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path: root/libm/k_tan.c
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 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 ``` ``````/* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* __kernel_tan( x, y, k ) * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. * Input k indicates whether tan (if k=1) or * -1/tan (if k= -1) is returned. * * Algorithm * 1. Since tan(-x) = -tan(x), we need only to consider positive x. * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. * 3. tan(x) is approximated by a odd polynomial of degree 27 on * [0,0.67434] * 3 27 * tan(x) ~ x + T1*x + ... + T13*x * where * * |tan(x) 2 4 26 | -59.2 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 * | x | * * Note: tan(x+y) = tan(x) + tan'(x)*y * ~ tan(x) + (1+x*x)*y * Therefore, for better accuracy in computing tan(x+y), let * 3 2 2 2 2 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) * then * 3 2 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) * * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) */ #include "math.h" #include "math_private.h" static const double one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ T[] = { 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ }; double attribute_hidden __kernel_tan(double x, double y, int iy) { double z,r,v,w,s; int32_t ix,hx; GET_HIGH_WORD(hx,x); ix = hx&0x7fffffff; /* high word of |x| */ if(ix<0x3e300000) /* x < 2**-28 */ {if((int)x==0) { /* generate inexact */ u_int32_t low; GET_LOW_WORD(low,x); if(((ix|low)|(iy+1))==0) return one/fabs(x); else return (iy==1)? x: -one/x; } } if(ix>=0x3FE59428) { /* |x|>=0.6744 */ if(hx<0) {x = -x; y = -y;} z = pio4-x; w = pio4lo-y; x = z+w; y = 0.0; } z = x*x; w = z*z; /* Break x^5*(T[1]+x^2*T[2]+...) into * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) */ r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); s = z*x; r = y + z*(s*(r+v)+y); r += T[0]*s; w = x+r; if(ix>=0x3FE59428) { v = (double)iy; return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); } if(iy==1) return w; else { /* if allow error up to 2 ulp, simply return -1.0/(x+r) here */ /* compute -1.0/(x+r) accurately */ double a,t; z = w; SET_LOW_WORD(z,0); v = r-(z - x); /* z+v = r+x */ t = a = -1.0/w; /* a = -1.0/w */ SET_LOW_WORD(t,0); s = 1.0+t*z; return t+a*(s+t*v); } } ``````