1 #pragma ident "@(#)k_tan.c 1.5 04/04/22 SMI" 2 3 /* 4 * ==================================================== 5 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 6 * 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 /* INDENT OFF */ 14 /* __kernel_tan( x, y, k ) 15 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 16 * Input x is assumed to be bounded by ~pi/4 in magnitude. 17 * Input y is the tail of x. 18 * Input k indicates whether ieee_tan (if k = 1) or -1/tan (if k = -1) is returned. 19 * 20 * Algorithm 21 * 1. Since ieee_tan(-x) = -ieee_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. ieee_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 * |ieee_tan(x) 2 4 26 | -59.2 30 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 31 * | x | 32 * 33 * Note: ieee_tan(x+y) = ieee_tan(x) + tan'(x)*y 34 * ~ ieee_tan(x) + (1+x*x)*y 35 * Therefore, for better accuracy in computing ieee_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) = ieee_tan(pi/4-y) = (1-ieee_tan(y))/(1+ieee_tan(y)) 44 * = 1 - 2*(ieee_tan(y) - (ieee_tan(y)^2)/(1+ieee_tan(y))) 45 */ 46 47 #include "fdlibm.h" 48 49 static const double xxx[] = { 50 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 51 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 52 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 53 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 54 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 55 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 56 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 57 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 58 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 59 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 60 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 61 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 62 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 63 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 64 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 65 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 66 }; 67 #define one xxx[13] 68 #define pio4 xxx[14] 69 #define pio4lo xxx[15] 70 #define T xxx 71 /* INDENT ON */ 72 73 double 74 __kernel_tan(double x, double y, int iy) { 75 double z, r, v, w, s; 76 int ix, hx; 77 78 hx = __HI(x); /* high word of x */ 79 ix = hx & 0x7fffffff; /* high word of |x| */ 80 if (ix < 0x3e300000) { /* x < 2**-28 */ 81 if ((int) x == 0) { /* generate inexact */ 82 if (((ix | __LO(x)) | (iy + 1)) == 0) 83 return one / ieee_fabs(x); 84 else { 85 if (iy == 1) 86 return x; 87 else { /* compute -1 / (x+y) carefully */ 88 double a, t; 89 90 z = w = x + y; 91 __LO(z) = 0; 92 v = y - (z - x); 93 t = a = -one / w; 94 __LO(t) = 0; 95 s = one + t * z; 96 return t + a * (s + t * v); 97 } 98 } 99 } 100 } 101 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 102 if (hx < 0) { 103 x = -x; 104 y = -y; 105 } 106 z = pio4 - x; 107 w = pio4lo - y; 108 x = z + w; 109 y = 0.0; 110 } 111 z = x * x; 112 w = z * z; 113 /* 114 * Break x^5*(T[1]+x^2*T[2]+...) into 115 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 116 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 117 */ 118 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 119 w * T[11])))); 120 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 121 w * T[12]))))); 122 s = z * x; 123 r = y + z * (s * (r + v) + y); 124 r += T[0] * s; 125 w = x + r; 126 if (ix >= 0x3FE59428) { 127 v = (double) iy; 128 return (double) (1 - ((hx >> 30) & 2)) * 129 (v - 2.0 * (x - (w * w / (w + v) - r))); 130 } 131 if (iy == 1) 132 return w; 133 else { 134 /* 135 * if allow error up to 2 ulp, simply return 136 * -1.0 / (x+r) here 137 */ 138 /* compute -1.0 / (x+r) accurately */ 139 double a, t; 140 z = w; 141 __LO(z) = 0; 142 v = r - (z - x); /* z+v = r+x */ 143 t = a = -1.0 / w; /* a = -1.0/w */ 144 __LO(t) = 0; 145 s = 1.0 + t * z; 146 return t + a * (s + t * v); 147 } 148 } 149
