k_tan.cpp (4012B)
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 //#include <sys/cdefs.h> 15 //__FBSDID("$FreeBSD$"); 16 17 /* __kernel_tan( x, y, k ) 18 * kernel tan function on ~[-pi/4, pi/4] (except on -0), 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 -1/tan (if k = -1) is returned. 22 * 23 * Algorithm 24 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 25 * 2. Callers must return tan(-0) = -0 without calling here since our 26 * odd polynomial is not evaluated in a way that preserves -0. 27 * Callers may do the optimization tan(x) ~ x for tiny x. 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_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 >= 0x3FE59428) { /* |x| >= 0.6744 */ 85 if (hx < 0) { 86 x = -x; 87 y = -y; 88 } 89 z = pio4 - x; 90 w = pio4lo - y; 91 x = z + w; 92 y = 0.0; 93 } 94 z = x * x; 95 w = z * z; 96 /* 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] + 102 w * T[11])))); 103 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 104 w * T[12]))))); 105 s = z * x; 106 r = y + z * (s * (r + v) + y); 107 r += T[0] * s; 108 w = x + r; 109 if (ix >= 0x3FE59428) { 110 v = (double) iy; 111 return (double) (1 - ((hx >> 30) & 2)) * 112 (v - 2.0 * (x - (w * w / (w + v) - r))); 113 } 114 if (iy == 1) 115 return w; 116 else { 117 /* 118 * if allow error up to 2 ulp, simply return 119 * -1.0 / (x+r) here 120 */ 121 /* compute -1.0 / (x+r) accurately */ 122 double a, t; 123 z = w; 124 SET_LOW_WORD(z,0); 125 v = r - (z - x); /* z+v = r+x */ 126 t = a = -1.0 / w; /* a = -1.0/w */ 127 SET_LOW_WORD(t,0); 128 s = 1.0 + t * z; 129 return t + a * (s + t * v); 130 } 131 }