Index: generic/lib/SOURCES =================================================================== --- generic/lib/SOURCES +++ generic/lib/SOURCES @@ -79,8 +79,6 @@ math/cosh.cl math/cospi.cl math/ep_log.cl -math/erf.cl -math/erfc.cl math/exp.cl math/exp_helper.cl math/exp2.cl Index: generic/lib/math/erf.cl =================================================================== --- generic/lib/math/erf.cl +++ /dev/null @@ -1,380 +0,0 @@ -#include - -#include "math.h" -#include "../clcmacro.h" - -/* - * ==================================================== - * 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. - * ==================================================== -*/ - -#define erx 8.4506291151e-01f /* 0x3f58560b */ - -// Coefficients for approximation to erf on [00.84375] - -#define efx 1.2837916613e-01f /* 0x3e0375d4 */ -#define efx8 1.0270333290e+00f /* 0x3f8375d4 */ - -#define pp0 1.2837916613e-01f /* 0x3e0375d4 */ -#define pp1 -3.2504209876e-01f /* 0xbea66beb */ -#define pp2 -2.8481749818e-02f /* 0xbce9528f */ -#define pp3 -5.7702702470e-03f /* 0xbbbd1489 */ -#define pp4 -2.3763017452e-05f /* 0xb7c756b1 */ -#define qq1 3.9791721106e-01f /* 0x3ecbbbce */ -#define qq2 6.5022252500e-02f /* 0x3d852a63 */ -#define qq3 5.0813062117e-03f /* 0x3ba68116 */ -#define qq4 1.3249473704e-04f /* 0x390aee49 */ -#define qq5 -3.9602282413e-06f /* 0xb684e21a */ - -// Coefficients for approximation to erf in [0.843751.25] - -#define pa0 -2.3621185683e-03f /* 0xbb1acdc6 */ -#define pa1 4.1485610604e-01f /* 0x3ed46805 */ -#define pa2 -3.7220788002e-01f /* 0xbebe9208 */ -#define pa3 3.1834661961e-01f /* 0x3ea2fe54 */ -#define pa4 -1.1089469492e-01f /* 0xbde31cc2 */ -#define pa5 3.5478305072e-02f /* 0x3d1151b3 */ -#define pa6 -2.1663755178e-03f /* 0xbb0df9c0 */ -#define qa1 1.0642088205e-01f /* 0x3dd9f331 */ -#define qa2 5.4039794207e-01f /* 0x3f0a5785 */ -#define qa3 7.1828655899e-02f /* 0x3d931ae7 */ -#define qa4 1.2617121637e-01f /* 0x3e013307 */ -#define qa5 1.3637083583e-02f /* 0x3c5f6e13 */ -#define qa6 1.1984500103e-02f /* 0x3c445aa3 */ - -// Coefficients for approximation to erfc in [1.251/0.35] - -#define ra0 -9.8649440333e-03f /* 0xbc21a093 */ -#define ra1 -6.9385856390e-01f /* 0xbf31a0b7 */ -#define ra2 -1.0558626175e+01f /* 0xc128f022 */ -#define ra3 -6.2375331879e+01f /* 0xc2798057 */ -#define ra4 -1.6239666748e+02f /* 0xc322658c */ -#define ra5 -1.8460508728e+02f /* 0xc3389ae7 */ -#define ra6 -8.1287437439e+01f /* 0xc2a2932b */ -#define ra7 -9.8143291473e+00f /* 0xc11d077e */ -#define sa1 1.9651271820e+01f /* 0x419d35ce */ -#define sa2 1.3765776062e+02f /* 0x4309a863 */ -#define sa3 4.3456588745e+02f /* 0x43d9486f */ -#define sa4 6.4538726807e+02f /* 0x442158c9 */ -#define sa5 4.2900814819e+02f /* 0x43d6810b */ -#define sa6 1.0863500214e+02f /* 0x42d9451f */ -#define sa7 6.5702495575e+00f /* 0x40d23f7c */ -#define sa8 -6.0424413532e-02f /* 0xbd777f97 */ - -// Coefficients for approximation to erfc in [1/.3528] - -#define rb0 -9.8649431020e-03f /* 0xbc21a092 */ -#define rb1 -7.9928326607e-01f /* 0xbf4c9dd4 */ -#define rb2 -1.7757955551e+01f /* 0xc18e104b */ -#define rb3 -1.6063638306e+02f /* 0xc320a2ea */ -#define rb4 -6.3756646729e+02f /* 0xc41f6441 */ -#define rb5 -1.0250950928e+03f /* 0xc480230b */ -#define rb6 -4.8351919556e+02f /* 0xc3f1c275 */ -#define sb1 3.0338060379e+01f /* 0x41f2b459 */ -#define sb2 3.2579251099e+02f /* 0x43a2e571 */ -#define sb3 1.5367296143e+03f /* 0x44c01759 */ -#define sb4 3.1998581543e+03f /* 0x4547fdbb */ -#define sb5 2.5530502930e+03f /* 0x451f90ce */ -#define sb6 4.7452853394e+02f /* 0x43ed43a7 */ -#define sb7 -2.2440952301e+01f /* 0xc1b38712 */ - -_CLC_OVERLOAD _CLC_DEF float erf(float x) { - int hx = as_uint(x); - int ix = hx & 0x7fffffff; - float absx = as_float(ix); - - float x2 = absx * absx; - float t = 1.0f / x2; - float tt = absx - 1.0f; - t = absx < 1.25f ? tt : t; - t = absx < 0.84375f ? x2 : t; - - float u, v, tu, tv; - - // |x| < 6 - u = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, rb6, rb5), rb4), rb3), rb2), rb1), rb0); - v = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sb7, sb6), sb5), sb4), sb3), sb2), sb1); - - tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, ra7, ra6), ra5), ra4), ra3), ra2), ra1), ra0); - tv = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sa8, sa7), sa6), sa5), sa4), sa3), sa2), sa1); - u = absx < 0x1.6db6dcp+1f ? tu : u; - v = absx < 0x1.6db6dcp+1f ? tv : v; - - tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, pa6, pa5), pa4), pa3), pa2), pa1), pa0); - tv = mad(t, mad(t, mad(t, mad(t, mad(t, qa6, qa5), qa4), qa3), qa2), qa1); - u = absx < 1.25f ? tu : u; - v = absx < 1.25f ? tv : v; - - tu = mad(t, mad(t, mad(t, mad(t, pp4, pp3), pp2), pp1), pp0); - tv = mad(t, mad(t, mad(t, mad(t, qq5, qq4), qq3), qq2), qq1); - u = absx < 0.84375f ? tu : u; - v = absx < 0.84375f ? tv : v; - - v = mad(t, v, 1.0f); - float q = MATH_DIVIDE(u, v); - - float ret = 1.0f; - - // |x| < 6 - float z = as_float(ix & 0xfffff000); - float r = exp(mad(-z, z, -0.5625f)) * exp(mad(z-absx, z+absx, q)); - r = 1.0f - MATH_DIVIDE(r, absx); - ret = absx < 6.0f ? r : ret; - - r = erx + q; - ret = absx < 1.25f ? r : ret; - - ret = as_float((hx & 0x80000000) | as_int(ret)); - - r = mad(x, q, x); - ret = absx < 0.84375f ? r : ret; - - // Prevent underflow - r = 0.125f * mad(8.0f, x, efx8 * x); - ret = absx < 0x1.0p-28f ? r : ret; - - ret = isnan(x) ? x : ret; - - return ret; -} - -_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, erf, float); - -#ifdef cl_khr_fp64 - -#pragma OPENCL EXTENSION cl_khr_fp64 : enable - -/* - * ==================================================== - * 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. - * ==================================================== - */ - -/* 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 >= 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 - */ - -#define AU0 -9.86494292470009928597e-03 -#define AU1 -7.99283237680523006574e-01 -#define AU2 -1.77579549177547519889e+01 -#define AU3 -1.60636384855821916062e+02 -#define AU4 -6.37566443368389627722e+02 -#define AU5 -1.02509513161107724954e+03 -#define AU6 -4.83519191608651397019e+02 - -#define AV1 3.03380607434824582924e+01 -#define AV2 3.25792512996573918826e+02 -#define AV3 1.53672958608443695994e+03 -#define AV4 3.19985821950859553908e+03 -#define AV5 2.55305040643316442583e+03 -#define AV6 4.74528541206955367215e+02 -#define AV7 -2.24409524465858183362e+01 - -#define BU0 -9.86494403484714822705e-03 -#define BU1 -6.93858572707181764372e-01 -#define BU2 -1.05586262253232909814e+01 -#define BU3 -6.23753324503260060396e+01 -#define BU4 -1.62396669462573470355e+02 -#define BU5 -1.84605092906711035994e+02 -#define BU6 -8.12874355063065934246e+01 -#define BU7 -9.81432934416914548592e+00 - -#define BV1 1.96512716674392571292e+01 -#define BV2 1.37657754143519042600e+02 -#define BV3 4.34565877475229228821e+02 -#define BV4 6.45387271733267880336e+02 -#define BV5 4.29008140027567833386e+02 -#define BV6 1.08635005541779435134e+02 -#define BV7 6.57024977031928170135e+00 -#define BV8 -6.04244152148580987438e-02 - -#define CU0 -2.36211856075265944077e-03 -#define CU1 4.14856118683748331666e-01 -#define CU2 -3.72207876035701323847e-01 -#define CU3 3.18346619901161753674e-01 -#define CU4 -1.10894694282396677476e-01 -#define CU5 3.54783043256182359371e-02 -#define CU6 -2.16637559486879084300e-03 - -#define CV1 1.06420880400844228286e-01 -#define CV2 5.40397917702171048937e-01 -#define CV3 7.18286544141962662868e-02 -#define CV4 1.26171219808761642112e-01 -#define CV5 1.36370839120290507362e-02 -#define CV6 1.19844998467991074170e-02 - -#define DU0 1.28379167095512558561e-01 -#define DU1 -3.25042107247001499370e-01 -#define DU2 -2.84817495755985104766e-02 -#define DU3 -5.77027029648944159157e-03 -#define DU4 -2.37630166566501626084e-05 - -#define DV1 3.97917223959155352819e-01 -#define DV2 6.50222499887672944485e-02 -#define DV3 5.08130628187576562776e-03 -#define DV4 1.32494738004321644526e-04 -#define DV5 -3.96022827877536812320e-06 - -_CLC_OVERLOAD _CLC_DEF double erf(double y) { - double x = fabs(y); - double x2 = x * x; - double xm1 = x - 1.0; - - // Poly variable - double t = 1.0 / x2; - t = x < 1.25 ? xm1 : t; - t = x < 0.84375 ? x2 : t; - - double u, ut, v, vt; - - // Evaluate rational poly - // XXX We need to see of we can grab 16 coefficents from a table - // faster than evaluating 3 of the poly pairs - // if (x < 6.0) - u = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AU6, AU5), AU4), AU3), AU2), AU1), AU0); - v = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AV7, AV6), AV5), AV4), AV3), AV2), AV1); - - ut = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BU7, BU6), BU5), BU4), BU3), BU2), BU1), BU0); - vt = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BV8, BV7), BV6), BV5), BV4), BV3), BV2), BV1); - u = x < 0x1.6db6ep+1 ? ut : u; - v = x < 0x1.6db6ep+1 ? vt : v; - - ut = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, CU6, CU5), CU4), CU3), CU2), CU1), CU0); - vt = fma(t, fma(t, fma(t, fma(t, fma(t, CV6, CV5), CV4), CV3), CV2), CV1); - u = x < 1.25 ? ut : u; - v = x < 1.25 ? vt : v; - - ut = fma(t, fma(t, fma(t, fma(t, DU4, DU3), DU2), DU1), DU0); - vt = fma(t, fma(t, fma(t, fma(t, DV5, DV4), DV3), DV2), DV1); - u = x < 0.84375 ? ut : u; - v = x < 0.84375 ? vt : v; - - v = fma(t, v, 1.0); - - // Compute rational approximation - double q = u / v; - - // Compute results - double z = as_double(as_long(x) & 0xffffffff00000000L); - double r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + q); - r = 1.0 - r / x; - - double ret = x < 6.0 ? r : 1.0; - - r = 8.45062911510467529297e-01 + q; - ret = x < 1.25 ? r : ret; - - q = x < 0x1.0p-28 ? 1.28379167095512586316e-01 : q; - - r = fma(x, q, x); - ret = x < 0.84375 ? r : ret; - - ret = isnan(x) ? x : ret; - - return y < 0.0 ? -ret : ret; -} - -_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, erf, double); - -#endif Index: generic/lib/math/erfc.cl =================================================================== --- generic/lib/math/erfc.cl +++ /dev/null @@ -1,391 +0,0 @@ -#include - -#include "math.h" -#include "../clcmacro.h" - -/* - * ==================================================== - * 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. - * ==================================================== - */ - -#define erx_f 8.4506291151e-01f /* 0x3f58560b */ - -// Coefficients for approximation to erf on [00.84375] - -#define efx 1.2837916613e-01f /* 0x3e0375d4 */ -#define efx8 1.0270333290e+00f /* 0x3f8375d4 */ - -#define pp0 1.2837916613e-01f /* 0x3e0375d4 */ -#define pp1 -3.2504209876e-01f /* 0xbea66beb */ -#define pp2 -2.8481749818e-02f /* 0xbce9528f */ -#define pp3 -5.7702702470e-03f /* 0xbbbd1489 */ -#define pp4 -2.3763017452e-05f /* 0xb7c756b1 */ -#define qq1 3.9791721106e-01f /* 0x3ecbbbce */ -#define qq2 6.5022252500e-02f /* 0x3d852a63 */ -#define qq3 5.0813062117e-03f /* 0x3ba68116 */ -#define qq4 1.3249473704e-04f /* 0x390aee49 */ -#define qq5 -3.9602282413e-06f /* 0xb684e21a */ - -// Coefficients for approximation to erf in [0.843751.25] - -#define pa0 -2.3621185683e-03f /* 0xbb1acdc6 */ -#define pa1 4.1485610604e-01f /* 0x3ed46805 */ -#define pa2 -3.7220788002e-01f /* 0xbebe9208 */ -#define pa3 3.1834661961e-01f /* 0x3ea2fe54 */ -#define pa4 -1.1089469492e-01f /* 0xbde31cc2 */ -#define pa5 3.5478305072e-02f /* 0x3d1151b3 */ -#define pa6 -2.1663755178e-03f /* 0xbb0df9c0 */ -#define qa1 1.0642088205e-01f /* 0x3dd9f331 */ -#define qa2 5.4039794207e-01f /* 0x3f0a5785 */ -#define qa3 7.1828655899e-02f /* 0x3d931ae7 */ -#define qa4 1.2617121637e-01f /* 0x3e013307 */ -#define qa5 1.3637083583e-02f /* 0x3c5f6e13 */ -#define qa6 1.1984500103e-02f /* 0x3c445aa3 */ - -// Coefficients for approximation to erfc in [1.251/0.35] - -#define ra0 -9.8649440333e-03f /* 0xbc21a093 */ -#define ra1 -6.9385856390e-01f /* 0xbf31a0b7 */ -#define ra2 -1.0558626175e+01f /* 0xc128f022 */ -#define ra3 -6.2375331879e+01f /* 0xc2798057 */ -#define ra4 -1.6239666748e+02f /* 0xc322658c */ -#define ra5 -1.8460508728e+02f /* 0xc3389ae7 */ -#define ra6 -8.1287437439e+01f /* 0xc2a2932b */ -#define ra7 -9.8143291473e+00f /* 0xc11d077e */ -#define sa1 1.9651271820e+01f /* 0x419d35ce */ -#define sa2 1.3765776062e+02f /* 0x4309a863 */ -#define sa3 4.3456588745e+02f /* 0x43d9486f */ -#define sa4 6.4538726807e+02f /* 0x442158c9 */ -#define sa5 4.2900814819e+02f /* 0x43d6810b */ -#define sa6 1.0863500214e+02f /* 0x42d9451f */ -#define sa7 6.5702495575e+00f /* 0x40d23f7c */ -#define sa8 -6.0424413532e-02f /* 0xbd777f97 */ - -// Coefficients for approximation to erfc in [1/.3528] - -#define rb0 -9.8649431020e-03f /* 0xbc21a092 */ -#define rb1 -7.9928326607e-01f /* 0xbf4c9dd4 */ -#define rb2 -1.7757955551e+01f /* 0xc18e104b */ -#define rb3 -1.6063638306e+02f /* 0xc320a2ea */ -#define rb4 -6.3756646729e+02f /* 0xc41f6441 */ -#define rb5 -1.0250950928e+03f /* 0xc480230b */ -#define rb6 -4.8351919556e+02f /* 0xc3f1c275 */ -#define sb1 3.0338060379e+01f /* 0x41f2b459 */ -#define sb2 3.2579251099e+02f /* 0x43a2e571 */ -#define sb3 1.5367296143e+03f /* 0x44c01759 */ -#define sb4 3.1998581543e+03f /* 0x4547fdbb */ -#define sb5 2.5530502930e+03f /* 0x451f90ce */ -#define sb6 4.7452853394e+02f /* 0x43ed43a7 */ -#define sb7 -2.2440952301e+01f /* 0xc1b38712 */ - -_CLC_OVERLOAD _CLC_DEF float erfc(float x) { - int hx = as_int(x); - int ix = hx & 0x7fffffff; - float absx = as_float(ix); - - // Argument for polys - float x2 = absx * absx; - float t = 1.0f / x2; - float tt = absx - 1.0f; - t = absx < 1.25f ? tt : t; - t = absx < 0.84375f ? x2 : t; - - // Evaluate polys - float tu, tv, u, v; - - u = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, rb6, rb5), rb4), rb3), rb2), rb1), rb0); - v = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sb7, sb6), sb5), sb4), sb3), sb2), sb1); - - tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, ra7, ra6), ra5), ra4), ra3), ra2), ra1), ra0); - tv = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sa8, sa7), sa6), sa5), sa4), sa3), sa2), sa1); - u = absx < 0x1.6db6dap+1f ? tu : u; - v = absx < 0x1.6db6dap+1f ? tv : v; - - tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, pa6, pa5), pa4), pa3), pa2), pa1), pa0); - tv = mad(t, mad(t, mad(t, mad(t, mad(t, qa6, qa5), qa4), qa3), qa2), qa1); - u = absx < 1.25f ? tu : u; - v = absx < 1.25f ? tv : v; - - tu = mad(t, mad(t, mad(t, mad(t, pp4, pp3), pp2), pp1), pp0); - tv = mad(t, mad(t, mad(t, mad(t, qq5, qq4), qq3), qq2), qq1); - u = absx < 0.84375f ? tu : u; - v = absx < 0.84375f ? tv : v; - - v = mad(t, v, 1.0f); - - float q = MATH_DIVIDE(u, v); - - float ret = 0.0f; - - float z = as_float(ix & 0xfffff000); - float r = exp(mad(-z, z, -0.5625f)) * exp(mad(z - absx, z + absx, q)); - r = MATH_DIVIDE(r, absx); - t = 2.0f - r; - r = x < 0.0f ? t : r; - ret = absx < 28.0f ? r : ret; - - r = 1.0f - erx_f - q; - t = erx_f + q + 1.0f; - r = x < 0.0f ? t : r; - ret = absx < 1.25f ? r : ret; - - r = 0.5f - mad(x, q, x - 0.5f); - ret = absx < 0.84375f ? r : ret; - - ret = x < -6.0f ? 2.0f : ret; - - ret = isnan(x) ? x : ret; - - return ret; -} - -_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, erfc, float); - -#ifdef cl_khr_fp64 - -#pragma OPENCL EXTENSION cl_khr_fp64 : enable - -/* - * ==================================================== - * 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. - * ==================================================== - */ - -/* 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 >= 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 - */ - -#define AU0 -9.86494292470009928597e-03 -#define AU1 -7.99283237680523006574e-01 -#define AU2 -1.77579549177547519889e+01 -#define AU3 -1.60636384855821916062e+02 -#define AU4 -6.37566443368389627722e+02 -#define AU5 -1.02509513161107724954e+03 -#define AU6 -4.83519191608651397019e+02 - -#define AV0 3.03380607434824582924e+01 -#define AV1 3.25792512996573918826e+02 -#define AV2 1.53672958608443695994e+03 -#define AV3 3.19985821950859553908e+03 -#define AV4 2.55305040643316442583e+03 -#define AV5 4.74528541206955367215e+02 -#define AV6 -2.24409524465858183362e+01 - -#define BU0 -9.86494403484714822705e-03 -#define BU1 -6.93858572707181764372e-01 -#define BU2 -1.05586262253232909814e+01 -#define BU3 -6.23753324503260060396e+01 -#define BU4 -1.62396669462573470355e+02 -#define BU5 -1.84605092906711035994e+02 -#define BU6 -8.12874355063065934246e+01 -#define BU7 -9.81432934416914548592e+00 - -#define BV0 1.96512716674392571292e+01 -#define BV1 1.37657754143519042600e+02 -#define BV2 4.34565877475229228821e+02 -#define BV3 6.45387271733267880336e+02 -#define BV4 4.29008140027567833386e+02 -#define BV5 1.08635005541779435134e+02 -#define BV6 6.57024977031928170135e+00 -#define BV7 -6.04244152148580987438e-02 - -#define CU0 -2.36211856075265944077e-03 -#define CU1 4.14856118683748331666e-01 -#define CU2 -3.72207876035701323847e-01 -#define CU3 3.18346619901161753674e-01 -#define CU4 -1.10894694282396677476e-01 -#define CU5 3.54783043256182359371e-02 -#define CU6 -2.16637559486879084300e-03 - -#define CV0 1.06420880400844228286e-01 -#define CV1 5.40397917702171048937e-01 -#define CV2 7.18286544141962662868e-02 -#define CV3 1.26171219808761642112e-01 -#define CV4 1.36370839120290507362e-02 -#define CV5 1.19844998467991074170e-02 - -#define DU0 1.28379167095512558561e-01 -#define DU1 -3.25042107247001499370e-01 -#define DU2 -2.84817495755985104766e-02 -#define DU3 -5.77027029648944159157e-03 -#define DU4 -2.37630166566501626084e-05 - -#define DV0 3.97917223959155352819e-01 -#define DV1 6.50222499887672944485e-02 -#define DV2 5.08130628187576562776e-03 -#define DV3 1.32494738004321644526e-04 -#define DV4 -3.96022827877536812320e-06 - -_CLC_OVERLOAD _CLC_DEF double erfc(double x) { - long lx = as_long(x); - long ax = lx & 0x7fffffffffffffffL; - double absx = as_double(ax); - int xneg = lx != ax; - - // Poly arg - double x2 = x * x; - double xm1 = absx - 1.0; - double t = 1.0 / x2; - t = absx < 1.25 ? xm1 : t; - t = absx < 0.84375 ? x2 : t; - - - // Evaluate rational poly - // XXX Need to evaluate if we can grab the 14 coefficients from a - // table faster than evaluating 3 pairs of polys - double tu, tv, u, v; - - // |x| < 28 - u = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AU6, AU5), AU4), AU3), AU2), AU1), AU0); - v = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AV6, AV5), AV4), AV3), AV2), AV1), AV0); - - tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BU7, BU6), BU5), BU4), BU3), BU2), BU1), BU0); - tv = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BV7, BV6), BV5), BV4), BV3), BV2), BV1), BV0); - u = absx < 0x1.6db6dp+1 ? tu : u; - v = absx < 0x1.6db6dp+1 ? tv : v; - - tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, CU6, CU5), CU4), CU3), CU2), CU1), CU0); - tv = fma(t, fma(t, fma(t, fma(t, fma(t, CV5, CV4), CV3), CV2), CV1), CV0); - u = absx < 1.25 ? tu : u; - v = absx < 1.25 ? tv : v; - - tu = fma(t, fma(t, fma(t, fma(t, DU4, DU3), DU2), DU1), DU0); - tv = fma(t, fma(t, fma(t, fma(t, DV4, DV3), DV2), DV1), DV0); - u = absx < 0.84375 ? tu : u; - v = absx < 0.84375 ? tv : v; - - v = fma(t, v, 1.0); - double q = u / v; - - - // Evaluate return value - - // |x| < 28 - double z = as_double(ax & 0xffffffff00000000UL); - double ret = exp(-z * z - 0.5625) * exp((z - absx) * (z + absx) + q) / absx; - t = 2.0 - ret; - ret = xneg ? t : ret; - - const double erx = 8.45062911510467529297e-01; - z = erx + q + 1.0; - t = 1.0 - erx - q; - t = xneg ? z : t; - ret = absx < 1.25 ? t : ret; - - // z = 1.0 - fma(x, q, x); - // t = 0.5 - fma(x, q, x - 0.5); - // t = xneg == 1 | absx < 0.25 ? z : t; - t = fma(-x, q, 1.0 - x); - ret = absx < 0.84375 ? t : ret; - - ret = x >= 28.0 ? 0.0 : ret; - ret = x <= -6.0 ? 2.0 : ret; - ret = ax > 0x7ff0000000000000UL ? x : ret; - - return ret; -} - -_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, erfc, double); - -#endif