| /* origin: FreeBSD /usr/src/lib/msun/ld80/k_cosl.c */ |
| /* origin: FreeBSD /usr/src/lib/msun/ld128/k_cosl.c */ |
| /* |
| * ==================================================== |
| * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. |
| * |
| * Developed at SunSoft, a Sun Microsystems, Inc. business. |
| * Permission to use, copy, modify, and distribute this |
| * software is freely granted, provided that this notice |
| * is preserved. |
| * ==================================================== |
| */ |
| |
| |
| #include "libm.h" |
| |
| #if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384 |
| #if LDBL_MANT_DIG == 64 |
| /* |
| * ld80 version of __cos.c. See __cos.c for most comments. |
| */ |
| /* |
| * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]: |
| * |cos(x) - c(x)| < 2**-75.1 |
| * |
| * The coefficients of c(x) were generated by a pari-gp script using |
| * a Remez algorithm that searches for the best higher coefficients |
| * after rounding leading coefficients to a specified precision. |
| * |
| * Simpler methods like Chebyshev or basic Remez barely suffice for |
| * cos() in 64-bit precision, because we want the coefficient of x^2 |
| * to be precisely -0.5 so that multiplying by it is exact, and plain |
| * rounding of the coefficients of a good polynomial approximation only |
| * gives this up to about 64-bit precision. Plain rounding also gives |
| * a mediocre approximation for the coefficient of x^4, but a rounding |
| * error of 0.5 ulps for this coefficient would only contribute ~0.01 |
| * ulps to the final error, so this is unimportant. Rounding errors in |
| * higher coefficients are even less important. |
| * |
| * In fact, coefficients above the x^4 one only need to have 53-bit |
| * precision, and this is more efficient. We get this optimization |
| * almost for free from the complications needed to search for the best |
| * higher coefficients. |
| */ |
| static const long double |
| C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */ |
| static const double |
| C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */ |
| C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */ |
| C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */ |
| C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */ |
| C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */ |
| C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */ |
| #define POLY(z) (z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7))))))) |
| #elif LDBL_MANT_DIG == 113 |
| /* |
| * ld128 version of __cos.c. See __cos.c for most comments. |
| */ |
| /* |
| * Domain [-0.7854, 0.7854], range ~[-1.80e-37, 1.79e-37]: |
| * |cos(x) - c(x))| < 2**-122.0 |
| * |
| * 113-bit precision requires more care than 64-bit precision, since |
| * simple methods give a minimax polynomial with coefficient for x^2 |
| * that is 1 ulp below 0.5, but we want it to be precisely 0.5. See |
| * above for more details. |
| */ |
| static const long double |
| C1 = 0.04166666666666666666666666666666658424671L, |
| C2 = -0.001388888888888888888888888888863490893732L, |
| C3 = 0.00002480158730158730158730158600795304914210L, |
| C4 = -0.2755731922398589065255474947078934284324e-6L, |
| C5 = 0.2087675698786809897659225313136400793948e-8L, |
| C6 = -0.1147074559772972315817149986812031204775e-10L, |
| C7 = 0.4779477332386808976875457937252120293400e-13L; |
| static const double |
| C8 = -0.1561920696721507929516718307820958119868e-15, |
| C9 = 0.4110317413744594971475941557607804508039e-18, |
| C10 = -0.8896592467191938803288521958313920156409e-21, |
| C11 = 0.1601061435794535138244346256065192782581e-23; |
| #define POLY(z) (z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*(C7+ \ |
| z*(C8+z*(C9+z*(C10+z*C11))))))))))) |
| #endif |
| |
| long double __cosl(long double x, long double y) |
| { |
| long double hz,z,r,w; |
| |
| z = x*x; |
| r = POLY(z); |
| hz = 0.5*z; |
| w = 1.0-hz; |
| return w + (((1.0-w)-hz) + (z*r-x*y)); |
| } |
| #endif |
