4239f144ce
libquadmath sources are mostly based on glibc sources at present, but derived from them by a manual editing / substitution process and with subsequent manual merges. The manual effort involved in merges means they are sometimes incomplete and long-delayed. Since libquadmath was first created, glibc's support for this format has undergone significant changes so that it can also be used in glibc to provide *f128 functions for the _Float128 type from TS 18661-3. This makes it significantly easier to use it for libquadmath in a more automated fashion, since glibc has a float128_private.h header that redefines many identifiers as macros as needed for building *f128 functions. Simply using float128_private.h directly in libquadmath, with unmodified glibc sources except for changing function names in that one header to be *q instead of *f128, would be tricky, given its dependence on lots of other glibc-internal headers (whereas libquadmath supports non-glibc systems), and also given how some libm functions in glibc are built from type-generic templates using a further set of macros rather than from separate function implementations for each type. So instead this patch adds a script update-quadmath.py to convert glibc sources into libquadmath ones, and the script reads float128_private.h to identify many of the substitutions it should make. quadmath-imp.h is updated with various new internal definitions, taken from glibc as needed; this is the main place expected to need updating manually when subsequent merges from glibc are done using the script. No attempt is made to make the script output match the details of existing formatting, although the differences are of a size that makes a rough comparison (ignoring whitespace) possible. Two new public interfaces are added to libquadmath, exp2q and issignalingq, at a new QUADMATH_1.2 symbol version, since those interfaces are used internally by some of the glibc sources being merged into libquadmath; although there is a new symbol version, no change however is made to the libtool version in the libtool-version file. Although there are various other interfaces now in glibc libm but not in libquadmath, this patch does nothing to add such interfaces (although adding many of them would in fact be easy to do, given the script). One internal file (not providing any public interfaces), math/isinf_nsq.c, is removed, as no longer used by anything in libquadmath after the merge. Conditionals in individual source files on <fenv.h> availability or features are moved into quadmath-imp.h (providing trivial macro versions of the functions if real implementations aren't available), to simplify the substitutions in individual source files. Note however that I haven't tested for any configurations lacking <fenv.h>, so further changes could well be needed there. Two files in libquadmath/math/ are based on glibc sources but not updated in this patch: fmaq.c and rem_pio2q.c. Both could be updated after further changes to the script (and quadmath-imp.h as needed); in the case of rem_pio2q.c, based on two separate glibc source files, those separate files would naturally be split out into separate libquadmath source files in the process (as done in this patch with expq_table.h and tanq_kernel.c, where previously two glibc source files had been merged into one libquadmath source file). complex.c, nanq.c and sqrtq.c are not based on glibc sources (though four of the (trivial) functions in complex.c could readily be replaced by instead using the four corresponding files from glibc, if desired). libquadmath also has printf/ and strtod/ sources based on glibc, also mostly not updated for a long time. Again the script could no doubt be made to generate those automatically, although that would be a larger change (effectively some completely separate logic in the script, not sharing much if anything with the existing code). Bootstrapped with no regressions on x86_64-pc-linux-gnu. PR libquadmath/68686 * Makefile.am: (libquadmath_la_SOURCES): Remove math/isinf_nsq.c. Add math/exp2q.c math/issignalingq.c math/lgammaq_neg.c math/lgammaq_product.c math/tanq_kernel.c math/tgammaq_product.c math/casinhq_kernel.c. * Makefile.in: Regenerate. * libquadmath.texi (exp2q, issignalingq): Document. * quadmath-imp.h: Include <errno.h>, <limits.h>, <stdbool.h> and <fenv.h>. (HIGH_ORDER_BIT_IS_SET_FOR_SNAN, FIX_FLT128_LONG_CONVERT_OVERFLOW) (FIX_FLT128_LLONG_CONVERT_OVERFLOW, __quadmath_kernel_tanq) (__quadmath_gamma_productq, __quadmath_gammaq_r) (__quadmath_lgamma_negq, __quadmath_lgamma_productq) (__quadmath_lgammaq_r, __quadmath_kernel_casinhq, mul_splitq) (math_check_force_underflow_complex, __glibc_likely) (__glibc_unlikely, struct rm_ctx, SET_RESTORE_ROUNDF128) (libc_feholdsetround_ctx, libc_feresetround_ctx): New. (feraiseexcept, fenv_t, feholdexcept, fesetround, feupdateenv) (fesetenv, fetestexcept, feclearexcept): Define if not supported through <fenv.h>. (__quadmath_isinf_nsq): Remove. * quadmath.h (exp2q, issignalingq): New. * quadmath.map (QUADMATH_1.2): New. * quadmath_weak.h (exp2q, issignalingq): New. * update-quadmath.py: New file. * math/isinf_nsq.c: Remove file. * math/casinhq_kernel.c, math/exp2q.c, math/expq_table.h, math/issignalingq.c, math/lgammaq_neg.c, math/lgammaq_product.c, math/tanq_kernel.c, math/tgammaq_product.c: New files. Generated from glibc sources with update-quadmath.py. * math/acoshq.c, math/acosq.c, math/asinhq.c, math/asinq.c, math/atan2q.c, math/atanhq.c, math/atanq.c, math/cacoshq.c, math/cacosq.c, math/casinhq.c, math/casinq.c, math/catanhq.c, math/catanq.c, math/cbrtq.c, math/ccoshq.c, math/ceilq.c, math/cexpq.c, math/cimagq.c, math/clog10q.c, math/clogq.c, math/conjq.c, math/copysignq.c, math/coshq.c, math/cosq.c, math/cosq_kernel.c, math/cprojq.c, math/crealq.c, math/csinhq.c, math/csinq.c, math/csqrtq.c, math/ctanhq.c, math/ctanq.c, math/erfq.c, math/expm1q.c, math/expq.c, math/fabsq.c, math/fdimq.c, math/finiteq.c, math/floorq.c, math/fmaxq.c, math/fminq.c, math/fmodq.c, math/frexpq.c, math/hypotq.c, math/ilogbq.c, math/isinfq.c, math/isnanq.c, math/j0q.c, math/j1q.c, math/jnq.c, math/ldexpq.c, math/lgammaq.c, math/llrintq.c, math/llroundq.c, math/log10q.c, math/log1pq.c, math/log2q.c, math/logbq.c, math/logq.c, math/lrintq.c, math/lroundq.c, math/modfq.c, math/nearbyintq.c, math/nextafterq.c, math/powq.c, math/remainderq.c, math/remquoq.c, math/rintq.c, math/roundq.c, math/scalblnq.c, math/scalbnq.c, math/signbitq.c, math/sincos_table.c, math/sincosq.c, math/sincosq_kernel.c, math/sinhq.c, math/sinq.c, math/sinq_kernel.c, math/tanhq.c, math/tanq.c, math/tgammaq.c, math/truncq.c, math/x2y2m1q.c: Regenerate from glibc sources with update-quadmath.py. From-SVN: r265822
944 lines
30 KiB
C
944 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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*
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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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* 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 "quadmath-imp.h"
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/* Evaluate P[n] x^n + P[n-1] x^(n-1) + ... + P[0] */
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static __float128
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neval (__float128 x, const __float128 *p, int n)
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{
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__float128 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 __float128
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deval (__float128 x, const __float128 *p, int n)
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{
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__float128 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 __float128
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tiny = 1e-4931Q,
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one = 1,
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two = 2,
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/* 2/sqrt(pi) - 1 */
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efx = 1.2837916709551257389615890312154517168810E-1Q;
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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 __float128 TN1[NTN1 + 1] =
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{
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-3.858252324254637124543172907442106422373E10Q,
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9.580319248590464682316366876952214879858E10Q,
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1.302170519734879977595901236693040544854E10Q,
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2.922956950426397417800321486727032845006E9Q,
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1.764317520783319397868923218385468729799E8Q,
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1.573436014601118630105796794840834145120E7Q,
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4.028077380105721388745632295157816229289E5Q,
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1.644056806467289066852135096352853491530E4Q,
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3.390868480059991640235675479463287886081E1Q
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};
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#define NTD1 8
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static const __float128 TD1[NTD1 + 1] =
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{
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-3.005357030696532927149885530689529032152E11Q,
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-1.342602283126282827411658673839982164042E11Q,
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-2.777153893355340961288511024443668743399E10Q,
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-3.483826391033531996955620074072768276974E9Q,
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-2.906321047071299585682722511260895227921E8Q,
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-1.653347985722154162439387878512427542691E7Q,
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-6.245520581562848778466500301865173123136E5Q,
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-1.402124304177498828590239373389110545142E4Q,
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-1.209368072473510674493129989468348633579E2Q
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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 __float128 erf_const = 0.845062911510467529296875Q;
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#define NTN2 8
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static const __float128 TN2[NTN2 + 1] =
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{
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-4.088889697077485301010486931817357000235E1Q,
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7.157046430681808553842307502826960051036E3Q,
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-2.191561912574409865550015485451373731780E3Q,
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2.180174916555316874988981177654057337219E3Q,
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2.848578658049670668231333682379720943455E2Q,
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1.630362490952512836762810462174798925274E2Q,
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6.317712353961866974143739396865293596895E0Q,
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2.450441034183492434655586496522857578066E1Q,
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5.127662277706787664956025545897050896203E-1Q
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};
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#define NTD2 8
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static const __float128 TD2[NTD2 + 1] =
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{
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1.731026445926834008273768924015161048885E4Q,
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1.209682239007990370796112604286048173750E4Q,
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1.160950290217993641320602282462976163857E4Q,
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5.394294645127126577825507169061355698157E3Q,
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2.791239340533632669442158497532521776093E3Q,
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8.989365571337319032943005387378993827684E2Q,
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2.974016493766349409725385710897298069677E2Q,
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6.148192754590376378740261072533527271947E1Q,
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1.178502892490738445655468927408440847480E1Q
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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 __float128 RNr13[NRNr13 + 1] =
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{
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-2.353707097641280550282633036456457014829E3Q,
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3.871159656228743599994116143079870279866E2Q,
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-3.888105134258266192210485617504098426679E2Q,
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-2.129998539120061668038806696199343094971E1Q,
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-8.125462263594034672468446317145384108734E1Q,
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8.151549093983505810118308635926270319660E0Q,
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-5.033362032729207310462422357772568553670E0Q,
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-4.253956621135136090295893547735851168471E-2Q,
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-8.098602878463854789780108161581050357814E-2Q
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};
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#define NRDr13 7
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static const __float128 RDr13[NRDr13 + 1] =
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{
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2.220448796306693503549505450626652881752E3Q,
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1.899133258779578688791041599040951431383E2Q,
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1.061906712284961110196427571557149268454E3Q,
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7.497086072306967965180978101974566760042E1Q,
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2.146796115662672795876463568170441327274E2Q,
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1.120156008362573736664338015952284925592E1Q,
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2.211014952075052616409845051695042741074E1Q,
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6.469655675326150785692908453094054988938E-1Q
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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 __float128 C13a = 0.723663330078125Q;
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static const __float128 C13b = 1.0279753638067014931732235184287934646022E-5Q;
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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 __float128 RNr14[NRNr14 + 1] =
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{
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-2.446164016404426277577283038988918202456E3Q,
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6.718753324496563913392217011618096698140E2Q,
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-4.581631138049836157425391886957389240794E2Q,
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-2.382844088987092233033215402335026078208E1Q,
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-7.119237852400600507927038680970936336458E1Q,
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1.313609646108420136332418282286454287146E1Q,
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-6.188608702082264389155862490056401365834E0Q,
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-2.787116601106678287277373011101132659279E-2Q,
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-2.230395570574153963203348263549700967918E-2Q
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};
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#define NRDr14 7
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static const __float128 RDr14[NRDr14 + 1] =
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{
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2.495187439241869732696223349840963702875E3Q,
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2.503549449872925580011284635695738412162E2Q,
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1.159033560988895481698051531263861842461E3Q,
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9.493751466542304491261487998684383688622E1Q,
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2.276214929562354328261422263078480321204E2Q,
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1.367697521219069280358984081407807931847E1Q,
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2.276988395995528495055594829206582732682E1Q,
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7.647745753648996559837591812375456641163E-1Q
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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 __float128 C14a = 0.5958709716796875Q;
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static const __float128 C14b = 1.2118885490201676174914080878232469565953E-5Q;
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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 __float128 RNr15[NRNr15 + 1] =
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{
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-2.624212418011181487924855581955853461925E3Q,
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8.473828904647825181073831556439301342756E2Q,
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-5.286207458628380765099405359607331669027E2Q,
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-3.895781234155315729088407259045269652318E1Q,
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-6.200857908065163618041240848728398496256E1Q,
|
|
1.469324610346924001393137895116129204737E1Q,
|
|
-6.961356525370658572800674953305625578903E0Q,
|
|
5.145724386641163809595512876629030548495E-3Q,
|
|
1.990253655948179713415957791776180406812E-2Q
|
|
};
|
|
#define NRDr15 7
|
|
static const __float128 RDr15[NRDr15 + 1] =
|
|
{
|
|
2.986190760847974943034021764693341524962E3Q,
|
|
5.288262758961073066335410218650047725985E2Q,
|
|
1.363649178071006978355113026427856008978E3Q,
|
|
1.921707975649915894241864988942255320833E2Q,
|
|
2.588651100651029023069013885900085533226E2Q,
|
|
2.628752920321455606558942309396855629459E1Q,
|
|
2.455649035885114308978333741080991380610E1Q,
|
|
1.378826653595128464383127836412100939126E0Q
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(0.5) = C15a + C15b to extra precision. */
|
|
static const __float128 C15a = 0.4794921875Q;
|
|
static const __float128 C15b = 7.9346869534623172533461080354712635484242E-6Q;
|
|
|
|
/* erfc(x + 0.625) = erfc(0.625) + x R(x)
|
|
0 <= x < 0.125
|
|
Peak relative error 5.1e-36 */
|
|
#define NRNr16 8
|
|
static const __float128 RNr16[NRNr16 + 1] =
|
|
{
|
|
-2.347887943200680563784690094002722906820E3Q,
|
|
8.008590660692105004780722726421020136482E2Q,
|
|
-5.257363310384119728760181252132311447963E2Q,
|
|
-4.471737717857801230450290232600243795637E1Q,
|
|
-4.849540386452573306708795324759300320304E1Q,
|
|
1.140885264677134679275986782978655952843E1Q,
|
|
-6.731591085460269447926746876983786152300E0Q,
|
|
1.370831653033047440345050025876085121231E-1Q,
|
|
2.022958279982138755020825717073966576670E-2Q,
|
|
};
|
|
#define NRDr16 7
|
|
static const __float128 RDr16[NRDr16 + 1] =
|
|
{
|
|
3.075166170024837215399323264868308087281E3Q,
|
|
8.730468942160798031608053127270430036627E2Q,
|
|
1.458472799166340479742581949088453244767E3Q,
|
|
3.230423687568019709453130785873540386217E2Q,
|
|
2.804009872719893612081109617983169474655E2Q,
|
|
4.465334221323222943418085830026979293091E1Q,
|
|
2.612723259683205928103787842214809134746E1Q,
|
|
2.341526751185244109722204018543276124997E0Q,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(0.625) = C16a + C16b to extra precision. */
|
|
static const __float128 C16a = 0.3767547607421875Q;
|
|
static const __float128 C16b = 4.3570693945275513594941232097252997287766E-6Q;
|
|
|
|
/* 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 __float128 RNr17[NRNr17 + 1] =
|
|
{
|
|
-1.767068734220277728233364375724380366826E3Q,
|
|
6.693746645665242832426891888805363898707E2Q,
|
|
-4.746224241837275958126060307406616817753E2Q,
|
|
-2.274160637728782675145666064841883803196E1Q,
|
|
-3.541232266140939050094370552538987982637E1Q,
|
|
6.988950514747052676394491563585179503865E0Q,
|
|
-5.807687216836540830881352383529281215100E0Q,
|
|
3.631915988567346438830283503729569443642E-1Q,
|
|
-1.488945487149634820537348176770282391202E-2Q
|
|
};
|
|
#define NRDr17 7
|
|
static const __float128 RDr17[NRDr17 + 1] =
|
|
{
|
|
2.748457523498150741964464942246913394647E3Q,
|
|
1.020213390713477686776037331757871252652E3Q,
|
|
1.388857635935432621972601695296561952738E3Q,
|
|
3.903363681143817750895999579637315491087E2Q,
|
|
2.784568344378139499217928969529219886578E2Q,
|
|
5.555800830216764702779238020065345401144E1Q,
|
|
2.646215470959050279430447295801291168941E1Q,
|
|
2.984905282103517497081766758550112011265E0Q,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(0.75) = C17a + C17b to extra precision. */
|
|
static const __float128 C17a = 0.2888336181640625Q;
|
|
static const __float128 C17b = 1.0748182422368401062165408589222625794046E-5Q;
|
|
|
|
|
|
/* 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 __float128 RNr18[NRNr18 + 1] =
|
|
{
|
|
-1.342044899087593397419622771847219619588E3Q,
|
|
6.127221294229172997509252330961641850598E2Q,
|
|
-4.519821356522291185621206350470820610727E2Q,
|
|
1.223275177825128732497510264197915160235E1Q,
|
|
-2.730789571382971355625020710543532867692E1Q,
|
|
4.045181204921538886880171727755445395862E0Q,
|
|
-4.925146477876592723401384464691452700539E0Q,
|
|
5.933878036611279244654299924101068088582E-1Q,
|
|
-5.557645435858916025452563379795159124753E-2Q
|
|
};
|
|
#define NRDr18 7
|
|
static const __float128 RDr18[NRDr18 + 1] =
|
|
{
|
|
2.557518000661700588758505116291983092951E3Q,
|
|
1.070171433382888994954602511991940418588E3Q,
|
|
1.344842834423493081054489613250688918709E3Q,
|
|
4.161144478449381901208660598266288188426E2Q,
|
|
2.763670252219855198052378138756906980422E2Q,
|
|
5.998153487868943708236273854747564557632E1Q,
|
|
2.657695108438628847733050476209037025318E1Q,
|
|
3.252140524394421868923289114410336976512E0Q,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(0.875) = C18a + C18b to extra precision. */
|
|
static const __float128 C18a = 0.215911865234375Q;
|
|
static const __float128 C18b = 1.3073705765341685464282101150637224028267E-5Q;
|
|
|
|
/* 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 __float128 RNr19[NRNr19 + 1] =
|
|
{
|
|
-1.139180936454157193495882956565663294826E3Q,
|
|
6.134903129086899737514712477207945973616E2Q,
|
|
-4.628909024715329562325555164720732868263E2Q,
|
|
4.165702387210732352564932347500364010833E1Q,
|
|
-2.286979913515229747204101330405771801610E1Q,
|
|
1.870695256449872743066783202326943667722E0Q,
|
|
-4.177486601273105752879868187237000032364E0Q,
|
|
7.533980372789646140112424811291782526263E-1Q,
|
|
-8.629945436917752003058064731308767664446E-2Q
|
|
};
|
|
#define NRDr19 7
|
|
static const __float128 RDr19[NRDr19 + 1] =
|
|
{
|
|
2.744303447981132701432716278363418643778E3Q,
|
|
1.266396359526187065222528050591302171471E3Q,
|
|
1.466739461422073351497972255511919814273E3Q,
|
|
4.868710570759693955597496520298058147162E2Q,
|
|
2.993694301559756046478189634131722579643E2Q,
|
|
6.868976819510254139741559102693828237440E1Q,
|
|
2.801505816247677193480190483913753613630E1Q,
|
|
3.604439909194350263552750347742663954481E0Q,
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(1.0) = C19a + C19b to extra precision. */
|
|
static const __float128 C19a = 0.15728759765625Q;
|
|
static const __float128 C19b = 1.1609394035130658779364917390740703933002E-5Q;
|
|
|
|
/* 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 __float128 RNr20[NRNr20 + 1] =
|
|
{
|
|
-9.652706916457973956366721379612508047640E2Q,
|
|
5.577066396050932776683469951773643880634E2Q,
|
|
-4.406335508848496713572223098693575485978E2Q,
|
|
5.202893466490242733570232680736966655434E1Q,
|
|
-1.931311847665757913322495948705563937159E1Q,
|
|
-9.364318268748287664267341457164918090611E-2Q,
|
|
-3.306390351286352764891355375882586201069E0Q,
|
|
7.573806045289044647727613003096916516475E-1Q,
|
|
-9.611744011489092894027478899545635991213E-2Q
|
|
};
|
|
#define NRDr20 7
|
|
static const __float128 RDr20[NRDr20 + 1] =
|
|
{
|
|
3.032829629520142564106649167182428189014E3Q,
|
|
1.659648470721967719961167083684972196891E3Q,
|
|
1.703545128657284619402511356932569292535E3Q,
|
|
6.393465677731598872500200253155257708763E2Q,
|
|
3.489131397281030947405287112726059221934E2Q,
|
|
8.848641738570783406484348434387611713070E1Q,
|
|
3.132269062552392974833215844236160958502E1Q,
|
|
4.430131663290563523933419966185230513168E0Q
|
|
/* 1.0E0 */
|
|
};
|
|
/* erfc(1.125) = C20a + C20b to extra precision. */
|
|
static const __float128 C20a = 0.111602783203125Q;
|
|
static const __float128 C20b = 8.9850951672359304215530728365232161564636E-6Q;
|
|
|
|
/* 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 __float128 RNr8[NRNr8 + 1] =
|
|
{
|
|
3.587451489255356250759834295199296936784E1Q,
|
|
5.406249749087340431871378009874875889602E2Q,
|
|
2.931301290625250886238822286506381194157E3Q,
|
|
7.359254185241795584113047248898753470923E3Q,
|
|
9.201031849810636104112101947312492532314E3Q,
|
|
5.749697096193191467751650366613289284777E3Q,
|
|
1.710415234419860825710780802678697889231E3Q,
|
|
2.150753982543378580859546706243022719599E2Q,
|
|
8.740953582272147335100537849981160931197E0Q,
|
|
4.876422978828717219629814794707963640913E-2Q
|
|
};
|
|
#define NRDr8 8
|
|
static const __float128 RDr8[NRDr8 + 1] =
|
|
{
|
|
6.358593134096908350929496535931630140282E1Q,
|
|
9.900253816552450073757174323424051765523E2Q,
|
|
5.642928777856801020545245437089490805186E3Q,
|
|
1.524195375199570868195152698617273739609E4Q,
|
|
2.113829644500006749947332935305800887345E4Q,
|
|
1.526438562626465706267943737310282977138E4Q,
|
|
5.561370922149241457131421914140039411782E3Q,
|
|
9.394035530179705051609070428036834496942E2Q,
|
|
6.147019596150394577984175188032707343615E1Q
|
|
/* 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 __float128 RNr7[NRNr7 + 1] =
|
|
{
|
|
1.686222193385987690785945787708644476545E1Q,
|
|
1.178224543567604215602418571310612066594E3Q,
|
|
1.764550584290149466653899886088166091093E4Q,
|
|
1.073758321890334822002849369898232811561E5Q,
|
|
3.132840749205943137619839114451290324371E5Q,
|
|
4.607864939974100224615527007793867585915E5Q,
|
|
3.389781820105852303125270837910972384510E5Q,
|
|
1.174042187110565202875011358512564753399E5Q,
|
|
1.660013606011167144046604892622504338313E4Q,
|
|
6.700393957480661937695573729183733234400E2Q
|
|
};
|
|
#define NRDr7 9
|
|
static const __float128 RDr7[NRDr7 + 1] =
|
|
{
|
|
-1.709305024718358874701575813642933561169E3Q,
|
|
-3.280033887481333199580464617020514788369E4Q,
|
|
-2.345284228022521885093072363418750835214E5Q,
|
|
-8.086758123097763971926711729242327554917E5Q,
|
|
-1.456900414510108718402423999575992450138E6Q,
|
|
-1.391654264881255068392389037292702041855E6Q,
|
|
-6.842360801869939983674527468509852583855E5Q,
|
|
-1.597430214446573566179675395199807533371E5Q,
|
|
-1.488876130609876681421645314851760773480E4Q,
|
|
-3.511762950935060301403599443436465645703E2Q
|
|
/* 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 __float128 RNr6[NRNr6 + 1] =
|
|
{
|
|
1.642076876176834390623842732352935761108E0Q,
|
|
1.207150003611117689000664385596211076662E2Q,
|
|
2.119260779316389904742873816462800103939E3Q,
|
|
1.562942227734663441801452930916044224174E4Q,
|
|
5.656779189549710079988084081145693580479E4Q,
|
|
1.052166241021481691922831746350942786299E5Q,
|
|
9.949798524786000595621602790068349165758E4Q,
|
|
4.491790734080265043407035220188849562856E4Q,
|
|
8.377074098301530326270432059434791287601E3Q,
|
|
4.506934806567986810091824791963991057083E2Q
|
|
};
|
|
#define NRDr6 9
|
|
static const __float128 RDr6[NRDr6 + 1] =
|
|
{
|
|
-1.664557643928263091879301304019826629067E2Q,
|
|
-3.800035902507656624590531122291160668452E3Q,
|
|
-3.277028191591734928360050685359277076056E4Q,
|
|
-1.381359471502885446400589109566587443987E5Q,
|
|
-3.082204287382581873532528989283748656546E5Q,
|
|
-3.691071488256738343008271448234631037095E5Q,
|
|
-2.300482443038349815750714219117566715043E5Q,
|
|
-6.873955300927636236692803579555752171530E4Q,
|
|
-8.262158817978334142081581542749986845399E3Q,
|
|
-2.517122254384430859629423488157361983661E2Q
|
|
/* 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 __float128 RNr5[NRNr5 + 1] =
|
|
{
|
|
-3.332258927455285458355550878136506961608E-3Q,
|
|
-2.697100758900280402659586595884478660721E-1Q,
|
|
-6.083328551139621521416618424949137195536E0Q,
|
|
-6.119863528983308012970821226810162441263E1Q,
|
|
-3.176535282475593173248810678636522589861E2Q,
|
|
-8.933395175080560925809992467187963260693E2Q,
|
|
-1.360019508488475978060917477620199499560E3Q,
|
|
-1.075075579828188621541398761300910213280E3Q,
|
|
-4.017346561586014822824459436695197089916E2Q,
|
|
-5.857581368145266249509589726077645791341E1Q,
|
|
-2.077715925587834606379119585995758954399E0Q
|
|
};
|
|
#define NRDr5 9
|
|
static const __float128 RDr5[NRDr5 + 1] =
|
|
{
|
|
3.377879570417399341550710467744693125385E-1Q,
|
|
1.021963322742390735430008860602594456187E1Q,
|
|
1.200847646592942095192766255154827011939E2Q,
|
|
7.118915528142927104078182863387116942836E2Q,
|
|
2.318159380062066469386544552429625026238E3Q,
|
|
4.238729853534009221025582008928765281620E3Q,
|
|
4.279114907284825886266493994833515580782E3Q,
|
|
2.257277186663261531053293222591851737504E3Q,
|
|
5.570475501285054293371908382916063822957E2Q,
|
|
5.142189243856288981145786492585432443560E1Q
|
|
/* 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 __float128 RNr4[NRNr4 + 1] =
|
|
{
|
|
3.258530712024527835089319075288494524465E-3Q,
|
|
2.987056016877277929720231688689431056567E-1Q,
|
|
8.738729089340199750734409156830371528862E0Q,
|
|
1.207211160148647782396337792426311125923E2Q,
|
|
8.997558632489032902250523945248208224445E2Q,
|
|
3.798025197699757225978410230530640879762E3Q,
|
|
9.113203668683080975637043118209210146846E3Q,
|
|
1.203285891339933238608683715194034900149E4Q,
|
|
8.100647057919140328536743641735339740855E3Q,
|
|
2.383888249907144945837976899822927411769E3Q,
|
|
2.127493573166454249221983582495245662319E2Q
|
|
};
|
|
#define NRDr4 10
|
|
static const __float128 RDr4[NRDr4 + 1] =
|
|
{
|
|
-3.303141981514540274165450687270180479586E-1Q,
|
|
-1.353768629363605300707949368917687066724E1Q,
|
|
-2.206127630303621521950193783894598987033E2Q,
|
|
-1.861800338758066696514480386180875607204E3Q,
|
|
-8.889048775872605708249140016201753255599E3Q,
|
|
-2.465888106627948210478692168261494857089E4Q,
|
|
-3.934642211710774494879042116768390014289E4Q,
|
|
-3.455077258242252974937480623730228841003E4Q,
|
|
-1.524083977439690284820586063729912653196E4Q,
|
|
-2.810541887397984804237552337349093953857E3Q,
|
|
-1.343929553541159933824901621702567066156E2Q
|
|
/* 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 __float128 RNr3[NRNr3 + 1] =
|
|
{
|
|
-1.952401126551202208698629992497306292987E-6Q,
|
|
-2.130881743066372952515162564941682716125E-4Q,
|
|
-8.376493958090190943737529486107282224387E-3Q,
|
|
-1.650592646560987700661598877522831234791E-1Q,
|
|
-1.839290818933317338111364667708678163199E0Q,
|
|
-1.216278715570882422410442318517814388470E1Q,
|
|
-4.818759344462360427612133632533779091386E1Q,
|
|
-1.120994661297476876804405329172164436784E2Q,
|
|
-1.452850765662319264191141091859300126931E2Q,
|
|
-9.485207851128957108648038238656777241333E1Q,
|
|
-2.563663855025796641216191848818620020073E1Q,
|
|
-1.787995944187565676837847610706317833247E0Q
|
|
};
|
|
#define NRDr3 10
|
|
static const __float128 RDr3[NRDr3 + 1] =
|
|
{
|
|
1.979130686770349481460559711878399476903E-4Q,
|
|
1.156941716128488266238105813374635099057E-2Q,
|
|
2.752657634309886336431266395637285974292E-1Q,
|
|
3.482245457248318787349778336603569327521E0Q,
|
|
2.569347069372696358578399521203959253162E1Q,
|
|
1.142279000180457419740314694631879921561E2Q,
|
|
3.056503977190564294341422623108332700840E2Q,
|
|
4.780844020923794821656358157128719184422E2Q,
|
|
4.105972727212554277496256802312730410518E2Q,
|
|
1.724072188063746970865027817017067646246E2Q,
|
|
2.815939183464818198705278118326590370435E1Q
|
|
/* 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 __float128 RNr2[NRNr2 + 1] =
|
|
{
|
|
-2.638914383420287212401687401284326363787E-8Q,
|
|
-3.479198370260633977258201271399116766619E-6Q,
|
|
-1.783985295335697686382487087502222519983E-4Q,
|
|
-4.777876933122576014266349277217559356276E-3Q,
|
|
-7.450634738987325004070761301045014986520E-2Q,
|
|
-7.068318854874733315971973707247467326619E-1Q,
|
|
-4.113919921935944795764071670806867038732E0Q,
|
|
-1.440447573226906222417767283691888875082E1Q,
|
|
-2.883484031530718428417168042141288943905E1Q,
|
|
-2.990886974328476387277797361464279931446E1Q,
|
|
-1.325283914915104866248279787536128997331E1Q,
|
|
-1.572436106228070195510230310658206154374E0Q
|
|
};
|
|
#define NRDr2 10
|
|
static const __float128 RDr2[NRDr2 + 1] =
|
|
{
|
|
2.675042728136731923554119302571867799673E-6Q,
|
|
2.170997868451812708585443282998329996268E-4Q,
|
|
7.249969752687540289422684951196241427445E-3Q,
|
|
1.302040375859768674620410563307838448508E-1Q,
|
|
1.380202483082910888897654537144485285549E0Q,
|
|
8.926594113174165352623847870299170069350E0Q,
|
|
3.521089584782616472372909095331572607185E1Q,
|
|
8.233547427533181375185259050330809105570E1Q,
|
|
1.072971579885803033079469639073292840135E2Q,
|
|
6.943803113337964469736022094105143158033E1Q,
|
|
1.775695341031607738233608307835017282662E1Q
|
|
/* 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 __float128 RNr1[NRNr1 + 1] =
|
|
{
|
|
-4.250780883202361946697751475473042685782E-8Q,
|
|
-5.375777053288612282487696975623206383019E-6Q,
|
|
-2.573645949220896816208565944117382460452E-4Q,
|
|
-6.199032928113542080263152610799113086319E-3Q,
|
|
-8.262721198693404060380104048479916247786E-2Q,
|
|
-6.242615227257324746371284637695778043982E-1Q,
|
|
-2.609874739199595400225113299437099626386E0Q,
|
|
-5.581967563336676737146358534602770006970E0Q,
|
|
-5.124398923356022609707490956634280573882E0Q,
|
|
-1.290865243944292370661544030414667556649E0Q
|
|
};
|
|
#define NRDr1 8
|
|
static const __float128 RDr1[NRDr1 + 1] =
|
|
{
|
|
4.308976661749509034845251315983612976224E-6Q,
|
|
3.265390126432780184125233455960049294580E-4Q,
|
|
9.811328839187040701901866531796570418691E-3Q,
|
|
1.511222515036021033410078631914783519649E-1Q,
|
|
1.289264341917429958858379585970225092274E0Q,
|
|
6.147640356182230769548007536914983522270E0Q,
|
|
1.573966871337739784518246317003956180750E1Q,
|
|
1.955534123435095067199574045529218238263E1Q,
|
|
9.472613121363135472247929109615785855865E0Q
|
|
/* 1.0E0 */
|
|
};
|
|
|
|
|
|
__float128
|
|
erfq (__float128 x)
|
|
{
|
|
__float128 a, y, z;
|
|
int32_t i, ix, sign;
|
|
ieee854_float128 u;
|
|
|
|
u.value = x;
|
|
sign = u.words32.w0;
|
|
ix = sign & 0x7fffffff;
|
|
|
|
if (ix >= 0x7fff0000)
|
|
{ /* erf(nan)=nan */
|
|
i = ((sign & 0xffff0000) >> 31) << 1;
|
|
return (__float128) (1 - i) + one / x; /* erf(+-inf)=+-1 */
|
|
}
|
|
|
|
if (ix >= 0x3fff0000) /* |x| >= 1.0 */
|
|
{
|
|
if (ix >= 0x40030000 && sign > 0)
|
|
return one; /* x >= 16, avoid spurious underflow from erfc. */
|
|
y = erfcq (x);
|
|
return (one - y);
|
|
/* return (one - erfcq (x)); */
|
|
}
|
|
u.words32.w0 = ix;
|
|
a = u.value;
|
|
z = x * x;
|
|
if (ix < 0x3ffec000) /* a < 0.875 */
|
|
{
|
|
if (ix < 0x3fc60000) /* |x|<2**-57 */
|
|
{
|
|
if (ix < 0x00080000)
|
|
{
|
|
/* Avoid spurious underflow. */
|
|
__float128 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 (sign & 0x80000000) /* x < 0 */
|
|
y = -y;
|
|
return( y );
|
|
}
|
|
|
|
|
|
__float128
|
|
erfcq (__float128 x)
|
|
{
|
|
__float128 y, z, p, r;
|
|
int32_t i, ix, sign;
|
|
ieee854_float128 u;
|
|
|
|
u.value = x;
|
|
sign = u.words32.w0;
|
|
ix = sign & 0x7fffffff;
|
|
u.words32.w0 = ix;
|
|
|
|
if (ix >= 0x7fff0000)
|
|
{ /* erfc(nan)=nan */
|
|
/* erfc(+-inf)=0,2 */
|
|
return (__float128) (((uint32_t) sign >> 31) << 1) + one / x;
|
|
}
|
|
|
|
if (ix < 0x3ffd0000) /* |x| <1/4 */
|
|
{
|
|
if (ix < 0x3f8d0000) /* |x|<2**-114 */
|
|
return one - x;
|
|
return one - erfq (x);
|
|
}
|
|
if (ix < 0x3fff4000) /* 1.25 */
|
|
{
|
|
x = u.value;
|
|
i = 8.0 * x;
|
|
switch (i)
|
|
{
|
|
case 2:
|
|
z = x - 0.25Q;
|
|
y = C13b + z * neval (z, RNr13, NRNr13) / deval (z, RDr13, NRDr13);
|
|
y += C13a;
|
|
break;
|
|
case 3:
|
|
z = x - 0.375Q;
|
|
y = C14b + z * neval (z, RNr14, NRNr14) / deval (z, RDr14, NRDr14);
|
|
y += C14a;
|
|
break;
|
|
case 4:
|
|
z = x - 0.5Q;
|
|
y = C15b + z * neval (z, RNr15, NRNr15) / deval (z, RDr15, NRDr15);
|
|
y += C15a;
|
|
break;
|
|
case 5:
|
|
z = x - 0.625Q;
|
|
y = C16b + z * neval (z, RNr16, NRNr16) / deval (z, RDr16, NRDr16);
|
|
y += C16a;
|
|
break;
|
|
case 6:
|
|
z = x - 0.75Q;
|
|
y = C17b + z * neval (z, RNr17, NRNr17) / deval (z, RDr17, NRDr17);
|
|
y += C17a;
|
|
break;
|
|
case 7:
|
|
z = x - 0.875Q;
|
|
y = C18b + z * neval (z, RNr18, NRNr18) / deval (z, RDr18, NRDr18);
|
|
y += C18a;
|
|
break;
|
|
case 8:
|
|
z = x - 1;
|
|
y = C19b + z * neval (z, RNr19, NRNr19) / deval (z, RDr19, NRDr19);
|
|
y += C19a;
|
|
break;
|
|
default: /* i == 9. */
|
|
z = x - 1.125Q;
|
|
y = C20b + z * neval (z, RNr20, NRNr20) / deval (z, RDr20, NRDr20);
|
|
y += C20a;
|
|
break;
|
|
}
|
|
if (sign & 0x80000000)
|
|
y = 2 - y;
|
|
return y;
|
|
}
|
|
/* 1.25 < |x| < 107 */
|
|
if (ix < 0x4005ac00)
|
|
{
|
|
/* x < -9 */
|
|
if ((ix >= 0x40022000) && (sign & 0x80000000))
|
|
return two - tiny;
|
|
|
|
x = fabsq (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;
|
|
}
|
|
u.value = x;
|
|
u.words32.w3 = 0;
|
|
u.words32.w2 &= 0xfe000000;
|
|
z = u.value;
|
|
r = expq (-z * z - 0.5625) *
|
|
expq ((z - x) * (z + x) + p);
|
|
if ((sign & 0x80000000) == 0)
|
|
{
|
|
__float128 ret = r / x;
|
|
if (ret == 0)
|
|
errno = ERANGE;
|
|
return ret;
|
|
}
|
|
else
|
|
return two - r / x;
|
|
}
|
|
else
|
|
{
|
|
if ((sign & 0x80000000) == 0)
|
|
{
|
|
errno = ERANGE;
|
|
return tiny * tiny;
|
|
}
|
|
else
|
|
return two - tiny;
|
|
}
|
|
}
|