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       1                 :             : /* From: @(#)k_cos.c 1.3 95/01/18 */
       2                 :             : /*
       3                 :             :  * ====================================================
       4                 :             :  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
       5                 :             :  * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
       6                 :             :  *
       7                 :             :  * Developed at SunSoft, a Sun Microsystems, Inc. business.
       8                 :             :  * Permission to use, copy, modify, and distribute this
       9                 :             :  * software is freely granted, provided that this notice 
      10                 :             :  * is preserved.
      11                 :             :  * ====================================================
      12                 :             :  */
      13                 :             : 
      14                 :             : #include "cdefs-compat.h"
      15                 :             : //__FBSDID("$FreeBSD: src/lib/msun/ld80/k_cosl.c,v 1.1 2008/02/17 07:32:14 das Exp $");
      16                 :             : 
      17                 :             : /*
      18                 :             :  * ld80 version of k_cos.c.  See ../src/k_cos.c for most comments.
      19                 :             :  */
      20                 :             : 
      21                 :             : #include "math_private.h"
      22                 :             : 
      23                 :             : /*
      24                 :             :  * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
      25                 :             :  * |cos(x) - c(x)| < 2**-75.1
      26                 :             :  *
      27                 :             :  * The coefficients of c(x) were generated by a pari-gp script using
      28                 :             :  * a Remez algorithm that searches for the best higher coefficients
      29                 :             :  * after rounding leading coefficients to a specified precision.
      30                 :             :  *
      31                 :             :  * Simpler methods like Chebyshev or basic Remez barely suffice for
      32                 :             :  * cos() in 64-bit precision, because we want the coefficient of x^2
      33                 :             :  * to be precisely -0.5 so that multiplying by it is exact, and plain
      34                 :             :  * rounding of the coefficients of a good polynomial approximation only
      35                 :             :  * gives this up to about 64-bit precision.  Plain rounding also gives
      36                 :             :  * a mediocre approximation for the coefficient of x^4, but a rounding
      37                 :             :  * error of 0.5 ulps for this coefficient would only contribute ~0.01
      38                 :             :  * ulps to the final error, so this is unimportant.  Rounding errors in
      39                 :             :  * higher coefficients are even less important.
      40                 :             :  *
      41                 :             :  * In fact, coefficients above the x^4 one only need to have 53-bit
      42                 :             :  * precision, and this is more efficient.  We get this optimization
      43                 :             :  * almost for free from the complications needed to search for the best
      44                 :             :  * higher coefficients.
      45                 :             :  */
      46                 :             : static const double
      47                 :             : one = 1.0;
      48                 :             : 
      49                 :             : #if defined(__amd64__) || defined(__i386__)
      50                 :             : /* Long double constants are slow on these arches, and broken on i386. */
      51                 :             : static const volatile double
      52                 :             : C1hi = 0.041666666666666664,            /*  0x15555555555555.0p-57 */
      53                 :             : C1lo = 2.2598839032744733e-18;          /*  0x14d80000000000.0p-111 */
      54                 :             : #define C1      ((long double)C1hi + C1lo)
      55                 :             : #else
      56                 :             : static const long double
      57                 :             : C1 =  0.0416666666666666666136L;        /*  0xaaaaaaaaaaaaaa9b.0p-68 */
      58                 :             : #endif
      59                 :             : 
      60                 :             : static const double
      61                 :             : C2 = -0.0013888888888888874,            /* -0x16c16c16c16c10.0p-62 */
      62                 :             : C3 =  0.000024801587301571716,          /*  0x1a01a01a018e22.0p-68 */
      63                 :             : C4 = -0.00000027557319215507120,        /* -0x127e4fb7602f22.0p-74 */
      64                 :             : C5 =  0.0000000020876754400407278,      /*  0x11eed8caaeccf1.0p-81 */
      65                 :             : C6 = -1.1470297442401303e-11,           /* -0x19393412bd1529.0p-89 */
      66                 :             : C7 =  4.7383039476436467e-14;           /*  0x1aac9d9af5c43e.0p-97 */
      67                 :             : 
      68                 :             : long double
      69                 :           0 : __kernel_cosl(long double x, long double y)
      70                 :             : {
      71                 :             :         long double hz,z,r,w;
      72                 :             : 
      73                 :           0 :         z  = x*x;
      74                 :           0 :         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7))))));
      75                 :           0 :         hz = 0.5*z;
      76                 :           0 :         w  = one-hz;
      77                 :           0 :         return w + (((one-w)-hz) + (z*r-x*y));
      78                 :             : }