1 /* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */ 2 /* 3 * ==================================================== 4 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. 5 * 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 /* __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 odd indicates whether tan (if odd = 0) or -1/tan (if odd = 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 "libm.h" 47 48 static const double T[] = { 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 }, 63 pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ 64 pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */ 65 66 double __tan(double x, double y, int odd) 67 { 68 double_t z, r, v, w, s, a; 69 double w0, a0; 70 uint32_t hx; 71 int big, sign; 72 73 GET_HIGH_WORD(hx,x); 74 big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */ 75 if (big) { 76 sign = hx>>31; 77 if (sign) { 78 x = -x; 79 y = -y; 80 } 81 x = (pio4 - x) + (pio4lo - y); 82 y = 0.0; 83 } 84 z = x * x; 85 w = z * z; 86 /* 87 * Break x^5*(T[1]+x^2*T[2]+...) into 88 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 89 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 90 */ 91 r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11])))); 92 v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12]))))); 93 s = z * x; 94 r = y + z*(s*(r + v) + y) + s*T[0]; 95 w = x + r; 96 if (big) { 97 s = 1 - 2*odd; 98 v = s - 2.0 * (x + (r - w*w/(w + s))); 99 return sign ? -v : v; 100 } 101 if (!odd) 102 return w; 103 /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */ 104 w0 = w; 105 SET_LOW_WORD(w0, 0); 106 v = r - (w0 - x); /* w0+v = r+x */ 107 a0 = a = -1.0 / w; 108 SET_LOW_WORD(a0, 0); 109 return a0 + a*(1.0 + a0*w0 + a0*v); 110 } 111