src/libm/k_cos.c
changeset 2756 a98604b691c8
child 3162 dc1eb82ffdaa
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2755:2a3ec308d995 2756:a98604b691c8
       
     1 /* @(#)k_cos.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 #if defined(LIBM_SCCS) && !defined(lint)
       
    14 static char rcsid[] = "$NetBSD: k_cos.c,v 1.8 1995/05/10 20:46:22 jtc Exp $";
       
    15 #endif
       
    16 
       
    17 /*
       
    18  * __kernel_cos( x,  y )
       
    19  * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
       
    20  * Input x is assumed to be bounded by ~pi/4 in magnitude.
       
    21  * Input y is the tail of x.
       
    22  *
       
    23  * Algorithm
       
    24  *	1. Since cos(-x) = cos(x), we need only to consider positive x.
       
    25  *	2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
       
    26  *	3. cos(x) is approximated by a polynomial of degree 14 on
       
    27  *	   [0,pi/4]
       
    28  *		  	                 4            14
       
    29  *	   	cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
       
    30  *	   where the remez error is
       
    31  *
       
    32  * 	|              2     4     6     8     10    12     14 |     -58
       
    33  * 	|cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  )| <= 2
       
    34  * 	|    					               |
       
    35  *
       
    36  * 	               4     6     8     10    12     14
       
    37  *	4. let r = C1*x +C2*x +C3*x +C4*x +C5*x  +C6*x  , then
       
    38  *	       cos(x) = 1 - x*x/2 + r
       
    39  *	   since cos(x+y) ~ cos(x) - sin(x)*y
       
    40  *			  ~ cos(x) - x*y,
       
    41  *	   a correction term is necessary in cos(x) and hence
       
    42  *		cos(x+y) = 1 - (x*x/2 - (r - x*y))
       
    43  *	   For better accuracy when x > 0.3, let qx = |x|/4 with
       
    44  *	   the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
       
    45  *	   Then
       
    46  *		cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
       
    47  *	   Note that 1-qx and (x*x/2-qx) is EXACT here, and the
       
    48  *	   magnitude of the latter is at least a quarter of x*x/2,
       
    49  *	   thus, reducing the rounding error in the subtraction.
       
    50  */
       
    51 
       
    52 #include "math.h"
       
    53 #include "math_private.h"
       
    54 
       
    55 #ifdef __STDC__
       
    56 static const double
       
    57 #else
       
    58 static double
       
    59 #endif
       
    60   one = 1.00000000000000000000e+00,     /* 0x3FF00000, 0x00000000 */
       
    61     C1 = 4.16666666666666019037e-02,    /* 0x3FA55555, 0x5555554C */
       
    62     C2 = -1.38888888888741095749e-03,   /* 0xBF56C16C, 0x16C15177 */
       
    63     C3 = 2.48015872894767294178e-05,    /* 0x3EFA01A0, 0x19CB1590 */
       
    64     C4 = -2.75573143513906633035e-07,   /* 0xBE927E4F, 0x809C52AD */
       
    65     C5 = 2.08757232129817482790e-09,    /* 0x3E21EE9E, 0xBDB4B1C4 */
       
    66     C6 = -1.13596475577881948265e-11;   /* 0xBDA8FAE9, 0xBE8838D4 */
       
    67 
       
    68 #ifdef __STDC__
       
    69 double attribute_hidden
       
    70 __kernel_cos(double x, double y)
       
    71 #else
       
    72 double attribute_hidden
       
    73 __kernel_cos(x, y)
       
    74      double x, y;
       
    75 #endif
       
    76 {
       
    77     double a, hz, z, r, qx;
       
    78     int32_t ix;
       
    79     GET_HIGH_WORD(ix, x);
       
    80     ix &= 0x7fffffff;           /* ix = |x|'s high word */
       
    81     if (ix < 0x3e400000) {      /* if x < 2**27 */
       
    82         if (((int) x) == 0)
       
    83             return one;         /* generate inexact */
       
    84     }
       
    85     z = x * x;
       
    86     r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
       
    87     if (ix < 0x3FD33333)        /* if |x| < 0.3 */
       
    88         return one - (0.5 * z - (z * r - x * y));
       
    89     else {
       
    90         if (ix > 0x3fe90000) {  /* x > 0.78125 */
       
    91             qx = 0.28125;
       
    92         } else {
       
    93             INSERT_WORDS(qx, ix - 0x00200000, 0);       /* x/4 */
       
    94         }
       
    95         hz = 0.5 * z - qx;
       
    96         a = one - qx;
       
    97         return a - (hz - (z * r - x * y));
       
    98     }
       
    99 }