libclc/generic/lib/math/sincosD_piby4.h

   1 /*
   2  * Copyright (c) 2014 Advanced Micro Devices, Inc.
   3  *
   4  * Permission is hereby granted, free of charge, to any person obtaining a copy
   5  * of this software and associated documentation files (the "Software"), to deal
   6  * in the Software without restriction, including without limitation the rights
   7  * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
   8  * copies of the Software, and to permit persons to whom the Software is
   9  * furnished to do so, subject to the following conditions:
  10  *
  11  * The above copyright notice and this permission notice shall be included in
  12  * all copies or substantial portions of the Software.
  13  *
  14  * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  15  * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  16  * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  17  * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  18  * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  19  * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
  20  * THE SOFTWARE.
  21  */
  22
  23 #pragma OPENCL EXTENSION cl_khr_fp64 : enable
  24
  25 _CLC_INLINE double2
  26 __libclc__sincos_piby4(double x, double xx)
  27 {
  28     // Taylor series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! ...
  29     //                      = x * (1 - x^2/3! + x^4/5! - x^6/7! ...
  30     //                      = x * f(w)
  31     // where w = x*x and f(w) = (1 - w/3! + w^2/5! - w^3/7! ...
  32     // We use a minimax approximation of (f(w) - 1) / w
  33     // because this produces an expansion in even powers of x.
  34     // If xx (the tail of x) is non-zero, we add a correction
  35     // term g(x,xx) = (1-x*x/2)*xx to the result, where g(x,xx)
  36     // is an approximation to cos(x)*sin(xx) valid because
  37     // xx is tiny relative to x.
  38
  39     // Taylor series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! ...
  40     //                      = f(w)
  41     // where w = x*x and f(w) = (1 - w/2! + w^2/4! - w^3/6! ...
  42     // We use a minimax approximation of (f(w) - 1 + w/2) / (w*w)
  43     // because this produces an expansion in even powers of x.
  44     // If xx (the tail of x) is non-zero, we subtract a correction
  45     // term g(x,xx) = x*xx to the result, where g(x,xx)
  46     // is an approximation to sin(x)*sin(xx) valid because
  47     // xx is tiny relative to x.
  48
  49     const double sc1 = -0.166666666666666646259241729;
  50     const double sc2 =  0.833333333333095043065222816e-2;
  51     const double sc3 = -0.19841269836761125688538679e-3;
  52     const double sc4 =  0.275573161037288022676895908448e-5;
  53     const double sc5 = -0.25051132068021699772257377197e-7;
  54     const double sc6 =  0.159181443044859136852668200e-9;
  55
  56     const double cc1 =  0.41666666666666665390037e-1;
  57     const double cc2 = -0.13888888888887398280412e-2;
  58     const double cc3 =  0.248015872987670414957399e-4;
  59     const double cc4 = -0.275573172723441909470836e-6;
  60     const double cc5 =  0.208761463822329611076335e-8;
  61     const double cc6 = -0.113826398067944859590880e-10;
  62
  63     double x2 = x * x;
  64     double x3 = x2 * x;
  65     double r = 0.5 * x2;
  66     double t = 1.0 - r;
  67
  68     double sp = fma(fma(fma(fma(sc6, x2, sc5), x2, sc4), x2, sc3), x2, sc2);
  69
  70     double cp = t + fma(fma(fma(fma(fma(fma(cc6, x2, cc5), x2, cc4), x2, cc3), x2, cc2), x2, cc1),
  71                         x2*x2, fma(x, xx, (1.0 - t) - r));
  72
  73     double2 ret;
  74     ret.lo = x - fma(-x3, sc1, fma(fma(-x3, sp, 0.5*xx), x2, -xx));
  75     ret.hi = cp;
  76
  77     return ret;
  78 }
  79
  80 _CLC_INLINE double2
  81 __clc_tan_piby4(double x, double xx)
  82 {
  83     const double piby4_lead = 7.85398163397448278999e-01; // 0x3fe921fb54442d18
  84     const double piby4_tail = 3.06161699786838240164e-17; // 0x3c81a62633145c06
  85
  86     // In order to maintain relative precision transform using the identity:
  87     // tan(pi/4-x) = (1-tan(x))/(1+tan(x)) for arguments close to pi/4.
  88     // Similarly use tan(x-pi/4) = (tan(x)-1)/(tan(x)+1) close to -pi/4.
  89
  90     int ca = x >  0.68;
  91     int cb = x < -0.68;
  92     double transform = ca ?  1.0 : 0.0;
  93     transform = cb ? -1.0 : transform;
  94
  95     double tx = fma(-transform, x, piby4_lead) + fma(-transform, xx, piby4_tail);
  96     int c = ca | cb;
  97     x = c ? tx : x;
  98     xx = c ? 0.0 : xx;
  99
 100     // Core Remez [2,3] approximation to tan(x+xx) on the interval [0,0.68].
 101     double t1 = x;
 102     double r = fma(2.0, x*xx, x*x);
 103
 104     double a = fma(r,
 105                    fma(r, 0.224044448537022097264602535574e-3, -0.229345080057565662883358588111e-1),
 106                    0.372379159759792203640806338901e0);
 107
 108     double b = fma(r,
 109                    fma(r,
 110                        fma(r, -0.232371494088563558304549252913e-3, 0.260656620398645407524064091208e-1),
 111                        -0.515658515729031149329237816945e0),
 112                    0.111713747927937668539901657944e1);
 113
 114     double t2 = fma(MATH_DIVIDE(a, b), x*r, xx);
 115
 116     double tp = t1 + t2;
 117
 118     // Compute -1.0/(t1 + t2) accurately
 119     double z1 = as_double(as_long(tp) & 0xffffffff00000000L);
 120     double z2 = t2 - (z1 - t1);
 121     double trec = -MATH_RECIP(tp);
 122     double trec_top = as_double(as_long(trec) & 0xffffffff00000000L);
 123
 124     double tpr = fma(fma(trec_top, z2, fma(trec_top, z1, 1.0)), trec, trec_top);
 125
 126     double tpt = transform * (1.0 - MATH_DIVIDE(2.0*tp, 1.0 + tp));
 127     double tptr = transform * (MATH_DIVIDE(2.0*tp, tp - 1.0) - 1.0);
 128
 129     double2 ret;
 130     ret.lo = c ? tpt : tp;
 131     ret.hi = c ? tptr : tpr;
 132     return ret;
 133 }