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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 #include  <LibConfig.h>
     13 #include  <sys/EfiCdefs.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