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jeremybenn |
#pragma ident "@(#)k_tan.c 1.5 04/04/22 SMI"
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/*
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* ====================================================
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* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
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*
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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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/* INDENT OFF */
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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 -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 "fdlibm.h"
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#ifndef _DOUBLE_IS_32BITS
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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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/* INDENT ON */
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double
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__kernel_tan(double x, double y, int iy) {
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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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uint32_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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}
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#endif /* defined(_DOUBLE_IS_32BITS) */
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