1#include "SDL_internal.h"
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 *
6 * Developed at SunPro, a Sun Microsystems, Inc. business.
7 * Permission to use, copy, modify, and distribute this
8 * software is freely granted, provided that this notice
9 * is preserved.
10 * ====================================================
11 */
12
13/* __kernel_tan( x, y, k )
14 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
15 * Input x is assumed to be bounded by ~pi/4 in magnitude.
16 * Input y is the tail of x.
17 * Input k indicates whether tan (if k=1) or
18 * -1/tan (if k= -1) is returned.
19 *
20 * Algorithm
21 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
22 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
23 * 3. tan(x) is approximated by a odd polynomial of degree 27 on
24 * [0,0.67434]
25 * 3 27
26 * tan(x) ~ x + T1*x + ... + T13*x
27 * where
28 *
29 * |tan(x) 2 4 26 | -59.2
30 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
31 * | x |
32 *
33 * Note: tan(x+y) = tan(x) + tan'(x)*y
34 * ~ tan(x) + (1+x*x)*y
35 * Therefore, for better accuracy in computing tan(x+y), let
36 * 3 2 2 2 2
37 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
38 * then
39 * 3 2
40 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
41 *
42 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
43 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
44 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
45 */
46
47#include "math_libm.h"
48#include "math_private.h"
49
50static const double
51one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
52pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
53pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
54T[] = {
55 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
56 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
57 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
58 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
59 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
60 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
61 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
62 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
63 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
64 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
65 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
66 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
67 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
68};
69
70double attribute_hidden __kernel_tan(double x, double y, int iy)
71{
72 double z,r,v,w,s;
73 int32_t ix,hx;
74 GET_HIGH_WORD(hx,x);
75 ix = hx&0x7fffffff; /* high word of |x| */
76 if(ix<0x3e300000) /* x < 2**-28 */
77 {if((int)x==0) { /* generate inexact */
78 u_int32_t low;
79 GET_LOW_WORD(low,x);
80 if(((ix|low)|(iy+1))==0) return one/fabs(x);
81 else return (iy==1)? x: -one/x;
82 }
83 }
84 if(ix>=0x3FE59428) { /* |x|>=0.6744 */
85 if(hx<0) {x = -x; y = -y;}
86 z = pio4-x;
87 w = pio4lo-y;
88 x = z+w; y = 0.0;
89 }
90 z = x*x;
91 w = z*z;
92 /* Break x^5*(T[1]+x^2*T[2]+...) into
93 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
94 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
95 */
96 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
97 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
98 s = z*x;
99 r = y + z*(s*(r+v)+y);
100 r += T[0]*s;
101 w = x+r;
102 if(ix>=0x3FE59428) {
103 v = (double)iy;
104 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
105 }
106 if(iy==1) return w;
107 else { /* if allow error up to 2 ulp,
108 simply return -1.0/(x+r) here */
109 /* compute -1.0/(x+r) accurately */
110 double a,t;
111 z = w;
112 SET_LOW_WORD(z,0);
113 v = r-(z - x); /* z+v = r+x */
114 t = a = -1.0/w; /* a = -1.0/w */
115 SET_LOW_WORD(t,0);
116 s = 1.0+t*z;
117 return t+a*(s+t*v);
118 }
119}
120