1 /* @(#)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 #ifndef lint 15 static char rcsid[] = "$FreeBSD$"; 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 -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 xxx[] = { 54 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 55 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 56 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 57 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 58 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 59 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 60 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 61 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 62 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 63 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 64 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 65 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 66 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 67 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 68 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 69 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 70 }; 71 #define one xxx[13] 72 #define pio4 xxx[14] 73 #define pio4lo xxx[15] 74 #define T xxx 75 /* INDENT ON */ 76 77 double 78 __kernel_tan(double x, double y, int iy) { 79 double z, r, v, w, s; 80 int32_t ix, hx; 81 82 GET_HIGH_WORD(hx,x); 83 ix = hx & 0x7fffffff; /* high word of |x| */ 84 if (ix < 0x3e300000) { /* x < 2**-28 */ 85 if ((int) x == 0) { /* generate inexact */ 86 u_int32_t low; 87 GET_LOW_WORD(low,x); 88 if (((ix | low) | (iy + 1)) == 0) 89 return one / fabs(x); 90 else { 91 if (iy == 1) 92 return x; 93 else { /* compute -1 / (x+y) carefully */ 94 double a, t; 95 96 z = w = x + y; 97 SET_LOW_WORD(z, 0); 98 v = y - (z - x); 99 t = a = -one / w; 100 SET_LOW_WORD(t, 0); 101 s = one + t * z; 102 return t + a * (s + t * v); 103 } 104 } 105 } 106 } 107 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 108 if (hx < 0) { 109 x = -x; 110 y = -y; 111 } 112 z = pio4 - x; 113 w = pio4lo - y; 114 x = z + w; 115 y = 0.0; 116 } 117 z = x * x; 118 w = z * z; 119 /* 120 * Break x^5*(T[1]+x^2*T[2]+...) into 121 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 122 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 123 */ 124 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 125 w * T[11])))); 126 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 127 w * T[12]))))); 128 s = z * x; 129 r = y + z * (s * (r + v) + y); 130 r += T[0] * s; 131 w = x + r; 132 if (ix >= 0x3FE59428) { 133 v = (double) iy; 134 return (double) (1 - ((hx >> 30) & 2)) * 135 (v - 2.0 * (x - (w * w / (w + v) - r))); 136 } 137 if (iy == 1) 138 return w; 139 else { 140 /* 141 * if allow error up to 2 ulp, simply return 142 * -1.0 / (x+r) here 143 */ 144 /* compute -1.0 / (x+r) accurately */ 145 double a, t; 146 z = w; 147 SET_LOW_WORD(z,0); 148 v = r - (z - x); /* z+v = r+x */ 149 t = a = -1.0 / w; /* a = -1.0/w */ 150 SET_LOW_WORD(t,0); 151 s = 1.0 + t * z; 152 return t + a * (s + t * v); 153 } 154 } 155