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/* @(#)s_erf.c 5.1 93/09/24 */ |
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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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#include <float.h> |
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#include <math.h> |
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#include "math_private.h" |
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static const double |
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tiny = 1e-300, |
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half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ |
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/* c = (float)0.84506291151 */ |
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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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efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ |
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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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double |
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erf(double x) |
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{ |
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int32_t hx,ix,i; |
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double R,S,P,Q,s,y,z,r; |
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25540 |
GET_HIGH_WORD(hx,x); |
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ix = hx&0x7fffffff; |
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✗✓ |
12770 |
if(ix>=0x7ff00000) { /* erf(nan)=nan */ |
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i = ((u_int32_t)hx>>31)<<1; |
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return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ |
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} |
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✓✓ |
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if(ix < 0x3feb0000) { /* |x|<0.84375 */ |
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✓✓ |
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if(ix < 0x3e300000) { /* |x|<2**-28 */ |
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✓✗ |
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if (ix < 0x00800000) |
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return 0.125*(8.0*x+efx8*x); /*avoid underflow */ |
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return x + efx*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 = one+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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✓✓ |
10670 |
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ |
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s = fabs(x)-one; |
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P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); |
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Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); |
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✓✓ |
2460 |
if(hx>=0) return erx + P/Q; else return -erx - P/Q; |
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} |
217 |
✓✓ |
9440 |
if (ix >= 0x40180000) { /* inf>|x|>=6 */ |
218 |
✓✓ |
1220 |
if(hx>=0) return one-tiny; else return tiny-one; |
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} |
220 |
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8830 |
x = fabs(x); |
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8830 |
s = one/(x*x); |
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✓✓ |
8830 |
if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ |
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2800 |
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( |
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ra5+s*(ra6+s*ra7)))))); |
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2800 |
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( |
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sa5+s*(sa6+s*(sa7+s*sa8))))))); |
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2800 |
} else { /* |x| >= 1/0.35 */ |
228 |
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6030 |
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( |
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rb5+s*rb6))))); |
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6030 |
S=one+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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8830 |
SET_LOW_WORD(z,0); |
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8830 |
r = exp(-z*z-0.5625)*exp((z-x)*(z+x)+R/S); |
236 |
✓✓ |
17660 |
if(hx>=0) return one-r/x; else return r/x-one; |
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12770 |
} |
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DEF_STD(erf); |
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LDBL_MAYBE_CLONE(erf); |
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double |
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erfc(double x) |
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{ |
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int32_t hx,ix; |
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double R,S,P,Q,s,y,z,r; |
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26360 |
GET_HIGH_WORD(hx,x); |
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13180 |
ix = hx&0x7fffffff; |
248 |
✗✓ |
13180 |
if(ix>=0x7ff00000) { /* erfc(nan)=nan */ |
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/* erfc(+-inf)=0,2 */ |
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return (double)(((u_int32_t)hx>>31)<<1)+one/x; |
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} |
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✓✓ |
13180 |
if(ix < 0x3feb0000) { /* |x|<0.84375 */ |
254 |
✓✓ |
940 |
if(ix < 0x3c700000) /* |x|<2**-56 */ |
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|
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return one-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 = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); |
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910 |
y = r/s; |
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✓✓ |
910 |
if(hx < 0x3fd00000) { /* x<1/4 */ |
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return one-(x+x*y); |
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} else { |
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r = x*y; |
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r += (x-half); |
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return half - r ; |
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} |
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} |
268 |
✓✓ |
12240 |
if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ |
269 |
|
1090 |
s = fabs(x)-one; |
270 |
|
1090 |
P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); |
271 |
|
1090 |
Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); |
272 |
✓✓ |
1090 |
if(hx>=0) { |
273 |
|
710 |
z = one-erx; return z - P/Q; |
274 |
|
|
} else { |
275 |
|
380 |
z = erx+P/Q; return one+z; |
276 |
|
|
} |
277 |
|
|
} |
278 |
✓✗ |
11150 |
if (ix < 0x403c0000) { /* |x|<28 */ |
279 |
|
11150 |
x = fabs(x); |
280 |
|
11150 |
s = one/(x*x); |
281 |
✓✓ |
11150 |
if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ |
282 |
|
2460 |
R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( |
283 |
|
2460 |
ra5+s*(ra6+s*ra7)))))); |
284 |
|
2460 |
S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( |
285 |
|
2460 |
sa5+s*(sa6+s*(sa7+s*sa8))))))); |
286 |
|
2460 |
} else { /* |x| >= 1/.35 ~ 2.857143 */ |
287 |
✓✓ |
8710 |
if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ |
288 |
|
8670 |
R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( |
289 |
|
8670 |
rb5+s*rb6))))); |
290 |
|
8670 |
S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( |
291 |
|
8670 |
sb5+s*(sb6+s*sb7)))))); |
292 |
|
|
} |
293 |
|
|
z = x; |
294 |
|
11130 |
SET_LOW_WORD(z,0); |
295 |
|
11130 |
r = exp(-z*z-0.5625) * exp((z-x)*(z+x)+R/S); |
296 |
✓✓ |
22260 |
if(hx>0) return r/x; else return two-r/x; |
297 |
|
|
} else { |
298 |
|
|
if(hx>0) return tiny*tiny; else return two-tiny; |
299 |
|
|
} |
300 |
|
13180 |
} |
301 |
|
|
DEF_STD(erfc); |
302 |
|
|
LDBL_MAYBE_CLONE(erfc); |