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| Rev | Author | Line No. | Line |
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| 2 | pj | 1 | /* @(#)k_tan.c 5.1 93/09/24 */ |
| 2 | /* |
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| 3 | * ==================================================== |
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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| 5 | * |
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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| 7 | * Permission to use, copy, modify, and distribute this |
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| 8 | * software is freely granted, provided that this notice |
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| 9 | * is preserved. |
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| 10 | * ==================================================== |
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| 11 | */ |
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| 12 | |||
| 13 | #ifndef lint |
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| 14 | static char rcsid[] = "$\Id: k_tan.c,v 1.2 1995/05/30 05:49:14 rgrimes Exp $"; |
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| 15 | #endif |
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| 16 | |||
| 17 | /* __kernel_tan( x, y, k ) |
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| 18 | * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
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| 19 | * Input x is assumed to be bounded by ~pi/4 in magnitude. |
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| 20 | * Input y is the tail of x. |
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| 21 | * Input k indicates whether tan (if k=1) or |
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| 22 | * -1/tan (if k= -1) is returned. |
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| 23 | * |
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| 24 | * Algorithm |
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| 25 | * 1. Since tan(-x) = -tan(x), we need only to consider positive x. |
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| 26 | * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
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| 27 | * 3. tan(x) is approximated by a odd polynomial of degree 27 on |
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| 28 | * [0,0.67434] |
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| 29 | * 3 27 |
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| 30 | * tan(x) ~ x + T1*x + ... + T13*x |
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| 31 | * where |
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| 32 | * |
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| 33 | * |tan(x) 2 4 26 | -59.2 |
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| 34 | * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
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| 35 | * | x | |
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| 36 | * |
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| 37 | * Note: tan(x+y) = tan(x) + tan'(x)*y |
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| 38 | * ~ tan(x) + (1+x*x)*y |
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| 39 | * Therefore, for better accuracy in computing tan(x+y), let |
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| 40 | * 3 2 2 2 2 |
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| 41 | * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
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| 42 | * then |
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| 43 | * 3 2 |
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| 44 | * tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
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| 45 | * |
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| 46 | * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
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| 47 | * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) |
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| 48 | * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) |
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| 49 | */ |
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| 50 | |||
| 51 | #include "math.h" |
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| 52 | #include "math_private.h" |
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| 53 | #ifdef __STDC__ |
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| 54 | static const double |
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| 55 | #else |
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| 56 | static double |
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| 57 | #endif |
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| 58 | one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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| 59 | pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ |
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| 60 | pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ |
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| 61 | T[] = { |
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| 62 | 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ |
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| 63 | 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ |
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| 64 | 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ |
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| 65 | 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ |
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| 66 | 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ |
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| 67 | 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ |
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| 68 | 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ |
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| 69 | 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ |
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| 70 | 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ |
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| 71 | 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ |
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| 72 | 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ |
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| 73 | -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ |
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| 74 | 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ |
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| 75 | }; |
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| 76 | |||
| 77 | #ifdef __STDC__ |
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| 78 | double __kernel_tan(double x, double y, int iy) |
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| 79 | #else |
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| 80 | double __kernel_tan(x, y, iy) |
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| 81 | double x,y; int iy; |
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| 82 | #endif |
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| 83 | { |
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| 84 | double z,r,v,w,s; |
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| 85 | int32_t ix,hx; |
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| 86 | GET_HIGH_WORD(hx,x); |
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| 87 | ix = hx&0x7fffffff; /* high word of |x| */ |
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| 88 | if(ix<0x3e300000) /* x < 2**-28 */ |
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| 89 | {if((int)x==0) { /* generate inexact */ |
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| 90 | u_int32_t low; |
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| 91 | GET_LOW_WORD(low,x); |
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| 92 | if(((ix|low)|(iy+1))==0) return one/fabs(x); |
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| 93 | else return (iy==1)? x: -one/x; |
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| 94 | } |
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| 95 | } |
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| 96 | if(ix>=0x3FE59428) { /* |x|>=0.6744 */ |
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| 97 | if(hx<0) {x = -x; y = -y;} |
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| 98 | z = pio4-x; |
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| 99 | w = pio4lo-y; |
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| 100 | x = z+w; y = 0.0; |
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| 101 | } |
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| 102 | z = x*x; |
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| 103 | w = z*z; |
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| 104 | /* Break x^5*(T[1]+x^2*T[2]+...) into |
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| 105 | * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + |
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| 106 | * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) |
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| 107 | */ |
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| 108 | r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); |
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| 109 | v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); |
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| 110 | s = z*x; |
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| 111 | r = y + z*(s*(r+v)+y); |
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| 112 | r += T[0]*s; |
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| 113 | w = x+r; |
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| 114 | if(ix>=0x3FE59428) { |
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| 115 | v = (double)iy; |
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| 116 | return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); |
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| 117 | } |
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| 118 | if(iy==1) return w; |
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| 119 | else { /* if allow error up to 2 ulp, |
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| 120 | simply return -1.0/(x+r) here */ |
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| 121 | /* compute -1.0/(x+r) accurately */ |
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| 122 | double a,t; |
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| 123 | z = w; |
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| 124 | SET_LOW_WORD(z,0); |
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| 125 | v = r-(z - x); /* z+v = r+x */ |
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| 126 | t = a = -1.0/w; /* a = -1.0/w */ |
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| 127 | SET_LOW_WORD(t,0); |
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| 128 | s = 1.0+t*z; |
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| 129 | return t+a*(s+t*v); |
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| 130 | } |
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| 131 | } |