github.com/afumu/libc@v0.0.6/musl/src/math/__cosl.c (about)

     1  /* origin: FreeBSD /usr/src/lib/msun/ld80/k_cosl.c */
     2  /* origin: FreeBSD /usr/src/lib/msun/ld128/k_cosl.c */
     3  /*
     4   * ====================================================
     5   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     6   * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
     7   *
     8   * Developed at SunSoft, a Sun Microsystems, Inc. business.
     9   * Permission to use, copy, modify, and distribute this
    10   * software is freely granted, provided that this notice
    11   * is preserved.
    12   * ====================================================
    13   */
    14  
    15  
    16  #include "libm.h"
    17  
    18  #if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
    19  #if LDBL_MANT_DIG == 64
    20  /*
    21   * ld80 version of __cos.c.  See __cos.c for most comments.
    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 long double
    47  C1 =  0.0416666666666666666136L;        /*  0xaaaaaaaaaaaaaa9b.0p-68 */
    48  static const double
    49  C2 = -0.0013888888888888874,            /* -0x16c16c16c16c10.0p-62 */
    50  C3 =  0.000024801587301571716,          /*  0x1a01a01a018e22.0p-68 */
    51  C4 = -0.00000027557319215507120,        /* -0x127e4fb7602f22.0p-74 */
    52  C5 =  0.0000000020876754400407278,      /*  0x11eed8caaeccf1.0p-81 */
    53  C6 = -1.1470297442401303e-11,           /* -0x19393412bd1529.0p-89 */
    54  C7 =  4.7383039476436467e-14;           /*  0x1aac9d9af5c43e.0p-97 */
    55  #define POLY(z) (z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7)))))))
    56  #elif LDBL_MANT_DIG == 113
    57  /*
    58   * ld128 version of __cos.c.  See __cos.c for most comments.
    59   */
    60  /*
    61   * Domain [-0.7854, 0.7854], range ~[-1.80e-37, 1.79e-37]:
    62   * |cos(x) - c(x))| < 2**-122.0
    63   *
    64   * 113-bit precision requires more care than 64-bit precision, since
    65   * simple methods give a minimax polynomial with coefficient for x^2
    66   * that is 1 ulp below 0.5, but we want it to be precisely 0.5.  See
    67   * above for more details.
    68   */
    69  static const long double
    70  C1 =  0.04166666666666666666666666666666658424671L,
    71  C2 = -0.001388888888888888888888888888863490893732L,
    72  C3 =  0.00002480158730158730158730158600795304914210L,
    73  C4 = -0.2755731922398589065255474947078934284324e-6L,
    74  C5 =  0.2087675698786809897659225313136400793948e-8L,
    75  C6 = -0.1147074559772972315817149986812031204775e-10L,
    76  C7 =  0.4779477332386808976875457937252120293400e-13L;
    77  static const double
    78  C8 = -0.1561920696721507929516718307820958119868e-15,
    79  C9 =  0.4110317413744594971475941557607804508039e-18,
    80  C10 = -0.8896592467191938803288521958313920156409e-21,
    81  C11 =  0.1601061435794535138244346256065192782581e-23;
    82  #define POLY(z) (z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*(C7+ \
    83  	z*(C8+z*(C9+z*(C10+z*C11)))))))))))
    84  #endif
    85  
    86  long double __cosl(long double x, long double y)
    87  {
    88  	long double hz,z,r,w;
    89  
    90  	z  = x*x;
    91  	r  = POLY(z);
    92  	hz = 0.5*z;
    93  	w  = 1.0-hz;
    94  	return w + (((1.0-w)-hz) + (z*r-x*y));
    95  }
    96  #endif