15b2ba9d3SPiotr Jasiukajtis /*
25b2ba9d3SPiotr Jasiukajtis * CDDL HEADER START
35b2ba9d3SPiotr Jasiukajtis *
45b2ba9d3SPiotr Jasiukajtis * The contents of this file are subject to the terms of the
55b2ba9d3SPiotr Jasiukajtis * Common Development and Distribution License (the "License").
65b2ba9d3SPiotr Jasiukajtis * You may not use this file except in compliance with the License.
75b2ba9d3SPiotr Jasiukajtis *
85b2ba9d3SPiotr Jasiukajtis * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
95b2ba9d3SPiotr Jasiukajtis * or http://www.opensolaris.org/os/licensing.
105b2ba9d3SPiotr Jasiukajtis * See the License for the specific language governing permissions
115b2ba9d3SPiotr Jasiukajtis * and limitations under the License.
125b2ba9d3SPiotr Jasiukajtis *
135b2ba9d3SPiotr Jasiukajtis * When distributing Covered Code, include this CDDL HEADER in each
145b2ba9d3SPiotr Jasiukajtis * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
155b2ba9d3SPiotr Jasiukajtis * If applicable, add the following below this CDDL HEADER, with the
165b2ba9d3SPiotr Jasiukajtis * fields enclosed by brackets "[]" replaced with your own identifying
175b2ba9d3SPiotr Jasiukajtis * information: Portions Copyright [yyyy] [name of copyright owner]
185b2ba9d3SPiotr Jasiukajtis *
195b2ba9d3SPiotr Jasiukajtis * CDDL HEADER END
205b2ba9d3SPiotr Jasiukajtis */
215b2ba9d3SPiotr Jasiukajtis
225b2ba9d3SPiotr Jasiukajtis /*
235b2ba9d3SPiotr Jasiukajtis * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
245b2ba9d3SPiotr Jasiukajtis */
255b2ba9d3SPiotr Jasiukajtis /*
265b2ba9d3SPiotr Jasiukajtis * Copyright 2006 Sun Microsystems, Inc. All rights reserved.
275b2ba9d3SPiotr Jasiukajtis * Use is subject to license terms.
285b2ba9d3SPiotr Jasiukajtis */
295b2ba9d3SPiotr Jasiukajtis
30*a9d3dcd5SRichard Lowe #pragma weak __erf = erf
31*a9d3dcd5SRichard Lowe #pragma weak __erfc = erfc
325b2ba9d3SPiotr Jasiukajtis
335b2ba9d3SPiotr Jasiukajtis /* INDENT OFF */
345b2ba9d3SPiotr Jasiukajtis /*
355b2ba9d3SPiotr Jasiukajtis * double erf(double x)
365b2ba9d3SPiotr Jasiukajtis * double erfc(double x)
375b2ba9d3SPiotr Jasiukajtis * x
385b2ba9d3SPiotr Jasiukajtis * 2 |\
395b2ba9d3SPiotr Jasiukajtis * erf(x) = --------- | exp(-t*t)dt
405b2ba9d3SPiotr Jasiukajtis * sqrt(pi) \|
415b2ba9d3SPiotr Jasiukajtis * 0
425b2ba9d3SPiotr Jasiukajtis *
435b2ba9d3SPiotr Jasiukajtis * erfc(x) = 1-erf(x)
445b2ba9d3SPiotr Jasiukajtis * Note that
455b2ba9d3SPiotr Jasiukajtis * erf(-x) = -erf(x)
465b2ba9d3SPiotr Jasiukajtis * erfc(-x) = 2 - erfc(x)
475b2ba9d3SPiotr Jasiukajtis *
485b2ba9d3SPiotr Jasiukajtis * Method:
495b2ba9d3SPiotr Jasiukajtis * 1. For |x| in [0, 0.84375]
505b2ba9d3SPiotr Jasiukajtis * erf(x) = x + x*R(x^2)
515b2ba9d3SPiotr Jasiukajtis * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
525b2ba9d3SPiotr Jasiukajtis * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
535b2ba9d3SPiotr Jasiukajtis * where R = P/Q where P is an odd poly of degree 8 and
545b2ba9d3SPiotr Jasiukajtis * Q is an odd poly of degree 10.
555b2ba9d3SPiotr Jasiukajtis * -57.90
565b2ba9d3SPiotr Jasiukajtis * | R - (erf(x)-x)/x | <= 2
575b2ba9d3SPiotr Jasiukajtis *
585b2ba9d3SPiotr Jasiukajtis *
595b2ba9d3SPiotr Jasiukajtis * Remark. The formula is derived by noting
605b2ba9d3SPiotr Jasiukajtis * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
615b2ba9d3SPiotr Jasiukajtis * and that
625b2ba9d3SPiotr Jasiukajtis * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
635b2ba9d3SPiotr Jasiukajtis * is close to one. The interval is chosen because the fix
645b2ba9d3SPiotr Jasiukajtis * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
655b2ba9d3SPiotr Jasiukajtis * near 0.6174), and by some experiment, 0.84375 is chosen to
665b2ba9d3SPiotr Jasiukajtis * guarantee the error is less than one ulp for erf.
675b2ba9d3SPiotr Jasiukajtis *
685b2ba9d3SPiotr Jasiukajtis * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
695b2ba9d3SPiotr Jasiukajtis * c = 0.84506291151 rounded to single (24 bits)
705b2ba9d3SPiotr Jasiukajtis * erf(x) = sign(x) * (c + P1(s)/Q1(s))
715b2ba9d3SPiotr Jasiukajtis * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
725b2ba9d3SPiotr Jasiukajtis * 1+(c+P1(s)/Q1(s)) if x < 0
735b2ba9d3SPiotr Jasiukajtis * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
745b2ba9d3SPiotr Jasiukajtis * Remark: here we use the taylor series expansion at x=1.
755b2ba9d3SPiotr Jasiukajtis * erf(1+s) = erf(1) + s*Poly(s)
765b2ba9d3SPiotr Jasiukajtis * = 0.845.. + P1(s)/Q1(s)
775b2ba9d3SPiotr Jasiukajtis * That is, we use rational approximation to approximate
785b2ba9d3SPiotr Jasiukajtis * erf(1+s) - (c = (single)0.84506291151)
795b2ba9d3SPiotr Jasiukajtis * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
805b2ba9d3SPiotr Jasiukajtis * where
815b2ba9d3SPiotr Jasiukajtis * P1(s) = degree 6 poly in s
825b2ba9d3SPiotr Jasiukajtis * Q1(s) = degree 6 poly in s
835b2ba9d3SPiotr Jasiukajtis *
845b2ba9d3SPiotr Jasiukajtis * 3. For x in [1.25,1/0.35(~2.857143)],
855b2ba9d3SPiotr Jasiukajtis * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
865b2ba9d3SPiotr Jasiukajtis * erf(x) = 1 - erfc(x)
875b2ba9d3SPiotr Jasiukajtis * where
885b2ba9d3SPiotr Jasiukajtis * R1(z) = degree 7 poly in z, (z=1/x^2)
895b2ba9d3SPiotr Jasiukajtis * S1(z) = degree 8 poly in z
905b2ba9d3SPiotr Jasiukajtis *
915b2ba9d3SPiotr Jasiukajtis * 4. For x in [1/0.35,28]
925b2ba9d3SPiotr Jasiukajtis * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
935b2ba9d3SPiotr Jasiukajtis * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
945b2ba9d3SPiotr Jasiukajtis * = 2.0 - tiny (if x <= -6)
955b2ba9d3SPiotr Jasiukajtis * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
965b2ba9d3SPiotr Jasiukajtis * erf(x) = sign(x)*(1.0 - tiny)
975b2ba9d3SPiotr Jasiukajtis * where
985b2ba9d3SPiotr Jasiukajtis * R2(z) = degree 6 poly in z, (z=1/x^2)
995b2ba9d3SPiotr Jasiukajtis * S2(z) = degree 7 poly in z
1005b2ba9d3SPiotr Jasiukajtis *
1015b2ba9d3SPiotr Jasiukajtis * Note1:
1025b2ba9d3SPiotr Jasiukajtis * To compute exp(-x*x-0.5625+R/S), let s be a single
1035b2ba9d3SPiotr Jasiukajtis * precision number and s := x; then
1045b2ba9d3SPiotr Jasiukajtis * -x*x = -s*s + (s-x)*(s+x)
1055b2ba9d3SPiotr Jasiukajtis * exp(-x*x-0.5626+R/S) =
1065b2ba9d3SPiotr Jasiukajtis * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
1075b2ba9d3SPiotr Jasiukajtis * Note2:
1085b2ba9d3SPiotr Jasiukajtis * Here 4 and 5 make use of the asymptotic series
1095b2ba9d3SPiotr Jasiukajtis * exp(-x*x)
1105b2ba9d3SPiotr Jasiukajtis * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
1115b2ba9d3SPiotr Jasiukajtis * x*sqrt(pi)
1125b2ba9d3SPiotr Jasiukajtis * We use rational approximation to approximate
1135b2ba9d3SPiotr Jasiukajtis * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
1145b2ba9d3SPiotr Jasiukajtis * Here is the error bound for R1/S1 and R2/S2
1155b2ba9d3SPiotr Jasiukajtis * |R1/S1 - f(x)| < 2**(-62.57)
1165b2ba9d3SPiotr Jasiukajtis * |R2/S2 - f(x)| < 2**(-61.52)
1175b2ba9d3SPiotr Jasiukajtis *
1185b2ba9d3SPiotr Jasiukajtis * 5. For inf > x >= 28
1195b2ba9d3SPiotr Jasiukajtis * erf(x) = sign(x) *(1 - tiny) (raise inexact)
1205b2ba9d3SPiotr Jasiukajtis * erfc(x) = tiny*tiny (raise underflow) if x > 0
1215b2ba9d3SPiotr Jasiukajtis * = 2 - tiny if x<0
1225b2ba9d3SPiotr Jasiukajtis *
1235b2ba9d3SPiotr Jasiukajtis * 7. Special case:
1245b2ba9d3SPiotr Jasiukajtis * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
1255b2ba9d3SPiotr Jasiukajtis * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
1265b2ba9d3SPiotr Jasiukajtis * erfc/erf(NaN) is NaN
1275b2ba9d3SPiotr Jasiukajtis */
1285b2ba9d3SPiotr Jasiukajtis /* INDENT ON */
1295b2ba9d3SPiotr Jasiukajtis
1305b2ba9d3SPiotr Jasiukajtis #include "libm_macros.h"
1315b2ba9d3SPiotr Jasiukajtis #include <math.h>
1325b2ba9d3SPiotr Jasiukajtis
1335b2ba9d3SPiotr Jasiukajtis static const double xxx[] = {
1345b2ba9d3SPiotr Jasiukajtis /* tiny */ 1e-300,
1355b2ba9d3SPiotr Jasiukajtis /* half */ 5.00000000000000000000e-01, /* 3FE00000, 00000000 */
1365b2ba9d3SPiotr Jasiukajtis /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */
1375b2ba9d3SPiotr Jasiukajtis /* two */ 2.00000000000000000000e+00, /* 40000000, 00000000 */
1385b2ba9d3SPiotr Jasiukajtis /* erx */ 8.45062911510467529297e-01, /* 3FEB0AC1, 60000000 */
1395b2ba9d3SPiotr Jasiukajtis /*
1405b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erf on [0,0.84375]
1415b2ba9d3SPiotr Jasiukajtis */
1425b2ba9d3SPiotr Jasiukajtis /* efx */ 1.28379167095512586316e-01, /* 3FC06EBA, 8214DB69 */
1435b2ba9d3SPiotr Jasiukajtis /* efx8 */ 1.02703333676410069053e+00, /* 3FF06EBA, 8214DB69 */
1445b2ba9d3SPiotr Jasiukajtis /* pp0 */ 1.28379167095512558561e-01, /* 3FC06EBA, 8214DB68 */
1455b2ba9d3SPiotr Jasiukajtis /* pp1 */ -3.25042107247001499370e-01, /* BFD4CD7D, 691CB913 */
1465b2ba9d3SPiotr Jasiukajtis /* pp2 */ -2.84817495755985104766e-02, /* BF9D2A51, DBD7194F */
1475b2ba9d3SPiotr Jasiukajtis /* pp3 */ -5.77027029648944159157e-03, /* BF77A291, 236668E4 */
1485b2ba9d3SPiotr Jasiukajtis /* pp4 */ -2.37630166566501626084e-05, /* BEF8EAD6, 120016AC */
1495b2ba9d3SPiotr Jasiukajtis /* qq1 */ 3.97917223959155352819e-01, /* 3FD97779, CDDADC09 */
1505b2ba9d3SPiotr Jasiukajtis /* qq2 */ 6.50222499887672944485e-02, /* 3FB0A54C, 5536CEBA */
1515b2ba9d3SPiotr Jasiukajtis /* qq3 */ 5.08130628187576562776e-03, /* 3F74D022, C4D36B0F */
1525b2ba9d3SPiotr Jasiukajtis /* qq4 */ 1.32494738004321644526e-04, /* 3F215DC9, 221C1A10 */
1535b2ba9d3SPiotr Jasiukajtis /* qq5 */ -3.96022827877536812320e-06, /* BED09C43, 42A26120 */
1545b2ba9d3SPiotr Jasiukajtis /*
1555b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erf in [0.84375,1.25]
1565b2ba9d3SPiotr Jasiukajtis */
1575b2ba9d3SPiotr Jasiukajtis /* pa0 */ -2.36211856075265944077e-03, /* BF6359B8, BEF77538 */
1585b2ba9d3SPiotr Jasiukajtis /* pa1 */ 4.14856118683748331666e-01, /* 3FDA8D00, AD92B34D */
1595b2ba9d3SPiotr Jasiukajtis /* pa2 */ -3.72207876035701323847e-01, /* BFD7D240, FBB8C3F1 */
1605b2ba9d3SPiotr Jasiukajtis /* pa3 */ 3.18346619901161753674e-01, /* 3FD45FCA, 805120E4 */
1615b2ba9d3SPiotr Jasiukajtis /* pa4 */ -1.10894694282396677476e-01, /* BFBC6398, 3D3E28EC */
1625b2ba9d3SPiotr Jasiukajtis /* pa5 */ 3.54783043256182359371e-02, /* 3FA22A36, 599795EB */
1635b2ba9d3SPiotr Jasiukajtis /* pa6 */ -2.16637559486879084300e-03, /* BF61BF38, 0A96073F */
1645b2ba9d3SPiotr Jasiukajtis /* qa1 */ 1.06420880400844228286e-01, /* 3FBB3E66, 18EEE323 */
1655b2ba9d3SPiotr Jasiukajtis /* qa2 */ 5.40397917702171048937e-01, /* 3FE14AF0, 92EB6F33 */
1665b2ba9d3SPiotr Jasiukajtis /* qa3 */ 7.18286544141962662868e-02, /* 3FB2635C, D99FE9A7 */
1675b2ba9d3SPiotr Jasiukajtis /* qa4 */ 1.26171219808761642112e-01, /* 3FC02660, E763351F */
1685b2ba9d3SPiotr Jasiukajtis /* qa5 */ 1.36370839120290507362e-02, /* 3F8BEDC2, 6B51DD1C */
1695b2ba9d3SPiotr Jasiukajtis /* qa6 */ 1.19844998467991074170e-02, /* 3F888B54, 5735151D */
1705b2ba9d3SPiotr Jasiukajtis /*
1715b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erfc in [1.25,1/0.35]
1725b2ba9d3SPiotr Jasiukajtis */
1735b2ba9d3SPiotr Jasiukajtis /* ra0 */ -9.86494403484714822705e-03, /* BF843412, 600D6435 */
1745b2ba9d3SPiotr Jasiukajtis /* ra1 */ -6.93858572707181764372e-01, /* BFE63416, E4BA7360 */
1755b2ba9d3SPiotr Jasiukajtis /* ra2 */ -1.05586262253232909814e+01, /* C0251E04, 41B0E726 */
1765b2ba9d3SPiotr Jasiukajtis /* ra3 */ -6.23753324503260060396e+01, /* C04F300A, E4CBA38D */
1775b2ba9d3SPiotr Jasiukajtis /* ra4 */ -1.62396669462573470355e+02, /* C0644CB1, 84282266 */
1785b2ba9d3SPiotr Jasiukajtis /* ra5 */ -1.84605092906711035994e+02, /* C067135C, EBCCABB2 */
1795b2ba9d3SPiotr Jasiukajtis /* ra6 */ -8.12874355063065934246e+01, /* C0545265, 57E4D2F2 */
1805b2ba9d3SPiotr Jasiukajtis /* ra7 */ -9.81432934416914548592e+00, /* C023A0EF, C69AC25C */
1815b2ba9d3SPiotr Jasiukajtis /* sa1 */ 1.96512716674392571292e+01, /* 4033A6B9, BD707687 */
1825b2ba9d3SPiotr Jasiukajtis /* sa2 */ 1.37657754143519042600e+02, /* 4061350C, 526AE721 */
1835b2ba9d3SPiotr Jasiukajtis /* sa3 */ 4.34565877475229228821e+02, /* 407B290D, D58A1A71 */
1845b2ba9d3SPiotr Jasiukajtis /* sa4 */ 6.45387271733267880336e+02, /* 40842B19, 21EC2868 */
1855b2ba9d3SPiotr Jasiukajtis /* sa5 */ 4.29008140027567833386e+02, /* 407AD021, 57700314 */
1865b2ba9d3SPiotr Jasiukajtis /* sa6 */ 1.08635005541779435134e+02, /* 405B28A3, EE48AE2C */
1875b2ba9d3SPiotr Jasiukajtis /* sa7 */ 6.57024977031928170135e+00, /* 401A47EF, 8E484A93 */
1885b2ba9d3SPiotr Jasiukajtis /* sa8 */ -6.04244152148580987438e-02, /* BFAEEFF2, EE749A62 */
1895b2ba9d3SPiotr Jasiukajtis /*
1905b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erfc in [1/.35,28]
1915b2ba9d3SPiotr Jasiukajtis */
1925b2ba9d3SPiotr Jasiukajtis /* rb0 */ -9.86494292470009928597e-03, /* BF843412, 39E86F4A */
1935b2ba9d3SPiotr Jasiukajtis /* rb1 */ -7.99283237680523006574e-01, /* BFE993BA, 70C285DE */
1945b2ba9d3SPiotr Jasiukajtis /* rb2 */ -1.77579549177547519889e+01, /* C031C209, 555F995A */
1955b2ba9d3SPiotr Jasiukajtis /* rb3 */ -1.60636384855821916062e+02, /* C064145D, 43C5ED98 */
1965b2ba9d3SPiotr Jasiukajtis /* rb4 */ -6.37566443368389627722e+02, /* C083EC88, 1375F228 */
1975b2ba9d3SPiotr Jasiukajtis /* rb5 */ -1.02509513161107724954e+03, /* C0900461, 6A2E5992 */
1985b2ba9d3SPiotr Jasiukajtis /* rb6 */ -4.83519191608651397019e+02, /* C07E384E, 9BDC383F */
1995b2ba9d3SPiotr Jasiukajtis /* sb1 */ 3.03380607434824582924e+01, /* 403E568B, 261D5190 */
2005b2ba9d3SPiotr Jasiukajtis /* sb2 */ 3.25792512996573918826e+02, /* 40745CAE, 221B9F0A */
2015b2ba9d3SPiotr Jasiukajtis /* sb3 */ 1.53672958608443695994e+03, /* 409802EB, 189D5118 */
2025b2ba9d3SPiotr Jasiukajtis /* sb4 */ 3.19985821950859553908e+03, /* 40A8FFB7, 688C246A */
2035b2ba9d3SPiotr Jasiukajtis /* sb5 */ 2.55305040643316442583e+03, /* 40A3F219, CEDF3BE6 */
2045b2ba9d3SPiotr Jasiukajtis /* sb6 */ 4.74528541206955367215e+02, /* 407DA874, E79FE763 */
2055b2ba9d3SPiotr Jasiukajtis /* sb7 */ -2.24409524465858183362e+01 /* C03670E2, 42712D62 */
2065b2ba9d3SPiotr Jasiukajtis };
2075b2ba9d3SPiotr Jasiukajtis
2085b2ba9d3SPiotr Jasiukajtis #define tiny xxx[0]
2095b2ba9d3SPiotr Jasiukajtis #define half xxx[1]
2105b2ba9d3SPiotr Jasiukajtis #define one xxx[2]
2115b2ba9d3SPiotr Jasiukajtis #define two xxx[3]
2125b2ba9d3SPiotr Jasiukajtis #define erx xxx[4]
2135b2ba9d3SPiotr Jasiukajtis /*
2145b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erf on [0,0.84375]
2155b2ba9d3SPiotr Jasiukajtis */
2165b2ba9d3SPiotr Jasiukajtis #define efx xxx[5]
2175b2ba9d3SPiotr Jasiukajtis #define efx8 xxx[6]
2185b2ba9d3SPiotr Jasiukajtis #define pp0 xxx[7]
2195b2ba9d3SPiotr Jasiukajtis #define pp1 xxx[8]
2205b2ba9d3SPiotr Jasiukajtis #define pp2 xxx[9]
2215b2ba9d3SPiotr Jasiukajtis #define pp3 xxx[10]
2225b2ba9d3SPiotr Jasiukajtis #define pp4 xxx[11]
2235b2ba9d3SPiotr Jasiukajtis #define qq1 xxx[12]
2245b2ba9d3SPiotr Jasiukajtis #define qq2 xxx[13]
2255b2ba9d3SPiotr Jasiukajtis #define qq3 xxx[14]
2265b2ba9d3SPiotr Jasiukajtis #define qq4 xxx[15]
2275b2ba9d3SPiotr Jasiukajtis #define qq5 xxx[16]
2285b2ba9d3SPiotr Jasiukajtis /*
2295b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erf in [0.84375,1.25]
2305b2ba9d3SPiotr Jasiukajtis */
2315b2ba9d3SPiotr Jasiukajtis #define pa0 xxx[17]
2325b2ba9d3SPiotr Jasiukajtis #define pa1 xxx[18]
2335b2ba9d3SPiotr Jasiukajtis #define pa2 xxx[19]
2345b2ba9d3SPiotr Jasiukajtis #define pa3 xxx[20]
2355b2ba9d3SPiotr Jasiukajtis #define pa4 xxx[21]
2365b2ba9d3SPiotr Jasiukajtis #define pa5 xxx[22]
2375b2ba9d3SPiotr Jasiukajtis #define pa6 xxx[23]
2385b2ba9d3SPiotr Jasiukajtis #define qa1 xxx[24]
2395b2ba9d3SPiotr Jasiukajtis #define qa2 xxx[25]
2405b2ba9d3SPiotr Jasiukajtis #define qa3 xxx[26]
2415b2ba9d3SPiotr Jasiukajtis #define qa4 xxx[27]
2425b2ba9d3SPiotr Jasiukajtis #define qa5 xxx[28]
2435b2ba9d3SPiotr Jasiukajtis #define qa6 xxx[29]
2445b2ba9d3SPiotr Jasiukajtis /*
2455b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erfc in [1.25,1/0.35]
2465b2ba9d3SPiotr Jasiukajtis */
2475b2ba9d3SPiotr Jasiukajtis #define ra0 xxx[30]
2485b2ba9d3SPiotr Jasiukajtis #define ra1 xxx[31]
2495b2ba9d3SPiotr Jasiukajtis #define ra2 xxx[32]
2505b2ba9d3SPiotr Jasiukajtis #define ra3 xxx[33]
2515b2ba9d3SPiotr Jasiukajtis #define ra4 xxx[34]
2525b2ba9d3SPiotr Jasiukajtis #define ra5 xxx[35]
2535b2ba9d3SPiotr Jasiukajtis #define ra6 xxx[36]
2545b2ba9d3SPiotr Jasiukajtis #define ra7 xxx[37]
2555b2ba9d3SPiotr Jasiukajtis #define sa1 xxx[38]
2565b2ba9d3SPiotr Jasiukajtis #define sa2 xxx[39]
2575b2ba9d3SPiotr Jasiukajtis #define sa3 xxx[40]
2585b2ba9d3SPiotr Jasiukajtis #define sa4 xxx[41]
2595b2ba9d3SPiotr Jasiukajtis #define sa5 xxx[42]
2605b2ba9d3SPiotr Jasiukajtis #define sa6 xxx[43]
2615b2ba9d3SPiotr Jasiukajtis #define sa7 xxx[44]
2625b2ba9d3SPiotr Jasiukajtis #define sa8 xxx[45]
2635b2ba9d3SPiotr Jasiukajtis /*
2645b2ba9d3SPiotr Jasiukajtis * Coefficients for approximation to erfc in [1/.35,28]
2655b2ba9d3SPiotr Jasiukajtis */
2665b2ba9d3SPiotr Jasiukajtis #define rb0 xxx[46]
2675b2ba9d3SPiotr Jasiukajtis #define rb1 xxx[47]
2685b2ba9d3SPiotr Jasiukajtis #define rb2 xxx[48]
2695b2ba9d3SPiotr Jasiukajtis #define rb3 xxx[49]
2705b2ba9d3SPiotr Jasiukajtis #define rb4 xxx[50]
2715b2ba9d3SPiotr Jasiukajtis #define rb5 xxx[51]
2725b2ba9d3SPiotr Jasiukajtis #define rb6 xxx[52]
2735b2ba9d3SPiotr Jasiukajtis #define sb1 xxx[53]
2745b2ba9d3SPiotr Jasiukajtis #define sb2 xxx[54]
2755b2ba9d3SPiotr Jasiukajtis #define sb3 xxx[55]
2765b2ba9d3SPiotr Jasiukajtis #define sb4 xxx[56]
2775b2ba9d3SPiotr Jasiukajtis #define sb5 xxx[57]
2785b2ba9d3SPiotr Jasiukajtis #define sb6 xxx[58]
2795b2ba9d3SPiotr Jasiukajtis #define sb7 xxx[59]
2805b2ba9d3SPiotr Jasiukajtis
2815b2ba9d3SPiotr Jasiukajtis double
erf(double x)2825b2ba9d3SPiotr Jasiukajtis erf(double x) {
2835b2ba9d3SPiotr Jasiukajtis int hx, ix, i;
2845b2ba9d3SPiotr Jasiukajtis double R, S, P, Q, s, y, z, r;
2855b2ba9d3SPiotr Jasiukajtis
2865b2ba9d3SPiotr Jasiukajtis hx = ((int *) &x)[HIWORD];
2875b2ba9d3SPiotr Jasiukajtis ix = hx & 0x7fffffff;
2885b2ba9d3SPiotr Jasiukajtis if (ix >= 0x7ff00000) { /* erf(nan)=nan */
2895b2ba9d3SPiotr Jasiukajtis #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
2905b2ba9d3SPiotr Jasiukajtis if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */
2915b2ba9d3SPiotr Jasiukajtis return (x);
2925b2ba9d3SPiotr Jasiukajtis #endif
2935b2ba9d3SPiotr Jasiukajtis i = ((unsigned) hx >> 31) << 1;
2945b2ba9d3SPiotr Jasiukajtis return ((double) (1 - i) + one / x); /* erf(+-inf)=+-1 */
2955b2ba9d3SPiotr Jasiukajtis }
2965b2ba9d3SPiotr Jasiukajtis
2975b2ba9d3SPiotr Jasiukajtis if (ix < 0x3feb0000) { /* |x|<0.84375 */
2985b2ba9d3SPiotr Jasiukajtis if (ix < 0x3e300000) { /* |x|<2**-28 */
2995b2ba9d3SPiotr Jasiukajtis if (ix < 0x00800000) /* avoid underflow */
3005b2ba9d3SPiotr Jasiukajtis return (0.125 * (8.0 * x + efx8 * x));
3015b2ba9d3SPiotr Jasiukajtis return (x + efx * x);
3025b2ba9d3SPiotr Jasiukajtis }
3035b2ba9d3SPiotr Jasiukajtis z = x * x;
3045b2ba9d3SPiotr Jasiukajtis r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
3055b2ba9d3SPiotr Jasiukajtis s = one +
3065b2ba9d3SPiotr Jasiukajtis z *(qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
3075b2ba9d3SPiotr Jasiukajtis y = r / s;
3085b2ba9d3SPiotr Jasiukajtis return (x + x * y);
3095b2ba9d3SPiotr Jasiukajtis }
3105b2ba9d3SPiotr Jasiukajtis if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
3115b2ba9d3SPiotr Jasiukajtis s = fabs(x) - one;
3125b2ba9d3SPiotr Jasiukajtis P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 +
3135b2ba9d3SPiotr Jasiukajtis s * (pa5 + s * pa6)))));
3145b2ba9d3SPiotr Jasiukajtis Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 +
3155b2ba9d3SPiotr Jasiukajtis s * (qa5 + s * qa6)))));
3165b2ba9d3SPiotr Jasiukajtis if (hx >= 0)
3175b2ba9d3SPiotr Jasiukajtis return (erx + P / Q);
3185b2ba9d3SPiotr Jasiukajtis else
3195b2ba9d3SPiotr Jasiukajtis return (-erx - P / Q);
3205b2ba9d3SPiotr Jasiukajtis }
3215b2ba9d3SPiotr Jasiukajtis if (ix >= 0x40180000) { /* inf > |x| >= 6 */
3225b2ba9d3SPiotr Jasiukajtis if (hx >= 0)
3235b2ba9d3SPiotr Jasiukajtis return (one - tiny);
3245b2ba9d3SPiotr Jasiukajtis else
3255b2ba9d3SPiotr Jasiukajtis return (tiny - one);
3265b2ba9d3SPiotr Jasiukajtis }
3275b2ba9d3SPiotr Jasiukajtis x = fabs(x);
3285b2ba9d3SPiotr Jasiukajtis s = one / (x * x);
3295b2ba9d3SPiotr Jasiukajtis if (ix < 0x4006DB6E) { /* |x| < 1/0.35 */
3305b2ba9d3SPiotr Jasiukajtis R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 +
3315b2ba9d3SPiotr Jasiukajtis s * (ra5 + s * (ra6 + s * ra7))))));
3325b2ba9d3SPiotr Jasiukajtis S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 +
3335b2ba9d3SPiotr Jasiukajtis s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
3345b2ba9d3SPiotr Jasiukajtis } else { /* |x| >= 1/0.35 */
3355b2ba9d3SPiotr Jasiukajtis R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 +
3365b2ba9d3SPiotr Jasiukajtis s * (rb5 + s * rb6)))));
3375b2ba9d3SPiotr Jasiukajtis S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 +
3385b2ba9d3SPiotr Jasiukajtis s * (sb5 + s * (sb6 + s * sb7))))));
3395b2ba9d3SPiotr Jasiukajtis }
3405b2ba9d3SPiotr Jasiukajtis z = x;
3415b2ba9d3SPiotr Jasiukajtis ((int *) &z)[LOWORD] = 0;
3425b2ba9d3SPiotr Jasiukajtis r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
3435b2ba9d3SPiotr Jasiukajtis if (hx >= 0)
3445b2ba9d3SPiotr Jasiukajtis return (one - r / x);
3455b2ba9d3SPiotr Jasiukajtis else
3465b2ba9d3SPiotr Jasiukajtis return (r / x - one);
3475b2ba9d3SPiotr Jasiukajtis }
3485b2ba9d3SPiotr Jasiukajtis
3495b2ba9d3SPiotr Jasiukajtis double
erfc(double x)3505b2ba9d3SPiotr Jasiukajtis erfc(double x) {
3515b2ba9d3SPiotr Jasiukajtis int hx, ix;
3525b2ba9d3SPiotr Jasiukajtis double R, S, P, Q, s, y, z, r;
3535b2ba9d3SPiotr Jasiukajtis
3545b2ba9d3SPiotr Jasiukajtis hx = ((int *) &x)[HIWORD];
3555b2ba9d3SPiotr Jasiukajtis ix = hx & 0x7fffffff;
3565b2ba9d3SPiotr Jasiukajtis if (ix >= 0x7ff00000) { /* erfc(nan)=nan */
3575b2ba9d3SPiotr Jasiukajtis #if defined(FPADD_TRAPS_INCOMPLETE_ON_NAN)
3585b2ba9d3SPiotr Jasiukajtis if (ix >= 0x7ff80000) /* assumes sparc-like QNaN */
3595b2ba9d3SPiotr Jasiukajtis return (x);
3605b2ba9d3SPiotr Jasiukajtis #endif
3615b2ba9d3SPiotr Jasiukajtis /* erfc(+-inf)=0,2 */
3625b2ba9d3SPiotr Jasiukajtis return ((double) (((unsigned) hx >> 31) << 1) + one / x);
3635b2ba9d3SPiotr Jasiukajtis }
3645b2ba9d3SPiotr Jasiukajtis
3655b2ba9d3SPiotr Jasiukajtis if (ix < 0x3feb0000) { /* |x| < 0.84375 */
3665b2ba9d3SPiotr Jasiukajtis if (ix < 0x3c700000) /* |x| < 2**-56 */
3675b2ba9d3SPiotr Jasiukajtis return (one - x);
3685b2ba9d3SPiotr Jasiukajtis z = x * x;
3695b2ba9d3SPiotr Jasiukajtis r = pp0 + z * (pp1 + z * (pp2 + z * (pp3 + z * pp4)));
3705b2ba9d3SPiotr Jasiukajtis s = one +
3715b2ba9d3SPiotr Jasiukajtis z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
3725b2ba9d3SPiotr Jasiukajtis y = r / s;
3735b2ba9d3SPiotr Jasiukajtis if (hx < 0x3fd00000) { /* x < 1/4 */
3745b2ba9d3SPiotr Jasiukajtis return (one - (x + x * y));
3755b2ba9d3SPiotr Jasiukajtis } else {
3765b2ba9d3SPiotr Jasiukajtis r = x * y;
3775b2ba9d3SPiotr Jasiukajtis r += (x - half);
3785b2ba9d3SPiotr Jasiukajtis return (half - r);
3795b2ba9d3SPiotr Jasiukajtis }
3805b2ba9d3SPiotr Jasiukajtis }
3815b2ba9d3SPiotr Jasiukajtis if (ix < 0x3ff40000) { /* 0.84375 <= |x| < 1.25 */
3825b2ba9d3SPiotr Jasiukajtis s = fabs(x) - one;
3835b2ba9d3SPiotr Jasiukajtis P = pa0 + s * (pa1 + s * (pa2 + s * (pa3 + s * (pa4 +
3845b2ba9d3SPiotr Jasiukajtis s * (pa5 + s * pa6)))));
3855b2ba9d3SPiotr Jasiukajtis Q = one + s * (qa1 + s * (qa2 + s * (qa3 + s * (qa4 +
3865b2ba9d3SPiotr Jasiukajtis s * (qa5 + s * qa6)))));
3875b2ba9d3SPiotr Jasiukajtis if (hx >= 0) {
3885b2ba9d3SPiotr Jasiukajtis z = one - erx;
3895b2ba9d3SPiotr Jasiukajtis return (z - P / Q);
3905b2ba9d3SPiotr Jasiukajtis } else {
3915b2ba9d3SPiotr Jasiukajtis z = erx + P / Q;
3925b2ba9d3SPiotr Jasiukajtis return (one + z);
3935b2ba9d3SPiotr Jasiukajtis }
3945b2ba9d3SPiotr Jasiukajtis }
3955b2ba9d3SPiotr Jasiukajtis if (ix < 0x403c0000) { /* |x|<28 */
3965b2ba9d3SPiotr Jasiukajtis x = fabs(x);
3975b2ba9d3SPiotr Jasiukajtis s = one / (x * x);
3985b2ba9d3SPiotr Jasiukajtis if (ix < 0x4006DB6D) { /* |x| < 1/.35 ~ 2.857143 */
3995b2ba9d3SPiotr Jasiukajtis R = ra0 + s * (ra1 + s * (ra2 + s * (ra3 + s * (ra4 +
4005b2ba9d3SPiotr Jasiukajtis s * (ra5 + s * (ra6 + s * ra7))))));
4015b2ba9d3SPiotr Jasiukajtis S = one + s * (sa1 + s * (sa2 + s * (sa3 + s * (sa4 +
4025b2ba9d3SPiotr Jasiukajtis s * (sa5 + s * (sa6 + s * (sa7 + s * sa8)))))));
4035b2ba9d3SPiotr Jasiukajtis } else {
4045b2ba9d3SPiotr Jasiukajtis /* |x| >= 1/.35 ~ 2.857143 */
4055b2ba9d3SPiotr Jasiukajtis if (hx < 0 && ix >= 0x40180000)
4065b2ba9d3SPiotr Jasiukajtis return (two - tiny); /* x < -6 */
4075b2ba9d3SPiotr Jasiukajtis
4085b2ba9d3SPiotr Jasiukajtis R = rb0 + s * (rb1 + s * (rb2 + s * (rb3 + s * (rb4 +
4095b2ba9d3SPiotr Jasiukajtis s * (rb5 + s * rb6)))));
4105b2ba9d3SPiotr Jasiukajtis S = one + s * (sb1 + s * (sb2 + s * (sb3 + s * (sb4 +
4115b2ba9d3SPiotr Jasiukajtis s * (sb5 + s * (sb6 + s * sb7))))));
4125b2ba9d3SPiotr Jasiukajtis }
4135b2ba9d3SPiotr Jasiukajtis z = x;
4145b2ba9d3SPiotr Jasiukajtis ((int *) &z)[LOWORD] = 0;
4155b2ba9d3SPiotr Jasiukajtis r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + R / S);
4165b2ba9d3SPiotr Jasiukajtis if (hx > 0)
4175b2ba9d3SPiotr Jasiukajtis return (r / x);
4185b2ba9d3SPiotr Jasiukajtis else
4195b2ba9d3SPiotr Jasiukajtis return (two - r / x);
4205b2ba9d3SPiotr Jasiukajtis } else {
4215b2ba9d3SPiotr Jasiukajtis if (hx > 0)
4225b2ba9d3SPiotr Jasiukajtis return (tiny * tiny);
4235b2ba9d3SPiotr Jasiukajtis else
4245b2ba9d3SPiotr Jasiukajtis return (two - tiny);
4255b2ba9d3SPiotr Jasiukajtis }
4265b2ba9d3SPiotr Jasiukajtis }
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