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
__libclc__sincos_piby4(double x,double xx)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
__clc_tan_piby4(double x,double xx)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 }
134