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/* @(#)k_cos.c 5.1 93/09/24 */ |
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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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* |
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* Developed at SunPro, 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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* __kernel_cos( x, y ) |
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* kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
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* Input x is assumed to be bounded by ~pi/4 in magnitude. |
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* Input y is the tail of x. |
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* |
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* Algorithm |
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* 1. Since cos(-x) = cos(x), we need only to consider positive x. |
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* 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. |
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* 3. cos(x) is approximated by a polynomial of degree 14 on |
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* [0,pi/4] |
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* 4 14 |
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* cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
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* where the Remes error is |
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* |
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* | 2 4 6 8 10 12 14 | -58 |
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* |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
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* | | |
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* |
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* 4 6 8 10 12 14 |
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* 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
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* cos(x) = 1 - x*x/2 + r |
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* since cos(x+y) ~ cos(x) - sin(x)*y |
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* ~ cos(x) - x*y, |
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* a correction term is necessary in cos(x) and hence |
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* cos(x+y) = 1 - (x*x/2 - (r - x*y)) |
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* For better accuracy when x > 0.3, let qx = |x|/4 with |
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* the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. |
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* Then |
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* cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). |
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* Note that 1-qx and (x*x/2-qx) is EXACT here, and the |
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* magnitude of the latter is at least a quarter of x*x/2, |
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* thus, reducing the rounding error in the subtraction. |
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*/ |
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#include "math.h" |
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#include "math_private.h" |
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static const double |
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ |
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C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ |
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C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ |
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C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ |
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C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ |
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C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ |
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double |
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__kernel_cos(double x, double y) |
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{ |
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double a,hz,z,r,qx; |
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int32_t ix; |
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GET_HIGH_WORD(ix,x); |
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ix &= 0x7fffffff; /* ix = |x|'s high word*/ |
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✗✓ |
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if(ix<0x3e400000) { /* if x < 2**27 */ |
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if(((int)x)==0) return one; /* generate inexact */ |
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} |
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z = x*x; |
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r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); |
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✓✓ |
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if(ix < 0x3FD33333) /* if |x| < 0.3 */ |
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return one - (0.5*z - (z*r - x*y)); |
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else { |
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✓✓ |
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if(ix > 0x3fe90000) { /* x > 0.78125 */ |
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qx = 0.28125; |
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} else { |
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INSERT_WORDS(qx,ix-0x00200000,0); /* x/4 */ |
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} |
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hz = 0.5*z-qx; |
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a = one-qx; |
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return a - (hz - (z*r-x*y)); |
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} |
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} |