| //===----------------------------------------------------------------------===// |
| // |
| // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. |
| // See https://llvm.org/LICENSE.txt for license information. |
| // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception |
| // |
| //===----------------------------------------------------------------------===// |
| |
| #include <clc/clc.h> |
| #include <clc/clcmacro.h> |
| #include <clc/math/math.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 [0, 0.84375] |
| |
| #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 [0.84375, 1.25] |
| |
| #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 [1.25, 1/0.35] |
| |
| #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 [1/0.35, 28] |
| |
| #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(-z * z) * exp(mad(z - absx, z + absx, q)); |
| r *= 0x1.23ba94p-1f; // exp(-0.5625) |
| 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<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 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); |
| |
| #ifdef cl_khr_fp16 |
| |
| #pragma OPENCL EXTENSION cl_khr_fp16 : enable |
| |
| _CLC_OVERLOAD _CLC_DEF half erfc(half h) { |
| return (half)erfc((float)h); |
| } |
| |
| _CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, half, erfc, half); |
| |
| #endif |
| |
| #endif |
