1 /* @(#)k_tan.c 5.1 93/09/24 */ 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 #include <sys/cdefs.h> 14 #if defined(LIBM_SCCS) && !defined(lint) 15 __RCSID("$NetBSD: k_tan.c,v 1.11 2002/05/26 22:01:54 wiz Exp $"); 16 #endif 17 18 /* __kernel_tan( x, y, k ) 19 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 20 * Input x is assumed to be bounded by ~pi/4 in magnitude. 21 * Input y is the tail of x. 22 * Input k indicates whether tan (if k=1) or 23 * -1/tan (if k= -1) is returned. 24 * 25 * Algorithm 26 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 27 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. 28 * 3. tan(x) is approximated by a odd polynomial of degree 27 on 29 * [0,0.67434] 30 * 3 27 31 * tan(x) ~ x + T1*x + ... + T13*x 32 * where 33 * 34 * |tan(x) 2 4 26 | -59.2 35 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 36 * | x | 37 * 38 * Note: tan(x+y) = tan(x) + tan'(x)*y 39 * ~ tan(x) + (1+x*x)*y 40 * Therefore, for better accuracy in computing tan(x+y), let 41 * 3 2 2 2 2 42 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 43 * then 44 * 3 2 45 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 46 * 47 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 48 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 49 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 50 */ 51 52 #include "math.h" 53 #include "math_private.h" 54 static const double 55 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 56 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 57 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ 58 T[] = { 59 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ 60 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ 61 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ 62 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ 63 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ 64 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ 65 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ 66 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ 67 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ 68 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ 69 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ 70 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ 71 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ 72 }; 73 74 double 75 __kernel_tan(double x, double y, int iy) 76 { 77 double z,r,v,w,s; 78 int32_t ix,hx; 79 GET_HIGH_WORD(hx,x); 80 ix = hx&0x7fffffff; /* high word of |x| */ 81 if(ix<0x3e300000) /* x < 2**-28 */ 82 {if((int)x==0) { /* generate inexact */ 83 u_int32_t low; 84 GET_LOW_WORD(low,x); 85 if(((ix|low)|(iy+1))==0) return one/fabs(x); 86 else return (iy==1)? x: -one/x; 87 } 88 } 89 if(ix>=0x3FE59428) { /* |x|>=0.6744 */ 90 if(hx<0) {x = -x; y = -y;} 91 z = pio4-x; 92 w = pio4lo-y; 93 x = z+w; y = 0.0; 94 } 95 z = x*x; 96 w = z*z; 97 /* Break x^5*(T[1]+x^2*T[2]+...) into 98 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 99 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 100 */ 101 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); 102 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); 103 s = z*x; 104 r = y + z*(s*(r+v)+y); 105 r += T[0]*s; 106 w = x+r; 107 if(ix>=0x3FE59428) { 108 v = (double)iy; 109 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); 110 } 111 if(iy==1) return w; 112 else { /* if allow error up to 2 ulp, 113 simply return -1.0/(x+r) here */ 114 /* compute -1.0/(x+r) accurately */ 115 double a,t; 116 z = w; 117 SET_LOW_WORD(z,0); 118 v = r-(z - x); /* z+v = r+x */ 119 t = a = -1.0/w; /* a = -1.0/w */ 120 SET_LOW_WORD(t,0); 121 s = 1.0+t*z; 122 return t+a*(s+t*v); 123 } 124 } 125