1 /* 2 * ==================================================== 3 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 4 * 5 * Permission to use, copy, modify, and distribute this 6 * software is freely granted, provided that this notice 7 * is preserved. 8 * ==================================================== 9 */ 10 11 /* __kernel_tan( x, y, k ) 12 * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 13 * Input x is assumed to be bounded by ~pi/4 in magnitude. 14 * Input y is the tail of x. 15 * Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. 16 * 17 * Algorithm 18 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. 19 * 2. Callers must return tan(-0) = -0 without calling here since our 20 * odd polynomial is not evaluated in a way that preserves -0. 21 * Callers may do the optimization tan(x) ~ x for tiny x. 22 * 3. tan(x) is approximated by a odd polynomial of degree 27 on 23 * [0,0.67434] 24 * 3 27 25 * tan(x) ~ x + T1*x + ... + T13*x 26 * where 27 * 28 * |tan(x) 2 4 26 | -59.2 29 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 30 * | x | 31 * 32 * Note: tan(x+y) = tan(x) + tan'(x)*y 33 * ~ tan(x) + (1+x*x)*y 34 * Therefore, for better accuracy in computing tan(x+y), let 35 * 3 2 2 2 2 36 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 37 * then 38 * 3 2 39 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 40 * 41 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 42 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 43 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 44 */ 45 46 #include "math.h" 47 #include "math_private.h" 48 static const double xxx[] = { 49 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ 50 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ 51 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ 52 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ 53 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ 54 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ 55 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ 56 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ 57 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ 58 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ 59 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ 60 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ 61 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ 62 /* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ 63 /* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 64 /* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ 65 }; 66 #define one xxx[13] 67 #define pio4 xxx[14] 68 #define pio4lo xxx[15] 69 #define T xxx 70 /* INDENT ON */ 71 72 double 73 __kernel_tan(double x, double y, int iy) { 74 double z, r, v, w, s; 75 int32_t ix, hx; 76 77 GET_HIGH_WORD(hx,x); 78 ix = hx & 0x7fffffff; /* high word of |x| */ 79 if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ 80 if (hx < 0) { 81 x = -x; 82 y = -y; 83 } 84 z = pio4 - x; 85 w = pio4lo - y; 86 x = z + w; 87 y = 0.0; 88 } 89 z = x * x; 90 w = z * z; 91 /* 92 * Break x^5*(T[1]+x^2*T[2]+...) into 93 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 94 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 95 */ 96 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + 97 w * T[11])))); 98 v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + 99 w * T[12]))))); 100 s = z * x; 101 r = y + z * (s * (r + v) + y); 102 r += T[0] * s; 103 w = x + r; 104 if (ix >= 0x3FE59428) { 105 v = (double) iy; 106 return (double) (1 - ((hx >> 30) & 2)) * 107 (v - 2.0 * (x - (w * w / (w + v) - r))); 108 } 109 if (iy == 1) 110 return w; 111 else { 112 /* 113 * if allow error up to 2 ulp, simply return 114 * -1.0 / (x+r) here 115 */ 116 /* compute -1.0 / (x+r) accurately */ 117 double a, t; 118 z = w; 119 SET_LOW_WORD(z,0); 120 v = r - (z - x); /* z+v = r+x */ 121 t = a = -1.0 / w; /* a = -1.0/w */ 122 SET_LOW_WORD(t,0); 123 s = 1.0 + t * z; 124 return t + a * (s + t * v); 125 } 126 } 127