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-/* @(#)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>
-#include <fix-int-fp-convert-zero.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 */
- 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)
- {
- /* Avoid spurious underflow. */
- double ret = 0.0625 * (16.0 * x + (16.0 * efx) * x);
- math_check_force_underflow (ret);
- return ret;
- }
- 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 */
- double ret = (double) (((u_int32_t) hx >> 31) << 1) + one / x;
- if (FIX_INT_FP_CONVERT_ZERO && ret == 0.0)
- return 0.0;
- return ret;
- }
-
- 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)
- {
- double ret = math_narrow_eval (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
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