121363135472247929109615785855865E0L
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
|
|
long double
|
|
__erfl (long double x)
|
|
{
|
|
long double a, y, z;
|
|
int32_t i, ix, hx;
|
|
double xhi;
|
|
|
|
xhi = ldbl_high (x);
|
|
GET_HIGH_WORD (hx, xhi);
|
|
ix = hx & 0x7fffffff;
|
|
|
|
if (ix >= 0x7ff00000)
|
|
{ /* erf(nan)=nan */
|
|
i = ((uint32_t) hx >> 31) << 1;
|
|
return (long double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
|
|
}
|
|
|
|
if (ix >= 0x3ff00000) /* |x| >= 1.0 */
|
|
{
|
|
if (ix >= 0x4039A0DE)
|
|
{
|
|
/* __erfcl (x) underflows if x > 25.6283 */
|
|
if ((hx & 0x80000000) == 0)
|
|
return one-tiny;
|
|
else
|
|
return tiny-one;
|
|
}
|
|
else
|
|
{
|
|
y = __erfcl (x);
|
|
return (one - y);
|
|
}
|
|
}
|
|
a = x;
|
|
if ((hx & 0x80000000) != 0)
|
|
a = -a;
|
|
z = x * x;
|
|
if (ix < 0x3fec0000) /* a < 0.875 */
|
|
{
|
|
if (ix < 0x3c600000) /* |x|<2**-57 */
|
|
{
|
|
if (ix < 0x00800000)
|
|
{
|
|
/* erf (-0) = -0. Unfortunately, for IBM extended double
|
|
0.125 * (8.0 * x + efx8 * x) for x = -0 evaluates to 0. */
|
|
if (x == 0)
|
|
return x;
|
|
return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */
|
|
}
|
|
return x + efx * x;
|
|
}
|
|
y = a + a * neval (z, TN1, NTN1) / deval (z, TD1, NTD1);
|
|
}
|
|
else
|
|
{
|
|
a = a - one;
|
|
y = erf_const + neval (a, TN2, NTN2) / deval (a, TD2, NTD2);
|
|
}
|
|
|
|
if (hx & 0x80000000) /* x < 0 */
|
|
y = -y;
|
|
return( y );
|
|
}
|
|
|
|
long_double_symbol (libm, __erfl, erfl);
|
|
long double
|
|
__erfcl (long double x)
|
|
{
|
|
long double y, z, p, r;
|
|
int32_t i, ix;
|
|
uint32_t hx;
|
|
double xhi;
|
|
|
|
xhi = ldbl_high (x);
|
|
GET_HIGH_WORD (hx, xhi);
|
|
ix = hx & 0x7fffffff;
|
|
|
|
if (ix >= 0x7ff00000)
|
|
{ /* erfc(nan)=nan */
|
|
/* erfc(+-inf)=0,2 */
|
|
return (long double) ((hx >> 31) << 1) + one / x;
|
|
}
|
|
|
|
if (ix < 0x3fd00000) /* |x| <1/4 */
|
|
{
|
|
if (ix < 0x38d00000) /* |x|<2**-114 */
|
|
return one - x;
|
|
return one - __erfl (x);
|
|
}
|
|
if (ix < 0x3ff40000) /* 1.25 */
|
|
{
|
|
if ((hx & 0x80000000) != 0)
|
|
x = -x;
|
|
i = 8.0 * x;
|
|
switch (i)
|
|
{
|
|
case 2:
|
|
z = x - 0.25L;
|
|
y = C13b + z * neval (z, RNr13, NRNr13) / deval (z, RDr13, NRDr13);
|
|
y += C13a;
|
|
break;
|
|
case 3:
|
|
z = x - 0.375L;
|
|
y = C14b + z * neval (z, RNr14, NRNr14) / deval (z, RDr14, NRDr14);
|
|
y += C14a;
|
|
break;
|
|
case 4:
|
|
z = x - 0.5L;
|
|
y = C15b + z * neval (z, RNr15, NRNr15) / deval (z, RDr15, NRDr15);
|
|
y += C15a;
|
|
break;
|
|
case 5:
|
|
z = x - 0.625L;
|
|
y = C16b + z * neval (z, RNr16, NRNr16) / deval (z, RDr16, NRDr16);
|
|
y += C16a;
|
|
break;
|
|
case 6:
|
|
z = x - 0.75L;
|
|
y = C17b + z * neval (z, RNr17, NRNr17) / deval (z, RDr17, NRDr17);
|
|
y += C17a;
|
|
break;
|
|
case 7:
|
|
z = x - 0.875L;
|
|
y = C18b + z * neval (z, RNr18, NRNr18) / deval (z, RDr18, NRDr18);
|
|
y += C18a;
|
|
break;
|
|
case 8:
|
|
z = x - 1.0L;
|
|
y = C19b + z * neval (z, RNr19, NRNr19) / deval (z, RDr19, NRDr19);
|
|
y += C19a;
|
|
break;
|
|
case 9:
|
|
z = x - 1.125L;
|
|
y = C20b + z * neval (z, RNr20, NRNr20) / deval (z, RDr20, NRDr20);
|
|
y += C20a;
|
|
break;
|
|
}
|
|
if (hx & 0x80000000)
|
|
y = 2.0L - y;
|
|
return y;
|
|
}
|
|
/* 1.25 < |x| < 107 */
|
|
if (ix < 0x405ac000)
|
|
{
|
|
/* x < -9 */
|
|
if (hx >= 0xc0220000)
|
|
return two - tiny;
|
|
|
|
if ((hx & 0x80000000) != 0)
|
|
x = -x;
|
|
z = one / (x * x);
|
|
i = 8.0 / x;
|
|
switch (i)
|
|
{
|
|
default:
|
|
case 0:
|
|
p = neval (z, RNr1, NRNr1) / deval (z, RDr1, NRDr1);
|
|
break;
|
|
case 1:
|
|
p = neval (z, RNr2, NRNr2) / deval (z, RDr2, NRDr2);
|
|
break;
|
|
case 2:
|
|
p = neval (z, RNr3, NRNr3) / deval (z, RDr3, NRDr3);
|
|
break;
|
|
case 3:
|
|
p = neval (z, RNr4, NRNr4) / deval (z, RDr4, NRDr4);
|
|
break;
|
|
case 4:
|
|
p = neval (z, RNr5, NRNr5) / deval (z, RDr5, NRDr5);
|
|
break;
|
|
case 5:
|
|
p = neval (z, RNr6, NRNr6) / deval (z, RDr6, NRDr6);
|
|
break;
|
|
case 6:
|
|
p = neval (z, RNr7, NRNr7) / deval (z, RDr7, NRDr7);
|
|
break;
|
|
case 7:
|
|
p = neval (z, RNr8, NRNr8) / deval (z, RDr8, NRDr8);
|
|
break;
|
|
}
|
|
z = (float) x;
|
|
r = __ieee754_expl (-z * z - 0.5625) *
|
|
__ieee754_expl ((z - x) * (z + x) + p);
|
|
if ((hx & 0x80000000) == 0)
|
|
return r / x;
|
|
else
|
|
return two - r / x;
|
|
}
|
|
else
|
|
{
|
|
if ((hx & 0x80000000) == 0)
|
|
return tiny * tiny;
|
|
else
|
|
return two - tiny;
|
|
}
|
|
}
|
|
|
|
long_double_symbol (libm, __erfcl, erfcl);
|