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 #if defined(LIBM_SCCS) && !defined(lint) 14 static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $"; 15 #endif 16 17 /* __kernel_tan( x, y, k ) 18 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 19 * Input x is assumed to be bounded by ~pi/4 in magnitude. 20 * Input y is the tail of x. 21 * Input k indicates whether tan (if k=1) or 22 * -1/tan (if k= -1) is returned. 23 * 24 * Algorithm 25 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 26 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. 27 * 3. tan(x) is approximated by a odd polynomial of degree 27 on 28 * [0,0.67434] 29 * 3 27 30 * tan(x) ~ x + T1*x + ... + T13*x 31 * where 32 * 33 * |tan(x) 2 4 26 | -59.2 34 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 35 * | x | 36 * 37 * Note: tan(x+y) = tan(x) + tan'(x)*y 38 * ~ tan(x) + (1+x*x)*y 39 * Therefore, for better accuracy in computing tan(x+y), let 40 * 3 2 2 2 2 41 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 42 * then 43 * 3 2 44 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 45 * 46 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 47 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 48 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 49 */ 50 51 #include "math.h" 52 #include "math_private.h" 53 static const double 54 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 55 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 56 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ 57 T[] = { 58 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ 59 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ 60 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ 61 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ 62 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ 63 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ 64 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ 65 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ 66 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ 67 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ 68 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ 69 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ 70 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ 71 }; 72 73 double 74 __kernel_tan(double x, double y, int iy) 75 { 76 double z,r,v,w,s; 77 int32_t ix,hx; 78 GET_HIGH_WORD(hx,x); 79 ix = hx&0x7fffffff; /* high word of |x| */ 80 if(ix<0x3e300000) /* x < 2**-28 */ 81 {if((int)x==0) { /* generate inexact */ 82 u_int32_t low; 83 GET_LOW_WORD(low,x); 84 if(((ix|low)|(iy+1))==0) return one/fabs(x); 85 else return (iy==1)? x: -one/x; 86 } 87 } 88 if(ix>=0x3FE59428) { /* |x|>=0.6744 */ 89 if(hx<0) {x = -x; y = -y;} 90 z = pio4-x; 91 w = pio4lo-y; 92 x = z+w; y = 0.0; 93 } 94 z = x*x; 95 w = z*z; 96 /* Break x^5*(T[1]+x^2*T[2]+...) into 97 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 98 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 99 */ 100 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); 101 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); 102 s = z*x; 103 r = y + z*(s*(r+v)+y); 104 r += T[0]*s; 105 w = x+r; 106 if(ix>=0x3FE59428) { 107 v = (double)iy; 108 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); 109 } 110 if(iy==1) return w; 111 else { /* if allow error up to 2 ulp, 112 simply return -1.0/(x+r) here */ 113 /* compute -1.0/(x+r) accurately */ 114 double a,t; 115 z = w; 116 SET_LOW_WORD(z,0); 117 v = r-(z - x); /* z+v = r+x */ 118 t = a = -1.0/w; /* a = -1.0/w */ 119 SET_LOW_WORD(t,0); 120 s = 1.0+t*z; 121 return t+a*(s+t*v); 122 } 123 } 124