11dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* @(#)s_erf.c 5.1 93/09/24 */ 21dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 31dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 41dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 51dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 61dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Developed at SunPro, a Sun Microsystems, Inc. business. 71dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Permission to use, copy, modify, and distribute this 81dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * software is freely granted, provided that this notice 91dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * is preserved. 101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * ==================================================== 111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 13a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes#include <sys/cdefs.h> 14a0ee07829a9ba7e99ef68e8c12551301cc797f0fElliott Hughes__FBSDID("$FreeBSD$"); 151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* double erf(double x) 171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * double erfc(double x) 181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * x 191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2 |\ 201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = --------- | exp(-t*t)dt 211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * sqrt(pi) \| 221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 0 231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(x) = 1-erf(x) 251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note that 261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(-x) = -erf(x) 271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(-x) = 2 - erfc(x) 281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Method: 301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1. For |x| in [0, 0.84375] 311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = x + x*R(x^2) 321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where R = P/Q where P is an odd poly of degree 8 and 351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Q is an odd poly of degree 10. 361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * -57.90 371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * | R - (erf(x)-x)/x | <= 2 381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Remark. The formula is derived by noting 411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * and that 431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * is close to one. The interval is chosen because the fix 451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * near 0.6174), and by some experiment, 0.84375 is chosen to 471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * guarantee the error is less than one ulp for erf. 481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * c = 0.84506291151 rounded to single (24 bits) 511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1+(c+P1(s)/Q1(s)) if x < 0 541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Remark: here we use the taylor series expansion at x=1. 561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(1+s) = erf(1) + s*Poly(s) 571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 0.845.. + P1(s)/Q1(s) 581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * That is, we use rational approximation to approximate 591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(1+s) - (c = (single)0.84506291151) 601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where 621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * P1(s) = degree 6 poly in s 631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Q1(s) = degree 6 poly in s 641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 3. For x in [1.25,1/0.35(~2.857143)], 661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = 1 - erfc(x) 681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where 691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * R1(z) = degree 7 poly in z, (z=1/x^2) 701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * S1(z) = degree 8 poly in z 711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 4. For x in [1/0.35,28] 731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 2.0 - tiny (if x <= -6) 761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 771dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = sign(x)*(1.0 - tiny) 781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * where 791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * R2(z) = degree 6 poly in z, (z=1/x^2) 801dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * S2(z) = degree 7 poly in z 811dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 821dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note1: 831dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * To compute exp(-x*x-0.5625+R/S), let s be a single 841dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * precision number and s := x; then 851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * -x*x = -s*s + (s-x)*(s+x) 861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * exp(-x*x-0.5626+R/S) = 871dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 881dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Note2: 891dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Here 4 and 5 make use of the asymptotic series 901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * exp(-x*x) 911dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * x*sqrt(pi) 931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * We use rational approximation to approximate 941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Here is the error bound for R1/S1 and R2/S2 961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |R1/S1 - f(x)| < 2**(-62.57) 971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * |R2/S2 - f(x)| < 2**(-61.52) 981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 5. For inf > x >= 28 1001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(x) = sign(x) *(1 - tiny) (raise inexact) 1011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(x) = tiny*tiny (raise underflow) if x > 0 1021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * = 2 - tiny if x<0 1031dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 1041dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * 7. Special case: 1051dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 1061dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 1071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * erfc/erf(NaN) is NaN 1081dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math.h" 1121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project#include "math_private.h" 1131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1141dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectstatic const double 1151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecttiny = 1e-300, 1161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecthalf= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ 1171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectone = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 1181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecttwo = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ 1191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* c = (float)0.84506291151 */ 1201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecterx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ 1211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 1221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Coefficients for approximation to erf on [0,0.84375] 1231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectefx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ 1251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectefx8= 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ 1261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpp0 = 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ 1271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpp1 = -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ 1281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpp2 = -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ 1291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpp3 = -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ 1301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpp4 = -2.37630166566501626084e-05, /* 0xBEF8EAD6, 0x120016AC */ 1311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqq1 = 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ 1321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqq2 = 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ 1331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqq3 = 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ 1341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqq4 = 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ 1351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqq5 = -3.96022827877536812320e-06, /* 0xBED09C43, 0x42A26120 */ 1361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 1371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Coefficients for approximation to erf in [0.84375,1.25] 1381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpa0 = -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ 1401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpa1 = 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ 1411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpa2 = -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ 1421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpa3 = 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ 1431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpa4 = -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ 1441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpa5 = 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ 1451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectpa6 = -2.16637559486879084300e-03, /* 0xBF61BF38, 0x0A96073F */ 1461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqa1 = 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ 1471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqa2 = 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ 1481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqa3 = 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ 1491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqa4 = 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ 1501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqa5 = 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ 1511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectqa6 = 1.19844998467991074170e-02, /* 0x3F888B54, 0x5735151D */ 1521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 1531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Coefficients for approximation to erfc in [1.25,1/0.35] 1541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra0 = -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ 1561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra1 = -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ 1571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra2 = -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ 1581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra3 = -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ 1591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra4 = -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ 1601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra5 = -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ 1611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra6 = -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ 1621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectra7 = -9.81432934416914548592e+00, /* 0xC023A0EF, 0xC69AC25C */ 1631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa1 = 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ 1641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa2 = 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ 1651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa3 = 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ 1661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa4 = 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ 1671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa5 = 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ 1681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa6 = 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ 1691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa7 = 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ 1701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsa8 = -6.04244152148580987438e-02, /* 0xBFAEEFF2, 0xEE749A62 */ 1711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project/* 1721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project * Coefficients for approximation to erfc in [1/.35,28] 1731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project */ 1741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectrb0 = -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ 1751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectrb1 = -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ 1761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectrb2 = -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ 1771dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectrb3 = -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ 1781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectrb4 = -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ 1791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectrb5 = -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ 1801dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectrb6 = -4.83519191608651397019e+02, /* 0xC07E384E, 0x9BDC383F */ 1811dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsb1 = 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ 1821dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsb2 = 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ 1831dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsb3 = 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ 1841dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsb4 = 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ 1851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsb5 = 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ 1861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsb6 = 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ 1871dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectsb7 = -2.24409524465858183362e+01; /* 0xC03670E2, 0x42712D62 */ 1881dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 1891dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectdouble 1901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecterf(double x) 1911dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project{ 1921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project int32_t hx,ix,i; 1931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project double R,S,P,Q,s,y,z,r; 1941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_HIGH_WORD(hx,x); 1951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ix = hx&0x7fffffff; 1961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>=0x7ff00000) { /* erf(nan)=nan */ 1971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project i = ((u_int32_t)hx>>31)<<1; 1981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (double)(1-i)+one/x; /* erf(+-inf)=+-1 */ 1991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix < 0x3feb0000) { /* |x|<0.84375 */ 2021dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix < 0x3e300000) { /* |x|<2**-28 */ 2031dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (ix < 0x00800000) 2041dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return 0.125*(8.0*x+efx8*x); /*avoid underflow */ 2051dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return x + efx*x; 2061dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2071dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = x*x; 2081dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 2091dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 2101dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project y = r/s; 2111dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return x + x*y; 2121dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2131dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 2141dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = fabs(x)-one; 2151dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 2161dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 2171dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>=0) return erx + P/Q; else return -erx - P/Q; 2181dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2191dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (ix >= 0x40180000) { /* inf>|x|>=6 */ 2201dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>=0) return one-tiny; else return tiny-one; 2211dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2221dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project x = fabs(x); 2231dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = one/(x*x); 2241dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix< 0x4006DB6E) { /* |x| < 1/0.35 */ 2251dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 2261dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ra5+s*(ra6+s*ra7)))))); 2271dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 2281dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project sa5+s*(sa6+s*(sa7+s*sa8))))))); 2291dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { /* |x| >= 1/0.35 */ 2301dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 2311dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project rb5+s*rb6))))); 2321dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 2331dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project sb5+s*(sb6+s*sb7)))))); 2341dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2351dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = x; 2361dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project SET_LOW_WORD(z,0); 2371dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = __ieee754_exp(-z*z-0.5625)*__ieee754_exp((z-x)*(z+x)+R/S); 2381dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>=0) return one-r/x; else return r/x-one; 2391dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 2401dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2411dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projectdouble 2421dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Projecterfc(double x) 2431dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project{ 2441dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project int32_t hx,ix; 2451dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project double R,S,P,Q,s,y,z,r; 2461dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project GET_HIGH_WORD(hx,x); 2471dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ix = hx&0x7fffffff; 2481dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix>=0x7ff00000) { /* erfc(nan)=nan */ 2491dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project /* erfc(+-inf)=0,2 */ 2501dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return (double)(((u_int32_t)hx>>31)<<1)+one/x; 2511dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2521dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project 2531dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix < 0x3feb0000) { /* |x|<0.84375 */ 2541dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix < 0x3c700000) /* |x|<2**-56 */ 2551dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return one-x; 2561dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = x*x; 2571dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = pp0+z*(pp1+z*(pp2+z*(pp3+z*pp4))); 2581dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = one+z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))); 2591dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project y = r/s; 2601dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx < 0x3fd00000) { /* x<1/4 */ 2611dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return one-(x+x*y); 2621dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { 2631dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = x*y; 2641dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r += (x-half); 2651dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project return half - r ; 2661dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2671dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2681dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ 2691dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = fabs(x)-one; 2701dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project P = pa0+s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))); 2711dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project Q = one+s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))); 2721dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>=0) { 2731dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = one-erx; return z - P/Q; 2741dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { 2751dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = erx+P/Q; return one+z; 2761dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2771dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2781dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if (ix < 0x403c0000) { /* |x|<28 */ 2791dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project x = fabs(x); 2801dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project s = one/(x*x); 2811dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(ix< 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143*/ 2821dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project R=ra0+s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*( 2831dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project ra5+s*(ra6+s*ra7)))))); 2841dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project S=one+s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*( 2851dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project sa5+s*(sa6+s*(sa7+s*sa8))))))); 2861dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { /* |x| >= 1/.35 ~ 2.857143 */ 2871dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx<0&&ix>=0x40180000) return two-tiny;/* x < -6 */ 2881dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project R=rb0+s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*( 2891dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project rb5+s*rb6))))); 2901dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project S=one+s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*( 2911dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project sb5+s*(sb6+s*sb7)))))); 2921dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 2931dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project z = x; 2941dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project SET_LOW_WORD(z,0); 2951dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project r = __ieee754_exp(-z*z-0.5625)* 2961dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project __ieee754_exp((z-x)*(z+x)+R/S); 2971dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>0) return r/x; else return two-r/x; 2981dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } else { 2991dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project if(hx>0) return tiny*tiny; else return two-tiny; 3001dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project } 3011dc9e472e19acfe6dc7f41e429236e7eef7ceda1The Android Open Source Project} 302