[444] | 1 | /* origin: FreeBSD /usr/src/lib/msun/ld80/k_cosl.c */
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| 2 | /* origin: FreeBSD /usr/src/lib/msun/ld128/k_cosl.c */
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| 3 | /*
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| 4 | * ====================================================
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| 5 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 6 | * Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
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| 7 | *
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| 8 | * Developed at SunSoft, a Sun Microsystems, Inc. business.
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| 9 | * Permission to use, copy, modify, and distribute this
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| 10 | * software is freely granted, provided that this notice
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| 11 | * is preserved.
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| 12 | * ====================================================
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| 13 | */
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| 14 |
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| 15 |
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| 16 | #include "libm.h"
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| 17 |
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| 18 | #if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
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| 19 | #if LDBL_MANT_DIG == 64
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| 20 | /*
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| 21 | * ld80 version of __cos.c. See __cos.c for most comments.
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| 22 | */
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| 23 | /*
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| 24 | * Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
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| 25 | * |cos(x) - c(x)| < 2**-75.1
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| 26 | *
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| 27 | * The coefficients of c(x) were generated by a pari-gp script using
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| 28 | * a Remez algorithm that searches for the best higher coefficients
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| 29 | * after rounding leading coefficients to a specified precision.
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| 30 | *
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| 31 | * Simpler methods like Chebyshev or basic Remez barely suffice for
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| 32 | * cos() in 64-bit precision, because we want the coefficient of x^2
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| 33 | * to be precisely -0.5 so that multiplying by it is exact, and plain
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| 34 | * rounding of the coefficients of a good polynomial approximation only
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| 35 | * gives this up to about 64-bit precision. Plain rounding also gives
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| 36 | * a mediocre approximation for the coefficient of x^4, but a rounding
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| 37 | * error of 0.5 ulps for this coefficient would only contribute ~0.01
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| 38 | * ulps to the final error, so this is unimportant. Rounding errors in
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| 39 | * higher coefficients are even less important.
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| 40 | *
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| 41 | * In fact, coefficients above the x^4 one only need to have 53-bit
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| 42 | * precision, and this is more efficient. We get this optimization
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| 43 | * almost for free from the complications needed to search for the best
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| 44 | * higher coefficients.
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| 45 | */
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| 46 | static const long double
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| 47 | C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */
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| 48 | static const double
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| 49 | C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */
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| 50 | C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */
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| 51 | C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */
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| 52 | C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */
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| 53 | C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */
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| 54 | C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */
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| 55 | #define POLY(z) (z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7)))))))
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| 56 | #elif LDBL_MANT_DIG == 113
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| 57 | /*
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| 58 | * ld128 version of __cos.c. See __cos.c for most comments.
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| 59 | */
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| 60 | /*
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| 61 | * Domain [-0.7854, 0.7854], range ~[-1.80e-37, 1.79e-37]:
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| 62 | * |cos(x) - c(x))| < 2**-122.0
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| 63 | *
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| 64 | * 113-bit precision requires more care than 64-bit precision, since
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| 65 | * simple methods give a minimax polynomial with coefficient for x^2
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| 66 | * that is 1 ulp below 0.5, but we want it to be precisely 0.5. See
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| 67 | * above for more details.
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| 68 | */
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| 69 | static const long double
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| 70 | C1 = 0.04166666666666666666666666666666658424671L,
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| 71 | C2 = -0.001388888888888888888888888888863490893732L,
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| 72 | C3 = 0.00002480158730158730158730158600795304914210L,
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| 73 | C4 = -0.2755731922398589065255474947078934284324e-6L,
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| 74 | C5 = 0.2087675698786809897659225313136400793948e-8L,
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| 75 | C6 = -0.1147074559772972315817149986812031204775e-10L,
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| 76 | C7 = 0.4779477332386808976875457937252120293400e-13L;
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| 77 | static const double
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| 78 | C8 = -0.1561920696721507929516718307820958119868e-15,
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| 79 | C9 = 0.4110317413744594971475941557607804508039e-18,
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| 80 | C10 = -0.8896592467191938803288521958313920156409e-21,
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| 81 | C11 = 0.1601061435794535138244346256065192782581e-23;
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| 82 | #define POLY(z) (z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*(C7+ \
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| 83 | z*(C8+z*(C9+z*(C10+z*C11)))))))))))
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| 84 | #endif
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| 85 |
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| 86 | long double __cosl(long double x, long double y)
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| 87 | {
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| 88 | long double hz,z,r,w;
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| 89 |
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| 90 | z = x*x;
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| 91 | r = POLY(z);
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| 92 | hz = 0.5*z;
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| 93 | w = 1.0-hz;
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| 94 | return w + (((1.0-w)-hz) + (z*r-x*y));
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| 95 | }
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| 96 | #endif
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