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      1 /* @(#)k_tan.c 5.1 93/09/24 */
      2 /*
      3  * ====================================================
      4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
      5  *
      6  * Developed at SunPro, a Sun Microsystems, Inc. business.
      7  * Permission to use, copy, modify, and distribute this
      8  * software is freely granted, provided that this notice
      9  * is preserved.
     10  * ====================================================
     11  */
     12 
     13 #include <sys/cdefs.h>
     14 #if defined(LIBM_SCCS) && !defined(lint)
     15 __RCSID("$NetBSD: k_tan.c,v 1.12 2004/07/22 18:24:09 drochner Exp $");
     16 #endif
     17 
     18 /* __kernel_tan( x, y, k )
     19  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
     20  * Input x is assumed to be bounded by ~pi/4 in magnitude.
     21  * Input y is the tail of x.
     22  * Input k indicates whether tan (if k=1) or
     23  * -1/tan (if k= -1) is returned.
     24  *
     25  * Algorithm
     26  *	1. Since tan(-x) = -tan(x), we need only to consider positive x.
     27  *	2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
     28  *	3. tan(x) is approximated by a odd polynomial of degree 27 on
     29  *	   [0,0.67434]
     30  *		  	         3             27
     31  *	   	tan(x) ~ x + T1*x + ... + T13*x
     32  *	   where
     33  *
     34  * 	        |tan(x)         2     4            26   |     -59.2
     35  * 	        |----- - (1+T1*x +T2*x +.... +T13*x    )| <= 2
     36  * 	        |  x 					|
     37  *
     38  *	   Note: tan(x+y) = tan(x) + tan'(x)*y
     39  *		          ~ tan(x) + (1+x*x)*y
     40  *	   Therefore, for better accuracy in computing tan(x+y), let
     41  *		     3      2      2       2       2
     42  *		r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
     43  *	   then
     44  *		 		    3    2
     45  *		tan(x+y) = x + (T1*x + (x *(r+y)+y))
     46  *
     47  *      4. For x in [0.67434,pi/4],  let y = pi/4 - x, then
     48  *		tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
     49  *		       = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
     50  */
     51 
     52 #include "math.h"
     53 #include "math_private.h"
     54 
     55 static const double xxx[] = {
     56 		 3.33333333333334091986e-01,	/* 3FD55555, 55555563 */
     57 		 1.33333333333201242699e-01,	/* 3FC11111, 1110FE7A */
     58 		 5.39682539762260521377e-02,	/* 3FABA1BA, 1BB341FE */
     59 		 2.18694882948595424599e-02,	/* 3F9664F4, 8406D637 */
     60 		 8.86323982359930005737e-03,	/* 3F8226E3, E96E8493 */
     61 		 3.59207910759131235356e-03,	/* 3F6D6D22, C9560328 */
     62 		 1.45620945432529025516e-03,	/* 3F57DBC8, FEE08315 */
     63 		 5.88041240820264096874e-04,	/* 3F4344D8, F2F26501 */
     64 		 2.46463134818469906812e-04,	/* 3F3026F7, 1A8D1068 */
     65 		 7.81794442939557092300e-05,	/* 3F147E88, A03792A6 */
     66 		 7.14072491382608190305e-05,	/* 3F12B80F, 32F0A7E9 */
     67 		-1.85586374855275456654e-05,	/* BEF375CB, DB605373 */
     68 		 2.59073051863633712884e-05,	/* 3EFB2A70, 74BF7AD4 */
     69 /* one */	 1.00000000000000000000e+00,	/* 3FF00000, 00000000 */
     70 /* pio4 */	 7.85398163397448278999e-01,	/* 3FE921FB, 54442D18 */
     71 /* pio4lo */	 3.06161699786838301793e-17	/* 3C81A626, 33145C07 */
     72 };
     73 #define	one	xxx[13]
     74 #define	pio4	xxx[14]
     75 #define	pio4lo	xxx[15]
     76 #define	T	xxx
     77 
     78 double
     79 __kernel_tan(double x, double y, int iy)
     80 {
     81 	double z, r, v, w, s;
     82 	int32_t ix, hx;
     83 
     84 	GET_HIGH_WORD(hx, x);	/* high word of x */
     85 	ix = hx & 0x7fffffff;			/* high word of |x| */
     86 	if (ix < 0x3e300000) {			/* x < 2**-28 */
     87 		if ((int) x == 0) {		/* generate inexact */
     88 			u_int32_t low;
     89 			GET_LOW_WORD(low, x);
     90 			if(((ix | low) | (iy + 1)) == 0)
     91 				return one / fabs(x);
     92 			else {
     93 				if (iy == 1)
     94 					return x;
     95 				else {	/* compute -1 / (x+y) carefully */
     96 					double a, t;
     97 
     98 					z = w = x + y;
     99 					SET_LOW_WORD(z, 0);
    100 					v = y - (z - x);
    101 					t = a = -one / w;
    102 					SET_LOW_WORD(t, 0);
    103 					s = one + t * z;
    104 					return t + a * (s + t * v);
    105 				}
    106 			}
    107 		}
    108 	}
    109 	if (ix >= 0x3FE59428) {	/* |x| >= 0.6744 */
    110 		if (hx < 0) {
    111 			x = -x;
    112 			y = -y;
    113 		}
    114 		z = pio4 - x;
    115 		w = pio4lo - y;
    116 		x = z + w;
    117 		y = 0.0;
    118 	}
    119 	z = x * x;
    120 	w = z * z;
    121 	/*
    122 	 * Break x^5*(T[1]+x^2*T[2]+...) into
    123 	 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
    124 	 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
    125 	 */
    126 	r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] +
    127 		w * T[11]))));
    128 	v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] +
    129 		w * T[12])))));
    130 	s = z * x;
    131 	r = y + z * (s * (r + v) + y);
    132 	r += T[0] * s;
    133 	w = x + r;
    134 	if (ix >= 0x3FE59428) {
    135 		v = (double) iy;
    136 		return (double) (1 - ((hx >> 30) & 2)) *
    137 			(v - 2.0 * (x - (w * w / (w + v) - r)));
    138 	}
    139 	if (iy == 1)
    140 		return w;
    141 	else {
    142 		/*
    143 		 * if allow error up to 2 ulp, simply return
    144 		 * -1.0 / (x+r) here
    145 		 */
    146 		/* compute -1.0 / (x+r) accurately */
    147 		double a, t;
    148 		z = w;
    149 		SET_LOW_WORD(z, 0);
    150 		v = r - (z - x);	/* z+v = r+x */
    151 		t = a = -1.0 / w;	/* a = -1.0/w */
    152 		SET_LOW_WORD(t, 0);
    153 		s = 1.0 + t * z;
    154 		return t + a * (s + t * v);
    155 	}
    156 }
    157