927 lines
29 KiB
C
927 lines
29 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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/*
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* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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*
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* Permission to use, copy, modify, and distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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*/
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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 <openlibm_math.h>
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#include "math_private.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-4931L,
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one = 1.0L,
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two = 2.0L,
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/* 2/sqrt(pi) - 1 */
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efx = 1.2837916709551257389615890312154517168810E-1L,
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/* 8 * (2/sqrt(pi) - 1) */
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efx8 = 1.0270333367641005911692712249723613735048E0L;
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/* erf(x) = x + x R(x^2)
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0 <= x <= 7/8
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Peak relative error 1.8e-35 */
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#define NTN1 8
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static const long double TN1[NTN1 + 1] =
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{
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-3.858252324254637124543172907442106422373E10L,
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9.580319248590464682316366876952214879858E10L,
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1.302170519734879977595901236693040544854E10L,
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2.922956950426397417800321486727032845006E9L,
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1.764317520783319397868923218385468729799E8L,
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1.573436014601118630105796794840834145120E7L,
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4.028077380105721388745632295157816229289E5L,
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1.644056806467289066852135096352853491530E4L,
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3.390868480059991640235675479463287886081E1L
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};
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#define NTD1 8
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static const long double TD1[NTD1 + 1] =
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{
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-3.005357030696532927149885530689529032152E11L,
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-1.342602283126282827411658673839982164042E11L,
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-2.777153893355340961288511024443668743399E10L,
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-3.483826391033531996955620074072768276974E9L,
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-2.906321047071299585682722511260895227921E8L,
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-1.653347985722154162439387878512427542691E7L,
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-6.245520581562848778466500301865173123136E5L,
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-1.402124304177498828590239373389110545142E4L,
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-1.209368072473510674493129989468348633579E2L
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/* 1.0E0 */
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};
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/* erf(z+1) = erf_const + P(z)/Q(z)
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-.125 <= z <= 0
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Peak relative error 7.3e-36 */
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static const long double erf_const = 0.845062911510467529296875L;
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#define NTN2 8
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static const long double TN2[NTN2 + 1] =
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{
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-4.088889697077485301010486931817357000235E1L,
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7.157046430681808553842307502826960051036E3L,
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-2.191561912574409865550015485451373731780E3L,
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2.180174916555316874988981177654057337219E3L,
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2.848578658049670668231333682379720943455E2L,
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1.630362490952512836762810462174798925274E2L,
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6.317712353961866974143739396865293596895E0L,
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2.450441034183492434655586496522857578066E1L,
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5.127662277706787664956025545897050896203E-1L
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};
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#define NTD2 8
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static const long double TD2[NTD2 + 1] =
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{
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1.731026445926834008273768924015161048885E4L,
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1.209682239007990370796112604286048173750E4L,
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1.160950290217993641320602282462976163857E4L,
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5.394294645127126577825507169061355698157E3L,
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2.791239340533632669442158497532521776093E3L,
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8.989365571337319032943005387378993827684E2L,
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2.974016493766349409725385710897298069677E2L,
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6.148192754590376378740261072533527271947E1L,
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1.178502892490738445655468927408440847480E1L
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/* 1.0E0 */
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};
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/* erfc(x + 0.25) = erfc(0.25) + x R(x)
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0 <= x < 0.125
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Peak relative error 1.4e-35 */
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#define NRNr13 8
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static const long double RNr13[NRNr13 + 1] =
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{
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-2.353707097641280550282633036456457014829E3L,
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3.871159656228743599994116143079870279866E2L,
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-3.888105134258266192210485617504098426679E2L,
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-2.129998539120061668038806696199343094971E1L,
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-8.125462263594034672468446317145384108734E1L,
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8.151549093983505810118308635926270319660E0L,
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-5.033362032729207310462422357772568553670E0L,
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-4.253956621135136090295893547735851168471E-2L,
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-8.098602878463854789780108161581050357814E-2L
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};
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#define NRDr13 7
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static const long double RDr13[NRDr13 + 1] =
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{
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2.220448796306693503549505450626652881752E3L,
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1.899133258779578688791041599040951431383E2L,
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1.061906712284961110196427571557149268454E3L,
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7.497086072306967965180978101974566760042E1L,
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2.146796115662672795876463568170441327274E2L,
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1.120156008362573736664338015952284925592E1L,
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2.211014952075052616409845051695042741074E1L,
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6.469655675326150785692908453094054988938E-1L
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/* 1.0E0 */
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};
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/* erfc(0.25) = C13a + C13b to extra precision. */
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static const long double C13a = 0.723663330078125L;
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static const long double C13b = 1.0279753638067014931732235184287934646022E-5L;
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|
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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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{
|
|
-2.347887943200680563784690094002722906820E3L,
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|
8.008590660692105004780722726421020136482E2L,
|
|
-5.257363310384119728760181252132311447963E2L,
|
|
-4.471737717857801230450290232600243795637E1L,
|
|
-4.849540386452573306708795324759300320304E1L,
|
|
1.140885264677134679275986782978655952843E1L,
|
|
-6.731591085460269447926746876983786152300E0L,
|
|
1.370831653033047440345050025876085121231E-1L,
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|
2.022958279982138755020825717073966576670E-2L,
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};
|
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#define NRDr16 7
|
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static const long double RDr16[NRDr16 + 1] =
|
|
{
|
|
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;
|
|
|
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/* erfc(x + 0.75) = erfc(0.75) + x R(x)
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|
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,
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|
-3.541232266140939050094370552538987982637E1L,
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|
6.988950514747052676394491563585179503865E0L,
|
|
-5.807687216836540830881352383529281215100E0L,
|
|
3.631915988567346438830283503729569443642E-1L,
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-1.488945487149634820537348176770282391202E-2L
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};
|
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#define NRDr17 7
|
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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 */
|
|
};
|
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/* erfc(0.75) = C17a + C17b to extra precision. */
|
|
static const long double C17a = 0.2888336181640625L;
|
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static const long double C17b = 1.0748182422368401062165408589222625794046E-5L;
|
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|
|
|
|
/* 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, sign;
|
|
ieee_quad_shape_type u;
|
|
|
|
u.value = x;
|
|
sign = u.parts32.mswhi;
|
|
ix = sign & 0x7fffffff;
|
|
|
|
if (ix >= 0x7fff0000)
|
|
{ /* erf(nan)=nan */
|
|
i = ((sign & 0xffff0000) >> 31) << 1;
|
|
return (long double) (1 - i) + one / x; /* erf(+-inf)=+-1 */
|
|
}
|
|
|
|
if (ix >= 0x3fff0000) /* |x| >= 1.0 */
|
|
{
|
|
y = erfcl (x);
|
|
return (one - y);
|
|
/* return (one - erfcl (x)); */
|
|
}
|
|
u.parts32.mswhi = ix;
|
|
a = u.value;
|
|
z = x * x;
|
|
if (ix < 0x3ffec000) /* a < 0.875 */
|
|
{
|
|
if (ix < 0x3fc60000) /* |x|<2**-57 */
|
|
{
|
|
if (ix < 0x00080000)
|
|
return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */
|
|
return x + efx * x;
|
|
}
|
|
y = a + a * neval (z, TN1, NTN1) / deval (z, TD1, NTD1);
|
|
}
|
|
else
|
|
{
|
|
a = a - one;
|
|
y = erf_const + neval (a, TN2, NTN2) / deval (a, TD2, NTD2);
|
|
}
|
|
|
|
if (sign & 0x80000000) /* x < 0 */
|
|
y = -y;
|
|
return( y );
|
|
}
|
|
|
|
long double
|
|
erfcl(long double x)
|
|
{
|
|
long double y, z, p, r;
|
|
int32_t i, ix, sign;
|
|
ieee_quad_shape_type u;
|
|
|
|
u.value = x;
|
|
sign = u.parts32.mswhi;
|
|
ix = sign & 0x7fffffff;
|
|
u.parts32.mswhi = ix;
|
|
|
|
if (ix >= 0x7fff0000)
|
|
{ /* erfc(nan)=nan */
|
|
/* erfc(+-inf)=0,2 */
|
|
return (long double) (((u_int32_t) sign >> 31) << 1) + one / x;
|
|
}
|
|
|
|
if (ix < 0x3ffd0000) /* |x| <1/4 */
|
|
{
|
|
if (ix < 0x3f8d0000) /* |x|<2**-114 */
|
|
return one - x;
|
|
return one - erfl (x);
|
|
}
|
|
if (ix < 0x3fff4000) /* 1.25 */
|
|
{
|
|
x = u.value;
|
|
i = 8.0 * x;
|
|
switch (i)
|
|
{
|
|
case 2:
|
|
z = x - 0.25L;
|
|
y = C13b + z * neval (z, RNr13, NRNr13) / deval (z, RDr13, NRDr13);
|
|
y += C13a;
|
|
break;
|
|
case 3:
|
|
z = x - 0.375L;
|
|
y = C14b + z * neval (z, RNr14, NRNr14) / deval (z, RDr14, NRDr14);
|
|
y += C14a;
|
|
break;
|
|
case 4:
|
|
z = x - 0.5L;
|
|
y = C15b + z * neval (z, RNr15, NRNr15) / deval (z, RDr15, NRDr15);
|
|
y += C15a;
|
|
break;
|
|
case 5:
|
|
z = x - 0.625L;
|
|
y = C16b + z * neval (z, RNr16, NRNr16) / deval (z, RDr16, NRDr16);
|
|
y += C16a;
|
|
break;
|
|
case 6:
|
|
z = x - 0.75L;
|
|
y = C17b + z * neval (z, RNr17, NRNr17) / deval (z, RDr17, NRDr17);
|
|
y += C17a;
|
|
break;
|
|
case 7:
|
|
z = x - 0.875L;
|
|
y = C18b + z * neval (z, RNr18, NRNr18) / deval (z, RDr18, NRDr18);
|
|
y += C18a;
|
|
break;
|
|
case 8:
|
|
z = x - 1.0L;
|
|
y = C19b + z * neval (z, RNr19, NRNr19) / deval (z, RDr19, NRDr19);
|
|
y += C19a;
|
|
break;
|
|
case 9:
|
|
z = x - 1.125L;
|
|
y = C20b + z * neval (z, RNr20, NRNr20) / deval (z, RDr20, NRDr20);
|
|
y += C20a;
|
|
break;
|
|
}
|
|
if (sign & 0x80000000)
|
|
y = 2.0L - y;
|
|
return y;
|
|
}
|
|
/* 1.25 < |x| < 107 */
|
|
if (ix < 0x4005ac00)
|
|
{
|
|
/* x < -9 */
|
|
if ((ix >= 0x40022000) && (sign & 0x80000000))
|
|
return two - tiny;
|
|
|
|
x = fabsl (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.parts32.lswlo = 0;
|
|
u.parts32.lswhi &= 0xfe000000;
|
|
z = u.value;
|
|
r = expl (-z * z - 0.5625) *
|
|
expl ((z - x) * (z + x) + p);
|
|
if ((sign & 0x80000000) == 0)
|
|
return r / x;
|
|
else
|
|
return two - r / x;
|
|
}
|
|
else
|
|
{
|
|
if ((sign & 0x80000000) == 0)
|
|
return tiny * tiny;
|
|
else
|
|
return two - tiny;
|
|
}
|
|
}
|