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annotate src/libm/k_cos.c @ 3632:da2a23b56493

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Updated 1.2 revisions that need to be revisited
author Sam Lantinga <slouken@libsdl.org>
date Sun, 10 Jan 2010 09:16:46 +0000
parents dc1eb82ffdaa
children
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1 /* @(#)k_cos.c 5.1 93/09/24 */
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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 SunPro, 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 #if defined(LIBM_SCCS) && !defined(lint)
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14 static const char rcsid[] =
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15 "$NetBSD: k_cos.c,v 1.8 1995/05/10 20:46:22 jtc Exp $";
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16 #endif
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17
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18 /*
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19 * __kernel_cos( x, y )
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20 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
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21 * Input x is assumed to be bounded by ~pi/4 in magnitude.
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22 * Input y is the tail of x.
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23 *
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24 * Algorithm
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25 * 1. Since cos(-x) = cos(x), we need only to consider positive x.
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26 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0.
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27 * 3. cos(x) is approximated by a polynomial of degree 14 on
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28 * [0,pi/4]
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29 * 4 14
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30 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x
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31 * where the remez error is
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32 *
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33 * | 2 4 6 8 10 12 14 | -58
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34 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2
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35 * | |
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36 *
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37 * 4 6 8 10 12 14
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38 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then
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39 * cos(x) = 1 - x*x/2 + r
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40 * since cos(x+y) ~ cos(x) - sin(x)*y
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41 * ~ cos(x) - x*y,
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42 * a correction term is necessary in cos(x) and hence
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43 * cos(x+y) = 1 - (x*x/2 - (r - x*y))
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44 * For better accuracy when x > 0.3, let qx = |x|/4 with
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45 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125.
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46 * Then
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47 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)).
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48 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the
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49 * magnitude of the latter is at least a quarter of x*x/2,
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50 * thus, reducing the rounding error in the subtraction.
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51 */
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52
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53 #include "math.h"
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54 #include "math_private.h"
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55
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56 #ifdef __STDC__
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57 static const double
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58 #else
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59 static double
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60 #endif
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61 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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62 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */
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63 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */
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64 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */
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65 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */
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66 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */
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67 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */
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68
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69 #ifdef __STDC__
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70 double attribute_hidden
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71 __kernel_cos(double x, double y)
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72 #else
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73 double attribute_hidden
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74 __kernel_cos(x, y)
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75 double x, y;
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76 #endif
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77 {
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78 double a, hz, z, r, qx;
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79 int32_t ix;
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80 GET_HIGH_WORD(ix, x);
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81 ix &= 0x7fffffff; /* ix = |x|'s high word */
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82 if (ix < 0x3e400000) { /* if x < 2**27 */
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83 if (((int) x) == 0)
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84 return one; /* generate inexact */
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85 }
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86 z = x * x;
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87 r = z * (C1 + z * (C2 + z * (C3 + z * (C4 + z * (C5 + z * C6)))));
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88 if (ix < 0x3FD33333) /* if |x| < 0.3 */
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89 return one - (0.5 * z - (z * r - x * y));
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90 else {
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91 if (ix > 0x3fe90000) { /* x > 0.78125 */
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92 qx = 0.28125;
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93 } else {
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94 INSERT_WORDS(qx, ix - 0x00200000, 0); /* x/4 */
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95 }
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96 hz = 0.5 * z - qx;
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97 a = one - qx;
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98 return a - (hz - (z * r - x * y));
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99 }
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100 }