Szabolcs Nagy
e216951f50
When FLT_EVAL_METHOD!=0 (only i386 with x87 fp) the excess
precision of an expression must be removed in an assignment.
(gcc needs -fexcess-precision=standard or -std=c99 for this)
This is done by extra load/store instructions which adds code
bloat when lot of temporaries are used and it makes the result
less precise in many cases.
Using double_t and float_t avoids these issues on i386 and
it makes no difference on other archs.
For now only a few functions are modified where the excess
precision is clearly beneficial (mostly polynomial evaluations
with temporaries).
object size differences on i386, gcc-4.8:
old new
__cosdf.o 123 95
__cos.o 199 169
__sindf.o 131 95
__sin.o 225 203
__tandf.o 207 151
__tan.o 605 499
erff.o 1470 1416
erf.o 1703 1649
j0f.o 1779 1745
j0.o 2308 2274
j1f.o 1602 1568
j1.o 2286 2252
tgamma.o 1431 1424
math/*.o 64164 63635
72 lines
2.8 KiB
C
72 lines
2.8 KiB
C
/* origin: FreeBSD /usr/src/lib/msun/src/k_cos.c */
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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 SunSoft, 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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* __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 remez 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, rearrange to
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* cos(x+y) ~ w + (tmp + (r-x*y))
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* where w = 1 - x*x/2 and tmp is a tiny correction term
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* (1 - x*x/2 == w + tmp exactly in infinite precision).
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* The exactness of w + tmp in infinite precision depends on w
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* and tmp having the same precision as x. If they have extra
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* precision due to compiler bugs, then the extra precision is
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* only good provided it is retained in all terms of the final
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* expression for cos(). Retention happens in all cases tested
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* under FreeBSD, so don't pessimize things by forcibly clipping
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* any extra precision in w.
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*/
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#include "libm.h"
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static const double
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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 __cos(double x, double y)
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{
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double_t hz,z,r,w;
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z = x*x;
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w = z*z;
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r = z*(C1+z*(C2+z*C3)) + w*w*(C4+z*(C5+z*C6));
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hz = 0.5*z;
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w = 1.0-hz;
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return w + (((1.0-w)-hz) + (z*r-x*y));
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}
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