/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_erfl.c */ /* * ==================================================== * 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. * ==================================================== */ /* * Copyright (c) 2008 Stephen L. Moshier * * Permission to use, copy, modify, and distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ /* 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] * 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 * 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) * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] * * 3. For x in [1.25,1/0.35(~2.857143)], * erfc(x) = (1/x)*exp(-x*x-0.5625+R1(z)/S1(z)) * z=1/x^2 * erf(x) = 1 - erfc(x) * * 4. For x in [1/0.35,107] * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2(z)/S2(z)) * if -6.666 x >= 107 * 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 "libm.h" #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 long double erfl(long double x) { return erf(x); } #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 static const long double tiny = 1e-4931L, half = 0.5L, one = 1.0L, two = 2.0L, /* c = (float)0.84506291151 */ erx = 0.845062911510467529296875L, /* * Coefficients for approximation to erf on [0,0.84375] */ /* 2/sqrt(pi) - 1 */ efx = 1.2837916709551257389615890312154517168810E-1L, /* 8 * (2/sqrt(pi) - 1) */ efx8 = 1.0270333367641005911692712249723613735048E0L, pp[6] = { 1.122751350964552113068262337278335028553E6L, -2.808533301997696164408397079650699163276E6L, -3.314325479115357458197119660818768924100E5L, -6.848684465326256109712135497895525446398E4L, -2.657817695110739185591505062971929859314E3L, -1.655310302737837556654146291646499062882E2L, }, qq[6] = { 8.745588372054466262548908189000448124232E6L, 3.746038264792471129367533128637019611485E6L, 7.066358783162407559861156173539693900031E5L, 7.448928604824620999413120955705448117056E4L, 4.511583986730994111992253980546131408924E3L, 1.368902937933296323345610240009071254014E2L, /* 1.000000000000000000000000000000000000000E0 */ }, /* * Coefficients for approximation to erf in [0.84375,1.25] */ /* erf(x+1) = 0.845062911510467529296875 + pa(x)/qa(x) -0.15625 <= x <= +.25 Peak relative error 8.5e-22 */ pa[8] = { -1.076952146179812072156734957705102256059E0L, 1.884814957770385593365179835059971587220E2L, -5.339153975012804282890066622962070115606E1L, 4.435910679869176625928504532109635632618E1L, 1.683219516032328828278557309642929135179E1L, -2.360236618396952560064259585299045804293E0L, 1.852230047861891953244413872297940938041E0L, 9.394994446747752308256773044667843200719E-2L, }, qa[7] = { 4.559263722294508998149925774781887811255E2L, 3.289248982200800575749795055149780689738E2L, 2.846070965875643009598627918383314457912E2L, 1.398715859064535039433275722017479994465E2L, 6.060190733759793706299079050985358190726E1L, 2.078695677795422351040502569964299664233E1L, 4.641271134150895940966798357442234498546E0L, /* 1.000000000000000000000000000000000000000E0 */ }, /* * Coefficients for approximation to erfc in [1.25,1/0.35] */ /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + ra(x^2)/sa(x^2)) 1/2.85711669921875 < 1/x < 1/1.25 Peak relative error 3.1e-21 */ ra[] = { 1.363566591833846324191000679620738857234E-1L, 1.018203167219873573808450274314658434507E1L, 1.862359362334248675526472871224778045594E2L, 1.411622588180721285284945138667933330348E3L, 5.088538459741511988784440103218342840478E3L, 8.928251553922176506858267311750789273656E3L, 7.264436000148052545243018622742770549982E3L, 2.387492459664548651671894725748959751119E3L, 2.220916652813908085449221282808458466556E2L, }, sa[] = { -1.382234625202480685182526402169222331847E1L, -3.315638835627950255832519203687435946482E2L, -2.949124863912936259747237164260785326692E3L, -1.246622099070875940506391433635999693661E4L, -2.673079795851665428695842853070996219632E4L, -2.880269786660559337358397106518918220991E4L, -1.450600228493968044773354186390390823713E4L, -2.874539731125893533960680525192064277816E3L, -1.402241261419067750237395034116942296027E2L, /* 1.000000000000000000000000000000000000000E0 */ }, /* * Coefficients for approximation to erfc in [1/.35,107] */ /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rb(x^2)/sb(x^2)) 1/6.6666259765625 < 1/x < 1/2.85711669921875 Peak relative error 4.2e-22 */ rb[] = { -4.869587348270494309550558460786501252369E-5L, -4.030199390527997378549161722412466959403E-3L, -9.434425866377037610206443566288917589122E-2L, -9.319032754357658601200655161585539404155E-1L, -4.273788174307459947350256581445442062291E0L, -8.842289940696150508373541814064198259278E0L, -7.069215249419887403187988144752613025255E0L, -1.401228723639514787920274427443330704764E0L, }, sb[] = { 4.936254964107175160157544545879293019085E-3L, 1.583457624037795744377163924895349412015E-1L, 1.850647991850328356622940552450636420484E0L, 9.927611557279019463768050710008450625415E0L, 2.531667257649436709617165336779212114570E1L, 2.869752886406743386458304052862814690045E1L, 1.182059497870819562441683560749192539345E1L, /* 1.000000000000000000000000000000000000000E0 */ }, /* erfc(1/x) = x exp (-1/x^2 - 0.5625 + rc(x^2)/sc(x^2)) 1/107 <= 1/x <= 1/6.6666259765625 Peak relative error 1.1e-21 */ rc[] = { -8.299617545269701963973537248996670806850E-5L, -6.243845685115818513578933902532056244108E-3L, -1.141667210620380223113693474478394397230E-1L, -7.521343797212024245375240432734425789409E-1L, -1.765321928311155824664963633786967602934E0L, -1.029403473103215800456761180695263439188E0L, }, sc[] = { 8.413244363014929493035952542677768808601E-3L, 2.065114333816877479753334599639158060979E-1L, 1.639064941530797583766364412782135680148E0L, 4.936788463787115555582319302981666347450E0L, 5.005177727208955487404729933261347679090E0L, /* 1.000000000000000000000000000000000000000E0 */ }; long double erfl(long double x) { long double R, S, P, Q, s, y, z, r; int32_t ix, i; uint32_t se, i0, i1; GET_LDOUBLE_WORDS (se, i0, i1, x); ix = se & 0x7fff; if (ix >= 0x7fff) { /* erf(nan)=nan */ i = ((se & 0xffff) >> 15) << 1; return (long double)(1 - i) + one / x; /* erf(+-inf)=+-1 */ } ix = (ix << 16) | (i0 >> 16); if (ix < 0x3ffed800) { /* |x| < 0.84375 */ if (ix < 0x3fde8000) { /* |x| < 2**-33 */ if (ix < 0x00080000) return 0.125 * (8.0 * x + efx8 * x); /* avoid underflow */ return x + efx * x; } z = x * x; r = pp[0] + z * (pp[1] + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5])))); s = qq[0] + z * (qq[1] + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z))))); y = r / s; return x + x * y; } if (ix < 0x3fffa000) { /* 0.84375 <= |x| < 1.25 */ s = fabsl (x) - one; P = pa[0] + s * (pa[1] + s * (pa[2] + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7])))))); Q = qa[0] + s * (qa[1] + s * (qa[2] + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s)))))); if ((se & 0x8000) == 0) return erx + P / Q; return -erx - P / Q; } if (ix >= 0x4001d555) { /* inf > |x| >= 6.6666259765625 */ if ((se & 0x8000) == 0) return one - tiny; return tiny - one; } x = fabsl (x); s = one / (x * x); if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */ R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] + s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8]))))))); S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] + s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s)))))))); } else { /* 2.857 <= |x| < 6.667 */ R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] + s * (rb[5] + s * (rb[6] + s * rb[7])))))); S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] + s * (sb[5] + s * (sb[6] + s)))))); } z = x; GET_LDOUBLE_WORDS(i, i0, i1, z); i1 = 0; SET_LDOUBLE_WORDS(z, i, i0, i1); r = expl(-z * z - 0.5625) * expl((z - x) * (z + x) + R / S); if ((se & 0x8000) == 0) return one - r / x; return r / x - one; } long double erfcl(long double x) { int32_t hx, ix; long double R, S, P, Q, s, y, z, r; uint32_t se, i0, i1; GET_LDOUBLE_WORDS (se, i0, i1, x); ix = se & 0x7fff; if (ix >= 0x7fff) { /* erfc(nan) = nan, erfc(+-inf) = 0,2 */ return (long double)(((se & 0xffff) >> 15) << 1) + one / x; } ix = (ix << 16) | (i0 >> 16); if (ix < 0x3ffed800) { /* |x| < 0.84375 */ if (ix < 0x3fbe0000) /* |x| < 2**-65 */ return one - x; z = x * x; r = pp[0] + z * (pp[1] + z * (pp[2] + z * (pp[3] + z * (pp[4] + z * pp[5])))); s = qq[0] + z * (qq[1] + z * (qq[2] + z * (qq[3] + z * (qq[4] + z * (qq[5] + z))))); y = r / s; if (ix < 0x3ffd8000) /* x < 1/4 */ return one - (x + x * y); r = x * y; r += x - half; return half - r; } if (ix < 0x3fffa000) { /* 0.84375 <= |x| < 1.25 */ s = fabsl (x) - one; P = pa[0] + s * (pa[1] + s * (pa[2] + s * (pa[3] + s * (pa[4] + s * (pa[5] + s * (pa[6] + s * pa[7])))))); Q = qa[0] + s * (qa[1] + s * (qa[2] + s * (qa[3] + s * (qa[4] + s * (qa[5] + s * (qa[6] + s)))))); if ((se & 0x8000) == 0) { z = one - erx; return z - P / Q; } z = erx + P / Q; return one + z; } if (ix < 0x4005d600) { /* |x| < 107 */ x = fabsl (x); s = one / (x * x); if (ix < 0x4000b6db) { /* 1.25 <= |x| < 2.85711669921875 ~ 1/.35 */ R = ra[0] + s * (ra[1] + s * (ra[2] + s * (ra[3] + s * (ra[4] + s * (ra[5] + s * (ra[6] + s * (ra[7] + s * ra[8]))))))); S = sa[0] + s * (sa[1] + s * (sa[2] + s * (sa[3] + s * (sa[4] + s * (sa[5] + s * (sa[6] + s * (sa[7] + s * (sa[8] + s)))))))); } else if (ix < 0x4001d555) { /* 6.6666259765625 > |x| >= 1/.35 ~ 2.857143 */ R = rb[0] + s * (rb[1] + s * (rb[2] + s * (rb[3] + s * (rb[4] + s * (rb[5] + s * (rb[6] + s * rb[7])))))); S = sb[0] + s * (sb[1] + s * (sb[2] + s * (sb[3] + s * (sb[4] + s * (sb[5] + s * (sb[6] + s)))))); } else { /* 107 > |x| >= 6.666 */ if (se & 0x8000) return two - tiny;/* x < -6.666 */ R = rc[0] + s * (rc[1] + s * (rc[2] + s * (rc[3] + s * (rc[4] + s * rc[5])))); S = sc[0] + s * (sc[1] + s * (sc[2] + s * (sc[3] + s * (sc[4] + s)))); } z = x; GET_LDOUBLE_WORDS (hx, i0, i1, z); i1 = 0; i0 &= 0xffffff00; SET_LDOUBLE_WORDS (z, hx, i0, i1); r = expl (-z * z - 0.5625) * expl ((z - x) * (z + x) + R / S); if ((se & 0x8000) == 0) return r / x; return two - r / x; } if ((se & 0x8000) == 0) return tiny * tiny; return two - tiny; } #endif