| /* |
| * 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 "math64.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. |
| * ==================================================== |
| */ |
| |
| /* 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 |
| |
| __attribute__((overloadable)) 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; |
| } |
| |
