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/* @(#)k_tan.c 5.1 93/09/24 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* |
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* Developed at SunPro, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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/* __kernel_tan( x, y, k ) |
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* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
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* Input x is assumed to be bounded by ~pi/4 in magnitude. |
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* Input y is the tail of x. |
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* Input k indicates whether tan (if k=1) or |
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* -1/tan (if k= -1) is returned. |
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* |
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* Algorithm |
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* 1. Since tan(-x) = -tan(x), we need only to consider positive x. |
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* 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
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* 3. tan(x) is approximated by a odd polynomial of degree 27 on |
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* [0,0.67434] |
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* 3 27 |
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* tan(x) ~ x + T1*x + ... + T13*x |
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* where |
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* |
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* |tan(x) 2 4 26 | -59.2 |
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* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
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* | x | |
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* |
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* Note: tan(x+y) = tan(x) + tan'(x)*y |
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* ~ tan(x) + (1+x*x)*y |
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* Therefore, for better accuracy in computing tan(x+y), let |
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* 3 2 2 2 2 |
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* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
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* then |
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* 3 2 |
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* tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
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* |
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* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
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* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) |
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* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) |
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*/ |
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#include "math.h" |
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#include "math_private.h" |
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static const double xxx[] = { |
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3.33333333333334091986e-01, /* 3FD55555, 55555563 */ |
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1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ |
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5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ |
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2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ |
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8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ |
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3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ |
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1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ |
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5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ |
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2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ |
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7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ |
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7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ |
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-1.85586374855275456654e-05, /* BEF375CB, DB605373 */ |
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2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ |
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/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ |
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/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ |
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/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ |
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}; |
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#define one xxx[13] |
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#define pio4 xxx[14] |
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#define pio4lo xxx[15] |
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#define T xxx |
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double |
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__kernel_tan(double x, double y, int iy) |
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{ |
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double z, r, v, w, s; |
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int32_t ix, hx; |
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GET_HIGH_WORD(hx, x); /* high word of x */ |
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ix = hx & 0x7fffffff; /* high word of |x| */ |
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if (ix < 0x3e300000) { /* x < 2**-28 */ |
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if ((int) x == 0) { /* generate inexact */ |
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u_int32_t low; |
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GET_LOW_WORD(low, x); |
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if(((ix | low) | (iy + 1)) == 0) |
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return one / fabs(x); |
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else { |
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if (iy == 1) |
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return x; |
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else { /* compute -1 / (x+y) carefully */ |
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double a, t; |
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z = w = x + y; |
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SET_LOW_WORD(z, 0); |
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v = y - (z - x); |
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t = a = -one / w; |
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SET_LOW_WORD(t, 0); |
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s = one + t * z; |
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return t + a * (s + t * v); |
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} |
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} |
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} |
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} |
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if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ |
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if (hx < 0) { |
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x = -x; |
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y = -y; |
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} |
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z = pio4 - x; |
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w = pio4lo - y; |
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x = z + w; |
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y = 0.0; |
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} |
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z = x * x; |
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w = z * z; |
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/* |
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* Break x^5*(T[1]+x^2*T[2]+...) into |
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* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + |
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* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) |
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*/ |
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r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + |
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w * T[11])))); |
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v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + |
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w * T[12]))))); |
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s = z * x; |
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r = y + z * (s * (r + v) + y); |
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r += T[0] * s; |
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w = x + r; |
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if (ix >= 0x3FE59428) { |
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v = (double) iy; |
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return (double) (1 - ((hx >> 30) & 2)) * |
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(v - 2.0 * (x - (w * w / (w + v) - r))); |
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} |
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if (iy == 1) |
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return w; |
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else { |
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/* |
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* if allow error up to 2 ulp, simply return |
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* -1.0 / (x+r) here |
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*/ |
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/* compute -1.0 / (x+r) accurately */ |
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double a, t; |
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z = w; |
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SET_LOW_WORD(z, 0); |
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v = r - (z - x); /* z+v = r+x */ |
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t = a = -1.0 / w; /* a = -1.0/w */ |
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SET_LOW_WORD(t, 0); |
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s = 1.0 + t * z; |
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return t + a * (s + t * v); |
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} |
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} |