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 /* __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.h" 48 #include "math_private.h" 49 50 static const double xxx[] = { 51 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 52 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 53 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 54 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 55 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 56 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 57 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 58 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 59 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 60 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 61 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 62 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 63 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 64 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 65 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 66 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 67 }; 68 #define one xxx[13] 69 #define pio4 xxx[14] 70 #define pio4lo xxx[15] 71 #define T xxx 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 79 GET_HIGH_WORD(hx, x); /* high word of 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) 86 return one / fabs(x); 87 else { 88 if (iy == 1) 89 return x; 90 else { /* compute -1 / (x+y) carefully */ 91 double a, t; 92 93 z = w = x + y; 94 SET_LOW_WORD(z, 0); 95 v = y - (z - x); 96 t = a = -one / w; 97 SET_LOW_WORD(t, 0); 98 s = one + t * z; 99 return t + a * (s + t * v); 100 } 101 } 102 } 103 } 104 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 105 if (hx < 0) { 106 x = -x; 107 y = -y; 108 } 109 z = pio4 - x; 110 w = pio4lo - y; 111 x = z + w; 112 y = 0.0; 113 } 114 z = x * x; 115 w = z * z; 116 /* 117 * Break x^5*(T[1]+x^2*T[2]+...) into 118 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 119 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 120 */ 121 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 122 w * T[11])))); 123 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 124 w * T[12]))))); 125 s = z * x; 126 r = y + z * (s * (r + v) + y); 127 r += T[0] * s; 128 w = x + r; 129 if (ix >= 0x3FE59428) { 130 v = (double) iy; 131 return (double) (1 - ((hx >> 30) & 2)) * 132 (v - 2.0 * (x - (w * w / (w + v) - r))); 133 } 134 if (iy == 1) 135 return w; 136 else { 137 /* 138 * if allow error up to 2 ulp, simply return 139 * -1.0 / (x+r) here 140 */ 141 /* compute -1.0 / (x+r) accurately */ 142 double a, t; 143 z = w; 144 SET_LOW_WORD(z, 0); 145 v = r - (z - x); /* z+v = r+x */ 146 t = a = -1.0 / w; /* a = -1.0/w */ 147 SET_LOW_WORD(t, 0); 148 s = 1.0 + t * z; 149 return t + a * (s + t * v); 150 } 151 } 152