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/* $OpenBSD: k_cosl.c,v 1.2 2017/01/21 08:29:13 krw Exp $ */ |
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/* From: @(#)k_cos.c 1.3 95/01/18 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans. |
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* |
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* Developed at SunSoft, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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/* |
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* ld80 version of k_cos.c. See ../k_cos.c for most comments. |
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*/ |
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#include "math_private.h" |
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/* |
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* Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]: |
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* |cos(x) - c(x)| < 2**-75.1 |
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* |
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* The coefficients of c(x) were generated by a pari-gp script using |
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* a Remez algorithm that searches for the best higher coefficients |
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* after rounding leading coefficients to a specified precision. |
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* |
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* Simpler methods like Chebyshev or basic Remez barely suffice for |
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* cos() in 64-bit precision, because we want the coefficient of x^2 |
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* to be precisely -0.5 so that multiplying by it is exact, and plain |
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* rounding of the coefficients of a good polynomial approximation only |
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* gives this up to about 64-bit precision. Plain rounding also gives |
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* a mediocre approximation for the coefficient of x^4, but a rounding |
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* error of 0.5 ulps for this coefficient would only contribute ~0.01 |
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* ulps to the final error, so this is unimportant. Rounding errors in |
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* higher coefficients are even less important. |
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* |
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* In fact, coefficients above the x^4 one only need to have 53-bit |
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* precision, and this is more efficient. We get this optimization |
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* almost for free from the complications needed to search for the best |
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* higher coefficients. |
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*/ |
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static const double |
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one = 1.0; |
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#if defined(__amd64__) || defined(__i386__) |
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/* Long double constants are slow on these arches, and broken on i386. */ |
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static const volatile double |
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C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */ |
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C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */ |
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#define C1 ((long double)C1hi + C1lo) |
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#else |
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static const long double |
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C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */ |
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#endif |
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static const double |
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C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */ |
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C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */ |
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C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */ |
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C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */ |
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C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */ |
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C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */ |
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long double |
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__kernel_cosl(long double x, long double y) |
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{ |
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long double hz,z,r,w; |
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z = x*x; |
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r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7)))))); |
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hz = 0.5*z; |
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w = one-hz; |
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return w + (((one-w)-hz) + (z*r-x*y)); |
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} |