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
__cosl(long double x,long double y)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
97