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Diffstat (limited to 'generic/lib/math/erf.cl')
| -rw-r--r-- | generic/lib/math/erf.cl | 402 |
1 files changed, 402 insertions, 0 deletions
diff --git a/generic/lib/math/erf.cl b/generic/lib/math/erf.cl new file mode 100644 index 0000000..3dc82d9 --- /dev/null +++ b/generic/lib/math/erf.cl @@ -0,0 +1,402 @@ +/* + * Copyright (c) 2014 Advanced Micro Devices, Inc. + * + * Permission is hereby granted, free of charge, to any person obtaining a copy + * of this software and associated documentation files (the "Software"), to deal + * in the Software without restriction, including without limitation the rights + * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell + * copies of the Software, and to permit persons to whom the Software is + * furnished to do so, subject to the following conditions: + * + * The above copyright notice and this permission notice shall be included in + * all copies or substantial portions of the Software. + * + * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR + * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, + * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE + * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER + * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, + * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN + * THE SOFTWARE. + */ + +#include <clc/clc.h> + +#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<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 + */ + +#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 |