[352] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/k_cos.c */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 5 | *
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| 6 | * Developed at SunSoft, a Sun Microsystems, Inc. business.
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| 7 | * Permission to use, copy, modify, and distribute this
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| 8 | * software is freely granted, provided that this notice
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| 9 | * is preserved.
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| 10 | * ====================================================
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| 11 | */
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| 12 | /*
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| 13 | * __cos( x, y )
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| 14 | * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
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| 15 | * Input x is assumed to be bounded by ~pi/4 in magnitude.
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| 16 | * Input y is the tail of x.
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| 17 | *
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| 18 | * Algorithm
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| 19 | * 1. Since cos(-x) = cos(x), we need only to consider positive x.
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| 20 | * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
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| 21 | * 3. cos(x) is approximated by a polynomial of degree 14 on
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| 22 | * [0,pi/4]
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| 23 | * 4 14
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| 24 | * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
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| 25 | * where the remez error is
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| 26 | *
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| 27 | * | 2 4 6 8 10 12 14 | -58
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| 28 | * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
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| 29 | * | |
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| 30 | *
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| 31 | * 4 6 8 10 12 14
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| 32 | * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
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| 33 | * cos(x) ~ 1 - x*x/2 + r
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| 34 | * since cos(x+y) ~ cos(x) - sin(x)*y
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| 35 | * ~ cos(x) - x*y,
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| 36 | * a correction term is necessary in cos(x) and hence
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| 37 | * cos(x+y) = 1 - (x*x/2 - (r - x*y))
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| 38 | * For better accuracy, rearrange to
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| 39 | * cos(x+y) ~ w + (tmp + (r-x*y))
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| 40 | * where w = 1 - x*x/2 and tmp is a tiny correction term
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| 41 | * (1 - x*x/2 == w + tmp exactly in infinite precision).
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| 42 | * The exactness of w + tmp in infinite precision depends on w
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| 43 | * and tmp having the same precision as x. If they have extra
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| 44 | * precision due to compiler bugs, then the extra precision is
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| 45 | * only good provided it is retained in all terms of the final
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| 46 | * expression for cos(). Retention happens in all cases tested
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| 47 | * under FreeBSD, so don't pessimize things by forcibly clipping
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| 48 | * any extra precision in w.
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| 49 | */
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| 50 |
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| 51 | #include "libm.h"
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| 52 |
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| 53 | static const double
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| 54 | C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
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| 55 | C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
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| 56 | C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
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| 57 | C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
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| 58 | C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
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| 59 | C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
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| 60 |
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| 61 | double __cos(double x, double y)
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| 62 | {
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| 63 | double_t hz,z,r,w;
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| 64 |
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| 65 | z = x*x;
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| 66 | w = z*z;
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| 67 | r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
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| 68 | hz = 0.5*z;
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| 69 | w = 1.0-hz;
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| 70 | return w + (((1.0-w)-hz) + (z*r-x*y));
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| 71 | }
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