/* * CDDL HEADER START * * The contents of this file are subject to the terms of the * Common Development and Distribution License (the "License"). * You may not use this file except in compliance with the License. * * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE * or http://www.opensolaris.org/os/licensing. * See the License for the specific language governing permissions * and limitations under the License. * * When distributing Covered Code, include this CDDL HEADER in each * file and include the License file at usr/src/OPENSOLARIS.LICENSE. * If applicable, add the following below this CDDL HEADER, with the * fields enclosed by brackets "[]" replaced with your own identifying * information: Portions Copyright [yyyy] [name of copyright owner] * * CDDL HEADER END */ /* * Copyright 2011 Nexenta Systems, Inc. All rights reserved. */ /* * Copyright 2006 Sun Microsystems, Inc. All rights reserved. * Use is subject to license terms. */ #pragma weak __erf = erf #pragma weak __erfc = erfc /* INDENT OFF */ /* * double erf(double x) * double erfc(double x) * x * 2 |\ * erf(x) = --------- | exp(-t*t)dt * sqrt(pi) \| * 0 * * erfc(x) = 1-erf(x) * Note that * erf(-x) = -erf(x) * erfc(-x) = 2 - erfc(x) * * Method: * 1. For |x| in [0, 0.84375] * erf(x) = x + x*R(x^2) * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] * where R = P/Q where P is an odd poly of degree 8 and * Q is an odd poly of degree 10. * -57.90 * | R - (erf(x)-x)/x | <= 2 * * * Remark. The formula is derived by noting * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) * and that * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 * is close to one. The interval is chosen because the fix * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is * near 0.6174), and by some experiment, 0.84375 is chosen to * guarantee the error is less than one ulp for erf. * * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and * c = 0.84506291151 rounded to single (24 bits) * erf(x) = sign(x) * (c + P1(s)/Q1(s)) * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 * 1+(c+P1(s)/Q1(s)) if x < 0 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 * Remark: here we use the taylor series expansion at x=1. * erf(1+s) = erf(1) + s*Poly(s) * = 0.845.. + P1(s)/Q1(s) * That is, we use rational approximation to approximate * erf(1+s) - (c = (single)0.84506291151) * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] * where * P1(s) = degree 6 poly in s * Q1(s) = degree 6 poly in s * * 3. For x in [1.25,1/0.35(~2.857143)], * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) * erf(x) = 1 - erfc(x) * where * R1(z) = degree 7 poly in z, (z=1/x^2) * S1(z) = degree 8 poly in z * * 4. For x in [1/0.35,28] * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 * = 2.0 - tiny (if x <= -6) * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else * erf(x) = sign(x)*(1.0 - tiny) * where * R2(z) = degree 6 poly in z, (z=1/x^2) * S2(z) = degree 7 poly in z * * Note1: * To compute exp(-x*x-0.5625+R/S), let s be a single * precision number and s := x; then * -x*x = -s*s + (s-x)*(s+x) * exp(-x*x-0.5626+R/S) = * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); * Note2: * Here 4 and 5 make use of the asymptotic series * exp(-x*x) * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) * x*sqrt(pi) * We use rational approximation to approximate * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 * Here is the error bound for R1/S1 and R2/S2 * |R1/S1 - f(x)| < 2**(-62.57) * |R2/S2 - f(x)| < 2**(-61.52) * * 5. For inf > x >= 28 * erf(x) = sign(x) *(1 - tiny) (raise inexact) * erfc(x) = tiny*tiny (raise underflow) if x > 0 * = 2 - tiny if x<0 * * 7. Special case: * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, * erfc/erf(NaN) is NaN */ /* INDENT ON */ #include "libm_macros.h" #include <math.h> static const double xxx[] = { /* tiny */ 1e-300, /* half */ 5.00000000000000000000e-01, /* 3FE00000, 00000000 */ /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ /* two */ 2.00000000000000000000e+00, /* 40000000, 00000000 */ /* erx */ 8.45062911510467529297e-01, /* 3FEB0AC1, 60000000 */ /* * Coefficients for approximation to erf on [0,0.84375] */ /* efx */ 1.28379167095512586316e-01, /* 3FC06EBA, 8214DB69 */ /* efx8 */ 1.02703333676410069053e+00, /* 3FF06EBA, 8214DB69 */ /* pp0 */ 1.28379167095512558561e-01, /* 3FC06EBA, 8214DB68 */ /* pp1 */ -3.25042107247001499370e-01, /* BFD4CD7D, 691CB913 */ /* pp2 */ -2.84817495755985104766e-02, /* BF9D2A51, DBD7194F */ /* pp3 */ -5.77027029648944159157e-03, /* BF77A291, 236668E4 */ /* pp4 */ -2.37630166566501626084e-05, /* BEF8EAD6, 120016AC */ /* qq1 */ 3.97917223959155352819e-01, /* 3FD97779, CDDADC09 */ /* qq2 */ 6.50222499887672944485e-02, /* 3FB0A54C, 5536CEBA */ /* qq3 */ 5.08130628187576562776e-03, /* 3F74D022, C4D36B0F */ /* qq4 */ 1.32494738004321644526e-04, /* 3F215DC9, 221C1A10 */ /* qq5 */ -3.96022827877536812320e-06, /* BED09C43, 42A26120 */ /* * Coefficients for approximation to erf in [0.84375,1.25] */ /* pa0 */ -2.36211856075265944077e-03, /* BF6359B8, BEF77538 */ /* pa1 */ 4.14856118683748331666e-01, /* 3FDA8D00, AD92B34D */ /* pa2 */ -3.72207876035701323847e-01, /* BFD7D240, FBB8C3F1 */ /* pa3 */ 3.18346619901161753674e-01, /* 3FD45FCA, 805120E4 */ /* pa4 */ -1.10894694282396677476e-01, /* BFBC6398, 3D3E28EC */ /* pa5 */ 3.54783043256182359371e-02, /* 3FA22A36, 599795EB */ /* pa6 */ -2.16637559486879084300e-03, /* BF61BF38, 0A96073F */ /* qa1 */ 1.06420880400844228286e-01, /* 3FBB3E66, 18EEE323 */ /* qa2 */ 5.40397917702171048937e-01, /* 3FE14AF0, 92EB6F33 */ /* qa3 */ 7.18286544141962662868e-02, /* 3FB2635C, D99FE9A7 */ /* qa4 */ 1.26171219808761642112e-01, /* 3FC02660, E763351F */ /* qa5 */ 1.36370839120290507362e-02, /* 3F8BEDC2, 6B51DD1C */ /* qa6 */ 1.19844998467991074170e-02, /* 3F888B54, 5735151D */ /* * Coefficients for approximation to erfc in [1.25,1/0.35] */ /* ra0 */ -9.86494403484714822705e-03, /* BF843412, 600D6435 */ /* ra1 */ -6.93858572707181764372e-01, /* BFE63416, E4BA7360 */ /* ra2 */ -1.05586262253232909814e+01, /* C0251E04, 41B0E726 */ /* ra3 */ -6.23753324503260060396e+01, /* C04F300A, E4CBA38D */ /* ra4 */ -1.62396669462573470355e+02, /* C0644CB1, 84282266 */ /* ra5 */ -1.84605092906711035994e+02, /* C067135C, EBCCABB2 */ /* ra6 */ -8.12874355063065934246e+01, /* C0545265, 57E4D2F2 */ /* ra7 */ -9.81432934416914548592e+00, /* C023A0EF, C69AC25C */ /* sa1 */ 1.96512716674392571292e+01, /* 4033A6B9, BD707687 */ /* sa2 */ 1.37657754143519042600e+02, /* 4061350C, 526AE721 */ /* sa3 */ 4.34565877475229228821e+02, /* 407B290D, D58A1A71 */ /* sa4 */ 6.45387271733267880336e+02, /* 40842B19, 21EC2868 */ /* sa5 */ 4.29008140027567833386e+02, /* 407AD021, 57700314 */ /* sa6 */ 1.08635005541779435134e+02, /* 405B28A3, EE48AE2C */ /* sa7 */ 6.57024977031928170135e+00, /* 401A47EF, 8E484A93 */ /* sa8 */ -6.04244152148580987438e-02, /* BFAEEFF2, EE749A62 */ /* * Coefficients for approximation to erfc in [1/.35,28] */ /* rb0 */ -9.86494292470009928597e-03, /* BF843412, 39E86F4A */ /* rb1 */ -7.99283237680523006574e-01, /* BFE993BA, 70C285DE */ /* rb2 */ -1.77579549177547519889e+01, /* C031C209, 555F995A */ /* rb3 */ -1.60636384855821916062e+02, /* C064145D, 43C5ED98 */ /* rb4 */ -6.37566443368389627722e+02, /* C083EC88, 1375F228 */ /* rb5 */ -1.02509513161107724954e+03, /* C0900461, 6A2E5992 */ /* rb6 */ -4.83519191608651397019e+02, /* C07E384E, 9BDC383F */ /* sb1 */ 3.03380607434824582924e+01, /* 403E568B, 261D5190 */ /* sb2 */ 3.25792512996573918826e+02, /* 40745CAE, 221B9F0A */ /* sb3 */ 1.53672958608443695994e+03, /* 409802EB, 189D5118 */ /* sb4 */ 3.19985821950859553908e+03, /* 40A8FFB7, 688C246A */ /* sb5 */ 2.55305040643316442583e+03, /* 40A3F219, CEDF3BE6 */ /* sb6 */ 4.74528541206955367215e+02, /* 407DA874, E79FE763 */ /* sb7 */ -2.24409524465858183362e+01 /* C03670E2, 42712D62 */ }; #define tiny xxx[0] #define half xxx[1] #define one xxx[2] #define two xxx[3] #define erx xxx[4] /* * Coefficients for approximation to erf on [0,0.84375] */ #define efx xxx[5] #define efx8 xxx[6] #define pp0 xxx[7] #define pp1 xxx[8] #define pp2 xxx[9] #define pp3 xxx[10] #define pp4 xxx[11] #define qq1 xxx[12] #define qq2 xxx[13] #define qq3 xxx[14] #define qq4 xxx[15] #define qq5 xxx[16] /* * Coefficients for approximation to erf in [0.84375,1.25] */ #define pa0 xxx[17] #define pa1 xxx[18] #define pa2 xxx[19] #define pa3 xxx[20] #define pa4 xxx[21] #define pa5 xxx[22] #define pa6 xxx[23] #define qa1 xxx[24] #define qa2 xxx[25] #define qa3 xxx[26] #define qa4 xxx[27] #define qa5 xxx[28] #define qa6 xxx[29] /* * Coefficients for approximation to erfc in [1.25,1/0.35] */ #define ra0 xxx[30] #define ra1 xxx[31] #define ra2 xxx[32] #define ra3 xxx[33] #define ra4 xxx[34] #define ra5 xxx[35] #define ra6 xxx[36] #define ra7 xxx[37] #define sa1 xxx[38] #define sa2 xxx[39] #define sa3 xxx[40] #define sa4 xxx[41] #define sa5 xxx[42] #define sa6 xxx[43] #define sa7 xxx[44] #define sa8 xxx[45] /* * Coefficients for approximation to erfc in [1/.35,28] */ #define rb0 xxx[46] #define rb1 xxx[47] #define rb2 xxx[48] #define rb3 xxx[49] #define rb4 xxx[50] #define rb5 xxx[51] #define rb6 xxx[52] #define sb1 xxx[53] #define sb2 xxx[54] #define sb3 xxx[55] #define sb4 xxx[56] #define sb5 xxx[57] #define sb6 xxx[58] #define sb7 xxx[59] double erf(double x) { int hx, ix, i; double R, S, P, Q, s, y, z, r; hx = ((int *) &x)[HIWORD]; ix = hx & 0x7fffffff; if (ix >= 0x7ff00000) { /* erf(nan)=nan */ #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN) if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */ return (x); #endif i = ((unsigned) hx >> 31) << 1; return ((double) (1 - i) + one / x); /* erf(+-inf)=+-1 */ } if (ix < 0x3feb0000) { /* |x|<0.84375 */ if (ix < 0x3e300000) { /* |x|<2**-28 */ if (ix < 0x00800000) /* avoid underflow */ return (0.125 * (8.0 * x + efx8 * x)); return (x + efx * x); } z = x * x; r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4))); s = one + z *(qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); y = r / s; return (x + x * y); } if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ s = fabs(x) - one; P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 + s * pa6))))); Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 + s * qa6))))); if (hx >= 0) return (erx + P / Q); else return (-erx - P / Q); } if (ix >= 0x40180000) { /* inf > |x| >= 6 */ if (hx >= 0) return (one - tiny); else return (tiny - one); } x = fabs(x); s = one / (x * x); if (ix < 0x4006DB6E) { /* |x| < 1/0.35 */ R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + s * (ra5 + s * (ra6 + s * ra7)))))); S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + s * (sa5 + s * (sa6 + s * (sa7 + s * sa8))))))); } else { /* |x| >= 1/0.35 */ R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 + s * rb6))))); S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 + s * (sb5 + s * (sb6 + s * sb7)))))); } z = x; ((int *) &z)[LOWORD] = 0; r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S); if (hx >= 0) return (one - r / x); else return (r / x - one); } double erfc(double x) { int hx, ix; double R, S, P, Q, s, y, z, r; hx = ((int *) &x)[HIWORD]; ix = hx & 0x7fffffff; if (ix >= 0x7ff00000) { /* erfc(nan)=nan */ #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN) if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */ return (x); #endif /* erfc(+-inf)=0,2 */ return ((double) (((unsigned) hx >> 31) << 1) + one / x); } if (ix < 0x3feb0000) { /* |x| < 0.84375 */ if (ix < 0x3c700000) /* |x| < 2**-56 */ return (one - x); z = x * x; r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4))); s = one + z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5)))); y = r / s; if (hx < 0x3fd00000) { /* x < 1/4 */ return (one - (x + x * y)); } else { r = x * y; r += (x - half); return (half - r); } } if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */ s = fabs(x) - one; P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 + s * (pa5 + s * pa6))))); Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 + s * (qa5 + s * qa6))))); if (hx >= 0) { z = one - erx; return (z - P / Q); } else { z = erx + P / Q; return (one + z); } } if (ix < 0x403c0000) { /* |x|<28 */ x = fabs(x); s = one / (x * x); if (ix < 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143 */ R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 + s * (ra5 + s * (ra6 + s * ra7)))))); S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 + s * (sa5 + s * (sa6 + s * (sa7 + s * sa8))))))); } else { /* |x| >= 1/.35 ~ 2.857143 */ if (hx < 0 && ix >= 0x40180000) return (two - tiny); /* x < -6 */ R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 + s * (rb5 + s * rb6))))); S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 + s * (sb5 + s * (sb6 + s * sb7)))))); } z = x; ((int *) &z)[LOWORD] = 0; r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S); if (hx > 0) return (r / x); else return (two - r / x); } else { if (hx > 0) return (tiny * tiny); else return (two - tiny); } }
