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394d998315
At least in neovim, `│vi:` is not recognized as a modeline because it has no preceding whitespace. After fixing this, opening a file yields an error because `net` is not an option. (`noet`, however, is.)
341 lines
14 KiB
C
341 lines
14 KiB
C
/*-*- mode:c;indent-tabs-mode:t;c-basic-offset:8;tab-width:8;coding:utf-8 -*-│
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│ vi: set et ft=c ts=8 tw=8 fenc=utf-8 :vi │
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╚──────────────────────────────────────────────────────────────────────────────╝
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│ │
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│ Musl Libc │
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│ Copyright © 2005-2014 Rich Felker, et al. │
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│ │
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│ Permission is hereby granted, free of charge, to any person obtaining │
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│ a copy of this software and associated documentation files (the │
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│ "Software"), to deal in the Software without restriction, including │
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│ without limitation the rights to use, copy, modify, merge, publish, │
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│ distribute, sublicense, and/or sell copies of the Software, and to │
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│ permit persons to whom the Software is furnished to do so, subject to │
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│ the following conditions: │
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│ │
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│ The above copyright notice and this permission notice shall be │
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│ included in all copies or substantial portions of the Software. │
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│ │
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│ THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, │
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│ EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF │
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│ MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. │
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│ IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY │
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│ CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, │
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│ TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE │
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│ SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. │
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│ │
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╚─────────────────────────────────────────────────────────────────────────────*/
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#include "libc/math.h"
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asm(".ident\t\"\\n\\n\
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fdlibm (fdlibm license)\\n\
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Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.\"");
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asm(".ident\t\"\\n\\n\
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Musl libc (MIT License)\\n\
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Copyright 2005-2014 Rich Felker, et. al.\"");
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asm(".include \"libc/disclaimer.inc\"");
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/* clang-format off */
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/* origin: FreeBSD /usr/src/lib/msun/src/s_erf.c */
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/*
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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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/* 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. For |x| in [0, 0.84375]
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* erf(x) = x + x*R(x^2)
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* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
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* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
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* where R = P/Q where P is an odd poly of degree 8 and
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* Q is an odd poly of degree 10.
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* -57.90
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* | R - (erf(x)-x)/x | <= 2
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*
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*
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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. The interval is chosen because the fix
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* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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* near 0.6174), and by some experiment, 0.84375 is chosen to
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* guarantee the error is less than one ulp for erf.
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*
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* 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(x) = sign(x) * (c + P1(s)/Q1(s))
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* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
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* 1+(c+P1(s)/Q1(s)) if x < 0
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* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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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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* That is, we use rational approximation to approximate
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* erf(1+s) - (c = (single)0.84506291151)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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* where
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* P1(s) = degree 6 poly in s
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* Q1(s) = degree 6 poly in s
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*
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* 3. For x in [1.25,1/0.35(~2.857143)],
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
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* erf(x) = 1 - erfc(x)
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* where
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* R1(z) = degree 7 poly in z, (z=1/x^2)
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* S1(z) = degree 8 poly in z
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*
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* 4. For x in [1/0.35,28]
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
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* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
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* = 2.0 - tiny (if x <= -6)
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* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
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* erf(x) = sign(x)*(1.0 - tiny)
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* where
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* R2(z) = degree 6 poly in z, (z=1/x^2)
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* S2(z) = degree 7 poly in z
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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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* We use rational approximation to approximate
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* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
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* Here is the error bound for R1/S1 and R2/S2
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* |R1/S1 - f(x)| < 2**(-62.57)
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* |R2/S2 - f(x)| < 2**(-61.52)
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*
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* 5. For inf > x >= 28
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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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static const double
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erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */
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/*
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* Coefficients for approximation to erf on [0,0.84375]
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*/
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efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */
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pp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */
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pp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */
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pp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */
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pp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */
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pp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */
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qq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */
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qq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */
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qq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */
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qq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */
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qq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */
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/*
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* Coefficients for approximation to erf in [0.84375,1.25]
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*/
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pa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */
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pa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */
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pa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */
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pa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */
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pa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */
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pa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */
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pa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */
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qa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */
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qa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */
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qa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */
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qa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */
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qa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */
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qa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */
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/*
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* Coefficients for approximation to erfc in [1.25,1/0.35]
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*/
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ra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */
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ra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */
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ra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */
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ra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */
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ra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */
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ra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */
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ra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */
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ra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */
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sa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */
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sa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */
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sa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */
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sa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */
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sa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */
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sa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */
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sa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */
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sa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */
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/*
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* Coefficients for approximation to erfc in [1/.35,28]
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*/
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rb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */
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rb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */
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rb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */
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rb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */
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rb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */
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rb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */
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rb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */
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sb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */
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sb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */
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sb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */
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sb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */
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sb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */
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sb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */
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sb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */
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#define asuint(f) ((union{float _f; uint32_t _i;}){f})._i
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#define asfloat(i) ((union{uint32_t _i; float _f;}){i})._f
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#define asuint64(f) ((union{double _f; uint64_t _i;}){f})._i
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#define asdouble(i) ((union{uint64_t _i; double _f;}){i})._f
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#define INSERT_WORDS(d,hi,lo) \
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do { \
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(d) = asdouble(((uint64_t)(hi)<<32) | (uint32_t)(lo)); \
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} while (0)
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#define GET_HIGH_WORD(hi,d) \
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do { \
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(hi) = asuint64(d) >> 32; \
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} while (0)
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#define GET_LOW_WORD(lo,d) \
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do { \
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(lo) = (uint32_t)asuint64(d); \
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} while (0)
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#define SET_HIGH_WORD(d,hi) \
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INSERT_WORDS(d, hi, (uint32_t)asuint64(d))
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#define SET_LOW_WORD(d,lo) \
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INSERT_WORDS(d, asuint64(d)>>32, lo)
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static double erfc1(double x)
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{
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double_t s,P,Q;
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s = fabs(x) - 1;
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P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))));
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Q = 1+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))));
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return 1 - erx - P/Q;
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}
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static double erfc2(uint32_t ix, double x)
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{
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double_t s,R,S;
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double z;
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if (ix < 0x3ff40000) /* |x| < 1.25 */
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return erfc1(x);
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x = fabs(x);
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s = 1/(x*x);
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if (ix < 0x4006db6d) { /* |x| < 1/.35 ~ 2.85714 */
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R = ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(
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ra5+s*(ra6+s*ra7))))));
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S = 1.0+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(
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sa5+s*(sa6+s*(sa7+s*sa8)))))));
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} else { /* |x| > 1/.35 */
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R = rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(
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rb5+s*rb6)))));
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S = 1.0+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(
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sb5+s*(sb6+s*sb7))))));
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}
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z = x;
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SET_LOW_WORD(z,0);
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return exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S)/x;
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}
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/**
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* Returns error function of 𝑥.
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*/
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double erf(double x)
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{
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double r,s,z,y;
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uint32_t ix;
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int sign;
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GET_HIGH_WORD(ix, x);
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sign = ix>>31;
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ix &= 0x7fffffff;
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if (ix >= 0x7ff00000) {
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/* erf(nan)=nan, erf(+-inf)=+-1 */
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return 1-2*sign + 1/x;
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}
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if (ix < 0x3feb0000) { /* |x| < 0.84375 */
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if (ix < 0x3e300000) { /* |x| < 2**-28 */
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/* avoid underflow */
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return 0.125*(8*x + efx8*x);
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}
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z = x*x;
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r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
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s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
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y = r/s;
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return x + x*y;
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}
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if (ix < 0x40180000) /* 0.84375 <= |x| < 6 */
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y = 1 - erfc2(ix,x);
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else
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y = 1 - 0x1p-1022;
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return sign ? -y : y;
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}
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/**
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* Returns complementary error function of 𝑥.
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*/
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double erfc(double x)
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{
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double r,s,z,y;
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uint32_t ix;
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int sign;
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GET_HIGH_WORD(ix, x);
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sign = ix>>31;
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ix &= 0x7fffffff;
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if (ix >= 0x7ff00000) {
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/* erfc(nan)=nan, erfc(+-inf)=0,2 */
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return 2*sign + 1/x;
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}
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if (ix < 0x3feb0000) { /* |x| < 0.84375 */
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if (ix < 0x3c700000) /* |x| < 2**-56 */
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return 1.0 - x;
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z = x*x;
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r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4)));
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s = 1.0+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))));
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y = r/s;
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if (sign || ix < 0x3fd00000) { /* x < 1/4 */
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return 1.0 - (x+x*y);
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}
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return 0.5 - (x - 0.5 + x*y);
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}
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if (ix < 0x403c0000) { /* 0.84375 <= |x| < 28 */
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return sign ? 2 - erfc2(ix,x) : erfc2(ix,x);
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}
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return sign ? 2 - 0x1p-1022 : 0x1p-1022*0x1p-1022;
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}
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#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
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__weak_reference(erf, erfl);
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__weak_reference(erfc, erfcl);
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#endif
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