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 "cdefs-compat.h" 15 //__FBSDID("$FreeBSD: src/lib/msun/src/k_tan.c,v 1.13 2008/02/22 02:30:35 das Exp $"); 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 <openlibm_math.h> 53 54 #include "math_private.h" 55 56 static const double xxx[] = { 57 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 58 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 59 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 60 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 61 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 62 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 63 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 64 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 65 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 66 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 67 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 68 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 69 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 70 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 71 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 72 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 73 }; 74 #define one xxx[13] 75 #define pio4 xxx[14] 76 #define pio4lo xxx[15] 77 #define T xxx 78 /* INDENT ON */ 79 80 double 81 __kernel_tan(double x, double y, int iy) { 82 double z, r, v, w, s; 83 int32_t ix, hx; 84 85 GET_HIGH_WORD(hx,x); 86 ix = hx & 0x7fffffff; /* high word of |x| */ 87 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 88 if (hx < 0) { 89 x = -x; 90 y = -y; 91 } 92 z = pio4 - x; 93 w = pio4lo - y; 94 x = z + w; 95 y = 0.0; 96 } 97 z = x * x; 98 w = z * z; 99 /* 100 * Break x^5*(T[1]+x^2*T[2]+...) into 101 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 102 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 103 */ 104 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 105 w * T[11])))); 106 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 107 w * T[12]))))); 108 s = z * x; 109 r = y + z * (s * (r + v) + y); 110 r += T[0] * s; 111 w = x + r; 112 if (ix >= 0x3FE59428) { 113 v = (double) iy; 114 return (double) (1 - ((hx >> 30) & 2)) * 115 (v - 2.0 * (x - (w * w / (w + v) - r))); 116 } 117 if (iy == 1) 118 return w; 119 else { 120 /* 121 * if allow error up to 2 ulp, simply return 122 * -1.0 / (x+r) here 123 */ 124 /* compute -1.0 / (x+r) accurately */ 125 double a, t; 126 z = w; 127 SET_LOW_WORD(z,0); 128 v = r - (z - x); /* z+v = r+x */ 129 t = a = -1.0 / w; /* a = -1.0/w */ 130 SET_LOW_WORD(t,0); 131 s = 1.0 + t * z; 132 return t + a * (s + t * v); 133 } 134 } 135