newlib/libm/ld80/k_cosl.c

   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 <sys/cdefs.h>
  15 __FBSDID("$FreeBSD$");
  16
  17 /*
  18  * ld80 version of k_cos.c.  See ../src/k_cos.c for most comments.
  19  */
  20 #include <math.h>
  21 #include "../ld/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 __kernel_cosl(long double x, long double y)
  70 {
  71         long double hz,z,r,w;
  72
  73         z  = x*x;
  74         r  = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7))))));
  75         hz = 0.5*z;
  76         w  = one-hz;
  77         return w + (((one-w)-hz) + (z*r-x*y));
  78 }