1 /* @(#)k_tan.c 5.1 93/09/24 */ 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 12 #ifndef lint 13 static char rcsid[] = "$FreeBSD$"; 14 #endif 15 16 /* __kernel_tan( x, y, k ) 17 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 18 * Input x is assumed to be bounded by ~pi/4 in magnitude. 19 * Input y is the tail of x. 20 * Input k indicates whether tan (if k=1) or 21 * -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. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. 26 * 3. tan(x) is approximated by an odd polynomial of degree 27 on 27 * [0,0.67434] 28 * 3 27 29 * tan(x) ~ x + T1*x + ... + T13*x 30 * where 31 * 32 * |tan(x) 2 4 26 | -59.2 33 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 34 * | x | 35 * 36 * Note: tan(x+y) = tan(x) + tan'(x)*y 37 * ~ tan(x) + (1+x*x)*y 38 * Therefore, for better accuracy in computing tan(x+y), let 39 * 3 2 2 2 2 40 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) 41 * then 42 * 3 2 43 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) 44 * 45 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then 46 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) 47 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) 48 */ 49 50 #include "math.h" 51 #include "math_private.h" 52 static const double 53 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ 54 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ 55 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ 56 T[] = { 57 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ 58 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ 59 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ 60 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ 61 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ 62 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ 63 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ 64 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ 65 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ 66 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ 67 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ 68 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ 69 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ 70 }; 71 72 double 73 __kernel_tan(double x, double y, int iy) 74 { 75 double z,r,v,w,s; 76 int32_t ix,hx; 77 GET_HIGH_WORD(hx,x); 78 ix = hx&0x7fffffff; /* high word of |x| */ 79 if(ix<0x3e300000) { /* x < 2**-28 */ 80 if ((int) x == 0) { /* generate inexact */ 81 u_int32_t low; 82 GET_LOW_WORD(low,x); 83 if (((ix | low) | (iy + 1)) == 0) 84 return one / fabs(x); 85 else { 86 if (iy == 1) 87 return x; 88 else { /* compute -1 / (x+y) carefully */ 89 double a, t; 90 91 z = w = x + y; 92 SET_LOW_WORD(z, 0); 93 v = y - (z - x); 94 t = a = -one / w; 95 SET_LOW_WORD(t, 0); 96 s = one + t * z; 97 return t + a * (s + t * v); 98 } 99 } 100 } 101 } 102 if(ix>=0x3FE59428) { /* |x|>=0.6744 */ 103 if(hx<0) {x = -x; y = -y;} 104 z = pio4-x; 105 w = pio4lo-y; 106 x = z+w; y = 0.0; 107 } 108 z = x*x; 109 w = z*z; 110 /* Break x^5*(T[1]+x^2*T[2]+...) into 111 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + 112 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) 113 */ 114 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); 115 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); 116 s = z*x; 117 r = y + z*(s*(r+v)+y); 118 r += T[0]*s; 119 w = x+r; 120 if(ix>=0x3FE59428) { 121 v = (double)iy; 122 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); 123 } 124 if(iy==1) return w; 125 else { /* if allow error up to 2 ulp, 126 simply return -1.0/(x+r) here */ 127 /* compute -1.0/(x+r) accurately */ 128 double a,t; 129 z = w; 130 SET_LOW_WORD(z,0); 131 v = r-(z - x); /* z+v = r+x */ 132 t = a = -1.0/w; /* a = -1.0/w */ 133 SET_LOW_WORD(t,0); 134 s = 1.0+t*z; 135 return t+a*(s+t*v); 136 } 137 } 138