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bc3753638a
On powerpc32 hard-float, older processors (ones where fcfid is not available for 32-bit code), GCC generates conversions from integers to floating point that wrongly convert integer 0 to -0 instead of +0 in FE_DOWNWARD mode. This in turn results in logb and a few other functions wrongly returning -0 when they should return +0. This patch works around this issue in glibc as I proposed in <https://sourceware.org/ml/libc-alpha/2015-09/msg00728.html>, so that the affected functions can be correct and the affected tests pass in the absence of a GCC fix for this longstanding issue (GCC bug 67771 - if fixed, of course we can put in GCC version conditionals, and eventually phase out the workarounds). A new macro FIX_INT_FP_CONVERT_ZERO is added in a new sysdeps header fix-int-fp-convert-zero.h, and the powerpc32/fpu version of that header defines the macro based on the results of a configure test for whether such conversions use the fcfid instruction. Tested for x86_64 (that installed stripped shared libraries are unchanged by the patch) and powerpc (that HAVE_PPC_FCFID comes out to 0 as expected and that the relevant tests are fixed). Also tested a build with GCC configured for -mcpu=power4 and verified that HAVE_PPC_FCFID comes out to 1 in that case. There are still some other issues to fix to get test-float and test-double passing cleanly for older powerpc32 processors (apart from the need for an ulps regeneration for powerpc). (test-ldouble will be harder to get passing cleanly, but with a combination of selected fixes to ldbl-128ibm code that don't involve significant performance issues, allowing spurious underflow and inexact exceptions for that format, and lots of XFAILing for the default case of unpatched libgcc, it should be doable.) [BZ #887] [BZ #19049] [BZ #19050] * sysdeps/generic/fix-int-fp-convert-zero.h: New file. * sysdeps/ieee754/dbl-64/e_log10.c: Include <fix-int-fp-convert-zero.h>. (__ieee754_log10): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/dbl-64/e_log2.c: Include <fix-int-fp-convert-zero.h>. (__ieee754_log2): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/dbl-64/s_erf.c: Include <fix-int-fp-convert-zero.h>. (__erfc): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/dbl-64/s_logb.c: Include <fix-int-fp-convert-zero.h>. (__logb): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/flt-32/e_log10f.c: Include <fix-int-fp-convert-zero.h>. (__ieee754_log10f): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/flt-32/e_log2f.c: Include <fix-int-fp-convert-zero.h>. (__ieee754_log2f): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/flt-32/s_erff.c: Include <fix-int-fp-convert-zero.h>. (__erfcf): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/flt-32/s_logbf.c: Include <fix-int-fp-convert-zero.h>. (__logbf): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/ldbl-128ibm/s_erfl.c: Include <fix-int-fp-convert-zero.h>. (__erfcl): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/ieee754/ldbl-128ibm/s_logbl.c: Include <fix-int-fp-convert-zero.h>. (__logbl): Adjust signs as needed if FIX_INT_FP_CONVERT_ZERO. * sysdeps/powerpc/powerpc32/fpu/configure.ac: New file. * sysdeps/powerpc/powerpc32/fpu/configure: New generated file. * sysdeps/powerpc/powerpc32/fpu/fix-int-fp-convert-zero.h: New file. * config.h.in [_LIBC] (HAVE_PPC_FCFID): New macro.
972 lines
30 KiB
C
972 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 <errno.h>
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#include <float.h>
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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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#include <fix-int-fp-convert-zero.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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/* 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,
|
|
-4.471737717857801230450290232600243795637E1L,
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|
-4.849540386452573306708795324759300320304E1L,
|
|
1.140885264677134679275986782978655952843E1L,
|
|
-6.731591085460269447926746876983786152300E0L,
|
|
1.370831653033047440345050025876085121231E-1L,
|
|
2.022958279982138755020825717073966576670E-2L,
|
|
};
|
|
#define NRDr16 7
|
|
static const long double RDr16[NRDr16 + 1] =
|
|
{
|
|
3.075166170024837215399323264868308087281E3L,
|
|
8.730468942160798031608053127270430036627E2L,
|
|
1.458472799166340479742581949088453244767E3L,
|
|
3.230423687568019709453130785873540386217E2L,
|
|
2.804009872719893612081109617983169474655E2L,
|
|
4.465334221323222943418085830026979293091E1L,
|
|
2.612723259683205928103787842214809134746E1L,
|
|
2.341526751185244109722204018543276124997E0L,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(0.625) = C16a + C16b to extra precision. */
|
|
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
|
|
0.0625 * (16.0 * x + (16.0 * efx) * x) for x = -0
|
|
evaluates to 0. */
|
|
if (x == 0)
|
|
return x;
|
|
long double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
|
|
math_check_force_underflow (ret);
|
|
return ret;
|
|
}
|
|
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 */
|
|
long double ret = (long double) ((hx >> 31) << 1) + one / x;
|
|
if (FIX_INT_FP_CONVERT_ZERO && ret == 0.0L)
|
|
return 0.0L;
|
|
return ret;
|
|
}
|
|
|
|
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;
|
|
default: /* i == 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)
|
|
{
|
|
long double ret = r / x;
|
|
if (ret == 0)
|
|
__set_errno (ERANGE);
|
|
return ret;
|
|
}
|
|
else
|
|
return two - r / x;
|
|
}
|
|
else
|
|
{
|
|
if ((hx & 0x80000000) == 0)
|
|
{
|
|
__set_errno (ERANGE);
|
|
return tiny * tiny;
|
|
}
|
|
else
|
|
return two - tiny;
|
|
}
|
|
}
|
|
|
|
long_double_symbol (libm, __erfcl, erfcl);
|