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