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[glibc.git] / sysdeps / libm-ieee754 / k_tan.c
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1 /* @(#)k_tan.c 5.1 93/09/24 */
2 /*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 * ====================================================
12 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
13 for performance improvement on pipelined processors.
16 #if defined(LIBM_SCCS) && !defined(lint)
17 static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $";
18 #endif
20 /* __kernel_tan( x, y, k )
21 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
22 * Input x is assumed to be bounded by ~pi/4 in magnitude.
23 * Input y is the tail of x.
24 * Input k indicates whether tan (if k=1) or
25 * -1/tan (if k= -1) is returned.
27 * Algorithm
28 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
29 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
30 * 3. tan(x) is approximated by a odd polynomial of degree 27 on
31 * [0,0.67434]
32 * 3 27
33 * tan(x) ~ x + T1*x + ... + T13*x
34 * where
36 * |tan(x) 2 4 26 | -59.2
37 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
38 * | x |
40 * Note: tan(x+y) = tan(x) + tan'(x)*y
41 * ~ tan(x) + (1+x*x)*y
42 * Therefore, for better accuracy in computing tan(x+y), let
43 * 3 2 2 2 2
44 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
45 * then
46 * 3 2
47 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
49 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
50 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
51 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
54 #include "math.h"
55 #include "math_private.h"
56 #ifdef __STDC__
57 static const double
58 #else
59 static double
60 #endif
61 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
62 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
63 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
64 T[] = {
65 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
66 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
67 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
68 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
69 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
70 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
71 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
72 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
73 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
74 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
75 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
76 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
77 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
80 #ifdef __STDC__
81 double __kernel_tan(double x, double y, int iy)
82 #else
83 double __kernel_tan(x, y, iy)
84 double x,y; int iy;
85 #endif
87 double z,r,v,w,s,r1,r2,r3,v1,v2,v3,w2,w4;
88 int32_t ix,hx;
89 GET_HIGH_WORD(hx,x);
90 ix = hx&0x7fffffff; /* high word of |x| */
91 if(ix<0x3e300000) /* x < 2**-28 */
92 {if((int)x==0) { /* generate inexact */
93 u_int32_t low;
94 GET_LOW_WORD(low,x);
95 if(((ix|low)|(iy+1))==0) return one/fabs(x);
96 else return (iy==1)? x: -one/x;
99 if(ix>=0x3FE59428) { /* |x|>=0.6744 */
100 if(hx<0) {x = -x; y = -y;}
101 z = pio4-x;
102 w = pio4lo-y;
103 x = z+w; y = 0.0;
105 z = x*x;
106 w = z*z;
107 /* Break x^5*(T[1]+x^2*T[2]+...) into
108 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
109 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
111 #ifdef DO_NOT_USE_THIS
112 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
113 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
114 #else
115 v1 = T[10]+w*T[12]; w2=w*w;
116 v2 = T[6]+w*T[8]; w4=w2*w2;
117 v3 = T[2]+w*T[4]; v1=z*v1;
118 r1 = T[9]+w*T[11]; v2=z*v2;
119 r2 = T[5]+w*T[7]; v3=z*v3;
120 r3 = T[1]+w*T[3];
121 v = v3 + w2*v2 + w4*v1;
122 r = r3 + w2*r2 + w4*r1;
123 #endif
124 s = z*x;
125 r = y + z*(s*(r+v)+y);
126 r += T[0]*s;
127 w = x+r;
128 if(ix>=0x3FE59428) {
129 v = (double)iy;
130 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
132 if(iy==1) return w;
133 else { /* if allow error up to 2 ulp,
134 simply return -1.0/(x+r) here */
135 /* compute -1.0/(x+r) accurately */
136 double a,t;
137 z = w;
138 SET_LOW_WORD(z,0);
139 v = r-(z - x); /* z+v = r+x */
140 t = a = -1.0/w; /* a = -1.0/w */
141 SET_LOW_WORD(t,0);
142 s = 1.0+t*z;
143 return t+a*(s+t*v);