| /* @(#)s_erf.c 5.1 93/09/24 */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * |
| * Developed at SunPro, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25, |
| for performance improvement on pipelined processors. |
| */ |
| |
| #if defined(LIBM_SCCS) && !defined(lint) |
| static char rcsid[] = "$NetBSD: s_erf.c,v 1.8 1995/05/10 20:47:05 jtc Exp $"; |
| #endif |
| |
| /* 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 |
| */ |
| |
| |
| #include <errno.h> |
| #include <float.h> |
| #include <math.h> |
| #include <math_private.h> |
| |
| static const double |
| tiny = 1e-300, |
| half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
| one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
| two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ |
| /* c = (float)0.84506291151 */ |
| erx = 8.45062911510467529297e-01, /* 0x3FEB0AC1, 0x60000000 */ |
| /* |
| * Coefficients for approximation to erf on [0,0.84375] |
| */ |
| efx = 1.28379167095512586316e-01, /* 0x3FC06EBA, 0x8214DB69 */ |
| efx8 = 1.02703333676410069053e+00, /* 0x3FF06EBA, 0x8214DB69 */ |
| pp[] = { 1.28379167095512558561e-01, /* 0x3FC06EBA, 0x8214DB68 */ |
| -3.25042107247001499370e-01, /* 0xBFD4CD7D, 0x691CB913 */ |
| -2.84817495755985104766e-02, /* 0xBF9D2A51, 0xDBD7194F */ |
| -5.77027029648944159157e-03, /* 0xBF77A291, 0x236668E4 */ |
| -2.37630166566501626084e-05 }, /* 0xBEF8EAD6, 0x120016AC */ |
| qq[] = { 0.0, 3.97917223959155352819e-01, /* 0x3FD97779, 0xCDDADC09 */ |
| 6.50222499887672944485e-02, /* 0x3FB0A54C, 0x5536CEBA */ |
| 5.08130628187576562776e-03, /* 0x3F74D022, 0xC4D36B0F */ |
| 1.32494738004321644526e-04, /* 0x3F215DC9, 0x221C1A10 */ |
| -3.96022827877536812320e-06 }, /* 0xBED09C43, 0x42A26120 */ |
| /* |
| * Coefficients for approximation to erf in [0.84375,1.25] |
| */ |
| pa[] = { -2.36211856075265944077e-03, /* 0xBF6359B8, 0xBEF77538 */ |
| 4.14856118683748331666e-01, /* 0x3FDA8D00, 0xAD92B34D */ |
| -3.72207876035701323847e-01, /* 0xBFD7D240, 0xFBB8C3F1 */ |
| 3.18346619901161753674e-01, /* 0x3FD45FCA, 0x805120E4 */ |
| -1.10894694282396677476e-01, /* 0xBFBC6398, 0x3D3E28EC */ |
| 3.54783043256182359371e-02, /* 0x3FA22A36, 0x599795EB */ |
| -2.16637559486879084300e-03 }, /* 0xBF61BF38, 0x0A96073F */ |
| qa[] = { 0.0, 1.06420880400844228286e-01, /* 0x3FBB3E66, 0x18EEE323 */ |
| 5.40397917702171048937e-01, /* 0x3FE14AF0, 0x92EB6F33 */ |
| 7.18286544141962662868e-02, /* 0x3FB2635C, 0xD99FE9A7 */ |
| 1.26171219808761642112e-01, /* 0x3FC02660, 0xE763351F */ |
| 1.36370839120290507362e-02, /* 0x3F8BEDC2, 0x6B51DD1C */ |
| 1.19844998467991074170e-02 }, /* 0x3F888B54, 0x5735151D */ |
| /* |
| * Coefficients for approximation to erfc in [1.25,1/0.35] |
| */ |
| ra[] = { -9.86494403484714822705e-03, /* 0xBF843412, 0x600D6435 */ |
| -6.93858572707181764372e-01, /* 0xBFE63416, 0xE4BA7360 */ |
| -1.05586262253232909814e+01, /* 0xC0251E04, 0x41B0E726 */ |
| -6.23753324503260060396e+01, /* 0xC04F300A, 0xE4CBA38D */ |
| -1.62396669462573470355e+02, /* 0xC0644CB1, 0x84282266 */ |
| -1.84605092906711035994e+02, /* 0xC067135C, 0xEBCCABB2 */ |
| -8.12874355063065934246e+01, /* 0xC0545265, 0x57E4D2F2 */ |
| -9.81432934416914548592e+00 }, /* 0xC023A0EF, 0xC69AC25C */ |
| sa[] = { 0.0, 1.96512716674392571292e+01, /* 0x4033A6B9, 0xBD707687 */ |
| 1.37657754143519042600e+02, /* 0x4061350C, 0x526AE721 */ |
| 4.34565877475229228821e+02, /* 0x407B290D, 0xD58A1A71 */ |
| 6.45387271733267880336e+02, /* 0x40842B19, 0x21EC2868 */ |
| 4.29008140027567833386e+02, /* 0x407AD021, 0x57700314 */ |
| 1.08635005541779435134e+02, /* 0x405B28A3, 0xEE48AE2C */ |
| 6.57024977031928170135e+00, /* 0x401A47EF, 0x8E484A93 */ |
| -6.04244152148580987438e-02 }, /* 0xBFAEEFF2, 0xEE749A62 */ |
| /* |
| * Coefficients for approximation to erfc in [1/.35,28] |
| */ |
| rb[] = { -9.86494292470009928597e-03, /* 0xBF843412, 0x39E86F4A */ |
| -7.99283237680523006574e-01, /* 0xBFE993BA, 0x70C285DE */ |
| -1.77579549177547519889e+01, /* 0xC031C209, 0x555F995A */ |
| -1.60636384855821916062e+02, /* 0xC064145D, 0x43C5ED98 */ |
| -6.37566443368389627722e+02, /* 0xC083EC88, 0x1375F228 */ |
| -1.02509513161107724954e+03, /* 0xC0900461, 0x6A2E5992 */ |
| -4.83519191608651397019e+02 }, /* 0xC07E384E, 0x9BDC383F */ |
| sb[] = { 0.0, 3.03380607434824582924e+01, /* 0x403E568B, 0x261D5190 */ |
| 3.25792512996573918826e+02, /* 0x40745CAE, 0x221B9F0A */ |
| 1.53672958608443695994e+03, /* 0x409802EB, 0x189D5118 */ |
| 3.19985821950859553908e+03, /* 0x40A8FFB7, 0x688C246A */ |
| 2.55305040643316442583e+03, /* 0x40A3F219, 0xCEDF3BE6 */ |
| 4.74528541206955367215e+02, /* 0x407DA874, 0xE79FE763 */ |
| -2.24409524465858183362e+01 }; /* 0xC03670E2, 0x42712D62 */ |
| |
| double |
| __erf (double x) |
| { |
| int32_t hx, ix, i; |
| double R, S, P, Q, s, y, z, r; |
| GET_HIGH_WORD (hx, x); |
| ix = hx & 0x7fffffff; |
| if (ix >= 0x7ff00000) /* erf(nan)=nan */ |
| { |
| i = ((u_int32_t) hx >> 31) << 1; |
| return (double) (1 - i) + one / x; /* erf(+-inf)=+-1 */ |
| } |
| |
| if (ix < 0x3feb0000) /* |x|<0.84375 */ |
| { |
| double r1, r2, s1, s2, s3, z2, z4; |
| if (ix < 0x3e300000) /* |x|<2**-28 */ |
| { |
| if (ix < 0x00800000) |
| return 0.125 * (8.0 * x + efx8 * x); /*avoid underflow */ |
| return x + efx * x; |
| } |
| z = x * x; |
| r1 = pp[0] + z * pp[1]; z2 = z * z; |
| r2 = pp[2] + z * pp[3]; z4 = z2 * z2; |
| s1 = one + z * qq[1]; |
| s2 = qq[2] + z * qq[3]; |
| s3 = qq[4] + z * qq[5]; |
| r = r1 + z2 * r2 + z4 * pp[4]; |
| s = s1 + z2 * s2 + z4 * s3; |
| y = r / s; |
| return x + x * y; |
| } |
| if (ix < 0x3ff40000) /* 0.84375 <= |x| < 1.25 */ |
| { |
| double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4; |
| s = fabs (x) - one; |
| P1 = pa[0] + s * pa[1]; s2 = s * s; |
| Q1 = one + s * qa[1]; s4 = s2 * s2; |
| P2 = pa[2] + s * pa[3]; s6 = s4 * s2; |
| Q2 = qa[2] + s * qa[3]; |
| P3 = pa[4] + s * pa[5]; |
| Q3 = qa[4] + s * qa[5]; |
| P4 = pa[6]; |
| Q4 = qa[6]; |
| P = P1 + s2 * P2 + s4 * P3 + s6 * P4; |
| Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4; |
| 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 */ |
| { |
| double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8; |
| R1 = ra[0] + s * ra[1]; s2 = s * s; |
| S1 = one + s * sa[1]; s4 = s2 * s2; |
| R2 = ra[2] + s * ra[3]; s6 = s4 * s2; |
| S2 = sa[2] + s * sa[3]; s8 = s4 * s4; |
| R3 = ra[4] + s * ra[5]; |
| S3 = sa[4] + s * sa[5]; |
| R4 = ra[6] + s * ra[7]; |
| S4 = sa[6] + s * sa[7]; |
| R = R1 + s2 * R2 + s4 * R3 + s6 * R4; |
| S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8]; |
| } |
| else /* |x| >= 1/0.35 */ |
| { |
| double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6; |
| R1 = rb[0] + s * rb[1]; s2 = s * s; |
| S1 = one + s * sb[1]; s4 = s2 * s2; |
| R2 = rb[2] + s * rb[3]; s6 = s4 * s2; |
| S2 = sb[2] + s * sb[3]; |
| R3 = rb[4] + s * rb[5]; |
| S3 = sb[4] + s * sb[5]; |
| S4 = sb[6] + s * sb[7]; |
| R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6]; |
| S = S1 + s2 * S2 + s4 * S3 + s6 * S4; |
| } |
| z = x; |
| SET_LOW_WORD (z, 0); |
| r = __ieee754_exp (-z * z - 0.5625) * |
| __ieee754_exp ((z - x) * (z + x) + R / S); |
| if (hx >= 0) |
| return one - r / x; |
| else |
| return r / x - one; |
| } |
| weak_alias (__erf, erf) |
| #ifdef NO_LONG_DOUBLE |
| strong_alias (__erf, __erfl) |
| weak_alias (__erf, erfl) |
| #endif |
| |
| double |
| __erfc (double x) |
| { |
| int32_t hx, ix; |
| double R, S, P, Q, s, y, z, r; |
| GET_HIGH_WORD (hx, x); |
| ix = hx & 0x7fffffff; |
| if (ix >= 0x7ff00000) /* erfc(nan)=nan */ |
| { /* erfc(+-inf)=0,2 */ |
| return (double) (((u_int32_t) hx >> 31) << 1) + one / x; |
| } |
| |
| if (ix < 0x3feb0000) /* |x|<0.84375 */ |
| { |
| double r1, r2, s1, s2, s3, z2, z4; |
| if (ix < 0x3c700000) /* |x|<2**-56 */ |
| return one - x; |
| z = x * x; |
| r1 = pp[0] + z * pp[1]; z2 = z * z; |
| r2 = pp[2] + z * pp[3]; z4 = z2 * z2; |
| s1 = one + z * qq[1]; |
| s2 = qq[2] + z * qq[3]; |
| s3 = qq[4] + z * qq[5]; |
| r = r1 + z2 * r2 + z4 * pp[4]; |
| s = s1 + z2 * s2 + z4 * s3; |
| 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 */ |
| { |
| double s2, s4, s6, P1, P2, P3, P4, Q1, Q2, Q3, Q4; |
| s = fabs (x) - one; |
| P1 = pa[0] + s * pa[1]; s2 = s * s; |
| Q1 = one + s * qa[1]; s4 = s2 * s2; |
| P2 = pa[2] + s * pa[3]; s6 = s4 * s2; |
| Q2 = qa[2] + s * qa[3]; |
| P3 = pa[4] + s * pa[5]; |
| Q3 = qa[4] + s * qa[5]; |
| P4 = pa[6]; |
| Q4 = qa[6]; |
| P = P1 + s2 * P2 + s4 * P3 + s6 * P4; |
| Q = Q1 + s2 * Q2 + s4 * Q3 + s6 * Q4; |
| 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*/ |
| { |
| double R1, R2, R3, R4, S1, S2, S3, S4, s2, s4, s6, s8; |
| R1 = ra[0] + s * ra[1]; s2 = s * s; |
| S1 = one + s * sa[1]; s4 = s2 * s2; |
| R2 = ra[2] + s * ra[3]; s6 = s4 * s2; |
| S2 = sa[2] + s * sa[3]; s8 = s4 * s4; |
| R3 = ra[4] + s * ra[5]; |
| S3 = sa[4] + s * sa[5]; |
| R4 = ra[6] + s * ra[7]; |
| S4 = sa[6] + s * sa[7]; |
| R = R1 + s2 * R2 + s4 * R3 + s6 * R4; |
| S = S1 + s2 * S2 + s4 * S3 + s6 * S4 + s8 * sa[8]; |
| } |
| else /* |x| >= 1/.35 ~ 2.857143 */ |
| { |
| double R1, R2, R3, S1, S2, S3, S4, s2, s4, s6; |
| if (hx < 0 && ix >= 0x40180000) |
| return two - tiny; /* x < -6 */ |
| R1 = rb[0] + s * rb[1]; s2 = s * s; |
| S1 = one + s * sb[1]; s4 = s2 * s2; |
| R2 = rb[2] + s * rb[3]; s6 = s4 * s2; |
| S2 = sb[2] + s * sb[3]; |
| R3 = rb[4] + s * rb[5]; |
| S3 = sb[4] + s * sb[5]; |
| S4 = sb[6] + s * sb[7]; |
| R = R1 + s2 * R2 + s4 * R3 + s6 * rb[6]; |
| S = S1 + s2 * S2 + s4 * S3 + s6 * S4; |
| } |
| z = x; |
| SET_LOW_WORD (z, 0); |
| r = __ieee754_exp (-z * z - 0.5625) * |
| __ieee754_exp ((z - x) * (z + x) + R / S); |
| if (hx > 0) |
| { |
| #if FLT_EVAL_METHOD != 0 |
| volatile |
| #endif |
| double ret = r / x; |
| if (ret == 0) |
| __set_errno (ERANGE); |
| return ret; |
| } |
| else |
| return two - r / x; |
| } |
| else |
| { |
| if (hx > 0) |
| { |
| __set_errno (ERANGE); |
| return tiny * tiny; |
| } |
| else |
| return two - tiny; |
| } |
| } |
| weak_alias (__erfc, erfc) |
| #ifdef NO_LONG_DOUBLE |
| strong_alias (__erfc, __erfcl) |
| weak_alias (__erfc, erfcl) |
| #endif |
