mirror of
https://github.com/jerryscript-project/jerryscript.git
synced 2025-12-15 16:29:21 +00:00
Re-style fdlibm to conform to jerry guidelines
* First re-style was done automatically by indent to minimize the
chance of errors during rewrite.
* Manual changes were applied to non-critical places only (comments
and spaces):
* Replaced all tabs with spaces.
* Fixed tab stops in formulae in function comments.
(Note: ASCII art for math formulae (especially for super- and
subscripts) is a terrible idea.)
* Unified the style of function comments.
* Moved some in-code comments to their right places, which indent
couldn't handle.
* Added spaces to formulae of in-code comments to make them more
readable.
* Added braces mandated by jerry style guidelines.
* Added parentheses to multiline #ifdef.
JerryScript-DCO-1.0-Signed-off-by: Akos Kiss akiss@inf.u-szeged.hu
This commit is contained in:
Akos Kiss
parent
b39474c746
commit
8dd5186a0d
30
third-party/fdlibm/include/fdlibm-math.h
vendored
30
third-party/fdlibm/include/fdlibm-math.h
vendored
@ -52,29 +52,29 @@ extern "C"
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#define M_2_SQRTPI 1.1283791670955125738961589031215452
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// Trigonometric functions
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double cos(double);
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double sin(double);
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double tan(double);
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double acos(double);
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double asin(double);
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double atan(double);
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double atan2(double, double);
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double cos (double);
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double sin (double);
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double tan (double);
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double acos (double);
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double asin (double);
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double atan (double);
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double atan2 (double, double);
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// Exponential and logarithmic functions
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double exp(double);
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double log(double);
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double exp (double);
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double log (double);
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// Power functions
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double pow(double, double);
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double sqrt(double);
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double pow (double, double);
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double sqrt (double);
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// Rounding and remainder functions
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double ceil(double);
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double floor(double);
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double ceil (double);
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double floor (double);
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// Other functions
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double fabs(double);
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double fmod(double, double);
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double fabs (double);
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double fmod (double, double);
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#ifdef __cplusplus
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}
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36
third-party/fdlibm/include/fdlibm.h
vendored
36
third-party/fdlibm/include/fdlibm.h
vendored
@ -13,43 +13,43 @@
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/* Sometimes it's necessary to define __LITTLE_ENDIAN explicitly
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but these catch some common cases. */
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#if defined(i386) || defined(__i386) || defined(__i386__) || \
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defined(i486) || defined(__i486) || defined(__i486__) || \
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defined(intel) || defined(x86) || defined(i86pc) || \
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defined(__alpha) || defined(__osf__) || \
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defined(__x86_64__) || defined(__arm__)
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#if (defined (i386) || defined (__i386) || defined (__i386__) || \
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defined (i486) || defined (__i486) || defined (__i486__) || \
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defined (intel) || defined (x86) || defined (i86pc) || \
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defined (__alpha) || defined (__osf__) || \
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defined (__x86_64__) || defined (__arm__))
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#define __LITTLE_ENDIAN
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#endif
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#ifdef __LITTLE_ENDIAN
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#define __HI(x) *(1+(int*)&x)
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#define __LO(x) *(int*)&x
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#define __HI(x) *(1 + (int *) &x)
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#define __LO(x) *(int *) &x
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#else
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#define __HI(x) *(int*)&x
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#define __LO(x) *(1+(int*)&x)
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#define __HI(x) *(int *) &x
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#define __LO(x) *(1 + (int *) &x)
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#endif
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/*
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* ANSI/POSIX
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*/
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#define MAXFLOAT ((float)3.40282346638528860e+38)
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#define MAXFLOAT ((float) 3.40282346638528860e+38)
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#define HUGE MAXFLOAT
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#define HUGE MAXFLOAT
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/*
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* set X_TLOSS = pi*2**52, which is possibly defined in <values.h>
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* (one may replace the following line by "#include <values.h>")
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*/
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#define X_TLOSS 1.41484755040568800000e+16
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#define X_TLOSS 1.41484755040568800000e+16
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#define DOMAIN 1
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#define SING 2
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#define OVERFLOW 3
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#define UNDERFLOW 4
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#define TLOSS 5
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#define PLOSS 6
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#define DOMAIN 1
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#define SING 2
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#define OVERFLOW 3
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#define UNDERFLOW 4
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#define TLOSS 5
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#define PLOSS 6
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/*
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* ANSI/POSIX
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125
third-party/fdlibm/s_acos.c
vendored
125
third-party/fdlibm/s_acos.c
vendored
@ -6,31 +6,32 @@
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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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* 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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/* acos(x)
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* Method :
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* acos(x) = pi/2 - asin(x)
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* acos(-x) = pi/2 + asin(x)
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*
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* Method:
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* acos(x) = pi/2 - asin(x)
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* acos(-x) = pi/2 + asin(x)
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* For |x|<=0.5
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* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
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* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
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* For x>0.5
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* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
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* = 2asin(sqrt((1-x)/2))
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* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
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* = 2f + (2c + 2s*z*R(z))
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* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
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* = 2asin(sqrt((1-x)/2))
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* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
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* = 2f + (2c + 2s*z*R(z))
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* where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
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* for f so that f+c ~ sqrt(z).
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* For x<-0.5
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* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
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* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
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* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
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* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
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*
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* Special cases:
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
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*
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* Function needed: sqrt
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*/
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@ -52,44 +53,62 @@
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#define qS3 -6.88283971605453293030e-01 /* 0xBFE6066C, 0x1B8D0159 */
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#define qS4 7.70381505559019352791e-02 /* 0x3FB3B8C5, 0xB12E9282 */
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double acos(double x)
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double
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acos (double x)
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{
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double z,p,q,r,w,s,c,df;
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int hx,ix;
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hx = __HI(x);
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ix = hx&0x7fffffff;
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if(ix>=0x3ff00000) { /* |x| >= 1 */
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if(((ix-0x3ff00000)|__LO(x))==0) { /* |x|==1 */
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if(hx>0) return 0.0; /* acos(1) = 0 */
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else return pi+2.0*pio2_lo; /* acos(-1)= pi */
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}
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return (x-x)/(x-x); /* acos(|x|>1) is NaN */
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}
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if(ix<0x3fe00000) { /* |x| < 0.5 */
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if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
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z = x*x;
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p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
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q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
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r = p/q;
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return pio2_hi - (x - (pio2_lo-x*r));
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} else if (hx<0) { /* x < -0.5 */
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z = (one+x)*0.5;
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p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
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q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
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s = sqrt(z);
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r = p/q;
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w = r*s-pio2_lo;
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return pi - 2.0*(s+w);
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} else { /* x > 0.5 */
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z = (one-x)*0.5;
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s = sqrt(z);
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df = s;
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__LO(df) = 0;
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c = (z-df*df)/(s+df);
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p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
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q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
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r = p/q;
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w = r*s+c;
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return 2.0*(df+w);
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}
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}
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double z, p, q, r, w, s, c, df;
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int hx, ix;
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hx = __HI (x);
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ix = hx & 0x7fffffff;
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if (ix >= 0x3ff00000) /* |x| >= 1 */
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{
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if (((ix - 0x3ff00000) | __LO (x)) == 0) /* |x| == 1 */
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{
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if (hx > 0) /* acos(1) = 0 */
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{
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return 0.0;
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}
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else /* acos(-1) = pi */
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{
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return pi + 2.0 * pio2_lo;
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}
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}
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return (x - x) / (x - x); /* acos(|x|>1) is NaN */
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}
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if (ix < 0x3fe00000) /* |x| < 0.5 */
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{
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if (ix <= 0x3c600000) /* if |x| < 2**-57 */
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{
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return pio2_hi + pio2_lo;
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}
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z = x * x;
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p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
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q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
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r = p / q;
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return pio2_hi - (x - (pio2_lo - x * r));
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}
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else if (hx < 0) /* x < -0.5 */
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{
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z = (one + x) * 0.5;
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p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
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q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
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s = sqrt (z);
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r = p / q;
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w = r * s - pio2_lo;
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return pi - 2.0 * (s + w);
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}
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else /* x > 0.5 */
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{
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z = (one - x) * 0.5;
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s = sqrt (z);
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df = s;
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__LO (df) = 0;
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c = (z - df * df) / (s + df);
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p = z * (pS0 + z * (pS1 + z * (pS2 + z * (pS3 + z * (pS4 + z * pS5)))));
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q = one + z * (qS1 + z * (qS2 + z * (qS3 + z * qS4)));
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r = p / q;
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w = r * s + c;
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return 2.0 * (df + w);
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}
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} /* acos */
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156
third-party/fdlibm/s_asin.c
vendored
156
third-party/fdlibm/s_asin.c
vendored
@ -6,42 +6,41 @@
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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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* 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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/* asin(x)
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* Method :
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* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
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* we approximate asin(x) on [0,0.5] by
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* asin(x) = x + x*x^2*R(x^2)
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* where
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* R(x^2) is a rational approximation of (asin(x)-x)/x^3
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* and its remez error is bounded by
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* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
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*
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* For x in [0.5,1]
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* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
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* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
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* then for x>0.98
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
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* For x<=0.98, let pio4_hi = pio2_hi/2, then
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* f = hi part of s;
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* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
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* and
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
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* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
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* Method:
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* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
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* we approximate asin(x) on [0,0.5] by
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* asin(x) = x + x*x^2*R(x^2)
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* where
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* R(x^2) is a rational approximation of (asin(x)-x)/x^3
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* and its remez error is bounded by
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* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
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*
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* For x in [0.5,1]
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* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
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* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
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* then for x>0.98
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
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* For x<=0.98, let pio4_hi = pio2_hi/2, then
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* f = hi part of s;
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* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
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* and
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
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* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
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*
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* Special cases:
|
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
|
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*
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
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*/
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#include "fdlibm.h"
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#define one 1.00000000000000000000e+00 /* 0x3FF00000, 0x00000000 */
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@ -49,7 +48,7 @@
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#define pio2_hi 1.57079632679489655800e+00 /* 0x3FF921FB, 0x54442D18 */
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#define pio2_lo 6.12323399573676603587e-17 /* 0x3C91A626, 0x33145C07 */
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#define pio4_hi 7.85398163397448278999e-01 /* 0x3FE921FB, 0x54442D18 */
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/* coefficient for R(x^2) */
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/* coefficient for R(x^2) */
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#define pS0 1.66666666666666657415e-01 /* 0x3FC55555, 0x55555555 */
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#define pS1 -3.25565818622400915405e-01 /* 0xBFD4D612, 0x03EB6F7D */
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#define pS2 2.01212532134862925881e-01 /* 0x3FC9C155, 0x0E884455 */
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@ -61,44 +60,67 @@
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#define qS3 -6.88283971605453293030e-01 /* 0xBFE6066C, 0x1B8D0159 */
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#define qS4 7.70381505559019352791e-02 /* 0x3FB3B8C5, 0xB12E9282 */
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double asin(double x)
|
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double
|
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asin (double x)
|
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{
|
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double t = 0,w,p,q,c,r,s;
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int hx,ix;
|
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hx = __HI(x);
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ix = hx&0x7fffffff;
|
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if(ix>= 0x3ff00000) { /* |x|>= 1 */
|
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if(((ix-0x3ff00000)|__LO(x))==0)
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/* asin(1)=+-pi/2 with inexact */
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return x*pio2_hi+x*pio2_lo;
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return (x-x)/(x-x); /* asin(|x|>1) is NaN */
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} else if (ix<0x3fe00000) { /* |x|<0.5 */
|
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if(ix<0x3e400000) { /* if |x| < 2**-27 */
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if(huge+x>one) return x;/* return x with inexact if x!=0*/
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} else
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t = x*x;
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p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
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q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
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w = p/q;
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return x+x*w;
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}
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/* 1> |x|>= 0.5 */
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w = one-fabs(x);
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t = w*0.5;
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p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
|
||||
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
|
||||
s = sqrt(t);
|
||||
if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
|
||||
w = p/q;
|
||||
t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
|
||||
} else {
|
||||
w = s;
|
||||
__LO(w) = 0;
|
||||
c = (t-w*w)/(s+w);
|
||||
r = p/q;
|
||||
p = 2.0*s*r-(pio2_lo-2.0*c);
|
||||
q = pio4_hi-2.0*w;
|
||||
t = pio4_hi-(p-q);
|
||||
}
|
||||
if(hx>0) return t; else return -t;
|
||||
}
|
||||
double t = 0, w, p, q, c, r, s;
|
||||
int hx, ix;
|
||||
|
||||
hx = __HI (x);
|
||||
ix = hx & 0x7fffffff;
|
||||
if (ix >= 0x3ff00000) /* |x| >= 1 */
|
||||
{
|
||||
if (((ix - 0x3ff00000) | __LO (x)) == 0) /* asin(1) = +-pi/2 with inexact */
|
||||
{
|
||||
return x * pio2_hi + x * pio2_lo;
|
||||
}
|
||||
return (x - x) / (x - x); /* asin(|x|>1) is NaN */
|
||||
}
|
||||
else if (ix < 0x3fe00000) /* |x| < 0.5 */
|
||||
{
|
||||
if (ix < 0x3e400000) /* if |x| < 2**-27 */
|
||||
{
|
||||
if (huge + x > one) /* return x with inexact if x != 0 */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
t = x * x;
|
||||
}
|
||||
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
|
||||
q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
|
||||
w = p / q;
|
||||
return x + x * w;
|
||||
}
|
||||
/* 1 > |x| >= 0.5 */
|
||||
w = one - fabs (x);
|
||||
t = w * 0.5;
|
||||
p = t * (pS0 + t * (pS1 + t * (pS2 + t * (pS3 + t * (pS4 + t * pS5)))));
|
||||
q = one + t * (qS1 + t * (qS2 + t * (qS3 + t * qS4)));
|
||||
s = sqrt (t);
|
||||
if (ix >= 0x3FEF3333) /* if |x| > 0.975 */
|
||||
{
|
||||
w = p / q;
|
||||
t = pio2_hi - (2.0 * (s + s * w) - pio2_lo);
|
||||
}
|
||||
else
|
||||
{
|
||||
w = s;
|
||||
__LO (w) = 0;
|
||||
c = (t - w * w) / (s + w);
|
||||
r = p / q;
|
||||
p = 2.0 * s * r - (pio2_lo - 2.0 * c);
|
||||
q = pio4_hi - 2.0 * w;
|
||||
t = pio4_hi - (p - q);
|
||||
}
|
||||
if (hx > 0)
|
||||
{
|
||||
return t;
|
||||
}
|
||||
else
|
||||
{
|
||||
return -t;
|
||||
}
|
||||
} /* asin */
|
||||
|
||||
133
third-party/fdlibm/s_atan.c
vendored
133
third-party/fdlibm/s_atan.c
vendored
@ -6,14 +6,14 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*
|
||||
*/
|
||||
|
||||
/* atan(x)
|
||||
* Method
|
||||
*
|
||||
* Method:
|
||||
* 1. Reduce x to positive by atan(x) = -atan(-x).
|
||||
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
|
||||
* is further reduced to one of the following intervals and the
|
||||
@ -34,14 +34,16 @@
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
static const double atanhi[] = {
|
||||
static const double atanhi[] =
|
||||
{
|
||||
4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
|
||||
7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
|
||||
9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
|
||||
1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
|
||||
};
|
||||
|
||||
static const double atanlo[] = {
|
||||
static const double atanlo[] =
|
||||
{
|
||||
2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
|
||||
3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
|
||||
1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
|
||||
@ -63,48 +65,83 @@ static const double atanlo[] = {
|
||||
#define one 1.0
|
||||
#define huge 1.0e300
|
||||
|
||||
double atan(double x)
|
||||
double
|
||||
atan (double x)
|
||||
{
|
||||
double w,s1,s2,z;
|
||||
int ix,hx,id;
|
||||
double w, s1, s2, z;
|
||||
int ix, hx, id;
|
||||
|
||||
hx = __HI(x);
|
||||
ix = hx&0x7fffffff;
|
||||
if(ix>=0x44100000) { /* if |x| >= 2^66 */
|
||||
if(ix>0x7ff00000||
|
||||
(ix==0x7ff00000&&(__LO(x)!=0)))
|
||||
return x+x; /* NaN */
|
||||
if(hx>0) return atanhi[3]+atanlo[3];
|
||||
else return -atanhi[3]-atanlo[3];
|
||||
} if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
|
||||
if (ix < 0x3e200000) { /* |x| < 2^-29 */
|
||||
if(huge+x>one) return x; /* raise inexact */
|
||||
}
|
||||
id = -1;
|
||||
} else {
|
||||
x = fabs(x);
|
||||
if (ix < 0x3ff30000) { /* |x| < 1.1875 */
|
||||
if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
|
||||
id = 0; x = (2.0*x-one)/(2.0+x);
|
||||
} else { /* 11/16<=|x|< 19/16 */
|
||||
id = 1; x = (x-one)/(x+one);
|
||||
}
|
||||
} else {
|
||||
if (ix < 0x40038000) { /* |x| < 2.4375 */
|
||||
id = 2; x = (x-1.5)/(one+1.5*x);
|
||||
} else { /* 2.4375 <= |x| < 2^66 */
|
||||
id = 3; x = -1.0/x;
|
||||
}
|
||||
}}
|
||||
/* end of argument reduction */
|
||||
z = x*x;
|
||||
w = z*z;
|
||||
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
|
||||
s1 = z*(aT0+w*(aT2+w*(aT4+w*(aT6+w*(aT8+w*aT10)))));
|
||||
s2 = w*(aT1+w*(aT3+w*(aT5+w*(aT7+w*aT9))));
|
||||
if (id<0) return x - x*(s1+s2);
|
||||
else {
|
||||
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
|
||||
return (hx<0)? -z:z;
|
||||
}
|
||||
}
|
||||
hx = __HI (x);
|
||||
ix = hx & 0x7fffffff;
|
||||
if (ix >= 0x44100000) /* if |x| >= 2^66 */
|
||||
{
|
||||
if (ix > 0x7ff00000 || (ix == 0x7ff00000 && (__LO (x) != 0)))
|
||||
{
|
||||
return x + x; /* NaN */
|
||||
}
|
||||
if (hx > 0)
|
||||
{
|
||||
return atanhi[3] + atanlo[3];
|
||||
}
|
||||
else
|
||||
{
|
||||
return -atanhi[3] - atanlo[3];
|
||||
}
|
||||
}
|
||||
if (ix < 0x3fdc0000) /* |x| < 0.4375 */
|
||||
{
|
||||
if (ix < 0x3e200000) /* |x| < 2^-29 */
|
||||
{
|
||||
if (huge + x > one) /* raise inexact */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
}
|
||||
id = -1;
|
||||
}
|
||||
else
|
||||
{
|
||||
x = fabs (x);
|
||||
if (ix < 0x3ff30000) /* |x| < 1.1875 */
|
||||
{
|
||||
if (ix < 0x3fe60000) /* 7/16 <= |x| < 11/16 */
|
||||
{
|
||||
id = 0;
|
||||
x = (2.0 * x - one) / (2.0 + x);
|
||||
}
|
||||
else /* 11/16 <= |x| < 19/16 */
|
||||
{
|
||||
id = 1;
|
||||
x = (x - one) / (x + one);
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
if (ix < 0x40038000) /* |x| < 2.4375 */
|
||||
{
|
||||
id = 2;
|
||||
x = (x - 1.5) / (one + 1.5 * x);
|
||||
}
|
||||
else /* 2.4375 <= |x| < 2^66 */
|
||||
{
|
||||
id = 3;
|
||||
x = -1.0 / x;
|
||||
}
|
||||
}
|
||||
}
|
||||
/* end of argument reduction */
|
||||
z = x * x;
|
||||
w = z * z;
|
||||
/* break sum from i=0 to 10 aT[i] z**(i+1) into odd and even poly */
|
||||
s1 = z * (aT0 + w * (aT2 + w * (aT4 + w * (aT6 + w * (aT8 + w * aT10)))));
|
||||
s2 = w * (aT1 + w * (aT3 + w * (aT5 + w * (aT7 + w * aT9))));
|
||||
if (id < 0)
|
||||
{
|
||||
return x - x * (s1 + s2);
|
||||
}
|
||||
else
|
||||
{
|
||||
z = atanhi[id] - ((x * (s1 + s2) - atanlo[id]) - x);
|
||||
return (hx < 0) ? -z : z;
|
||||
}
|
||||
} /* atan */
|
||||
|
||||
231
third-party/fdlibm/s_atan2.c
vendored
231
third-party/fdlibm/s_atan2.c
vendored
@ -6,31 +6,30 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*
|
||||
*/
|
||||
|
||||
/* atan2(y,x)
|
||||
* Method :
|
||||
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
|
||||
* 2. Reduce x to positive by (if x and y are unexceptional):
|
||||
* ARG (x+iy) = arctan(y/x) ... if x > 0,
|
||||
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
|
||||
*
|
||||
* Method:
|
||||
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
|
||||
* 2. Reduce x to positive by (if x and y are unexceptional):
|
||||
* ARG (x+iy) = arctan(y/x) ... if x > 0,
|
||||
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
|
||||
*
|
||||
* Special cases:
|
||||
*
|
||||
* ATAN2((anything), NaN ) is NaN;
|
||||
* ATAN2(NAN , (anything) ) is NaN;
|
||||
* ATAN2(+-0, +(anything but NaN)) is +-0 ;
|
||||
* ATAN2(+-0, -(anything but NaN)) is +-pi ;
|
||||
* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
|
||||
* ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
|
||||
* ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
|
||||
* ATAN2(+-INF,+INF ) is +-pi/4 ;
|
||||
* ATAN2(+-INF,-INF ) is +-3pi/4;
|
||||
* ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
|
||||
* ATAN2((anything), NaN ) is NaN;
|
||||
* ATAN2(NAN , (anything) ) is NaN;
|
||||
* ATAN2(+-0, +(anything but NaN)) is +-0 ;
|
||||
* ATAN2(+-0, -(anything but NaN)) is +-pi ;
|
||||
* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
|
||||
* ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
|
||||
* ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
|
||||
* ATAN2(+-INF,+INF ) is +-pi/4 ;
|
||||
* ATAN2(+-INF,-INF ) is +-3pi/4;
|
||||
* ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
@ -48,66 +47,142 @@
|
||||
#define pi 3.1415926535897931160E+00 /* 0x400921FB, 0x54442D18 */
|
||||
#define pi_lo 1.2246467991473531772E-16 /* 0x3CA1A626, 0x33145C07 */
|
||||
|
||||
double atan2(double y, double x)
|
||||
{
|
||||
double z;
|
||||
int k,m,hx,hy,ix,iy;
|
||||
unsigned lx,ly;
|
||||
double
|
||||
atan2 (double y, double x)
|
||||
{
|
||||
double z;
|
||||
int k, m, hx, hy, ix, iy;
|
||||
unsigned lx, ly;
|
||||
|
||||
hx = __HI(x); ix = hx&0x7fffffff;
|
||||
lx = __LO(x);
|
||||
hy = __HI(y); iy = hy&0x7fffffff;
|
||||
ly = __LO(y);
|
||||
if(((ix|((lx|-lx)>>31))>0x7ff00000)||
|
||||
((iy|((ly|-ly)>>31))>0x7ff00000)) /* x or y is NaN */
|
||||
return x+y;
|
||||
if((hx-0x3ff00000|lx)==0) return atan(y); /* x=1.0 */
|
||||
m = ((hy>>31)&1)|((hx>>30)&2); /* 2*sign(x)+sign(y) */
|
||||
hx = __HI (x);
|
||||
ix = hx & 0x7fffffff;
|
||||
lx = __LO (x);
|
||||
hy = __HI (y);
|
||||
iy = hy & 0x7fffffff;
|
||||
ly = __LO (y);
|
||||
if (((ix | ((lx | -lx) >> 31)) > 0x7ff00000) || ((iy | ((ly | -ly) >> 31)) > 0x7ff00000)) /* x or y is NaN */
|
||||
{
|
||||
return x + y;
|
||||
}
|
||||
if ((hx - 0x3ff00000 | lx) == 0) /* x = 1.0 */
|
||||
{
|
||||
return atan (y);
|
||||
}
|
||||
m = ((hy >> 31) & 1) | ((hx >> 30) & 2); /* 2 * sign(x) + sign(y) */
|
||||
|
||||
/* when y = 0 */
|
||||
if((iy|ly)==0) {
|
||||
switch(m) {
|
||||
case 0:
|
||||
case 1: return y; /* atan(+-0,+anything)=+-0 */
|
||||
case 2: return pi+tiny;/* atan(+0,-anything) = pi */
|
||||
case 3: return -pi-tiny;/* atan(-0,-anything) =-pi */
|
||||
}
|
||||
}
|
||||
/* when x = 0 */
|
||||
if((ix|lx)==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
|
||||
|
||||
/* when x is INF */
|
||||
if(ix==0x7ff00000) {
|
||||
if(iy==0x7ff00000) {
|
||||
switch(m) {
|
||||
case 0: return pi_o_4+tiny;/* atan(+INF,+INF) */
|
||||
case 1: return -pi_o_4-tiny;/* atan(-INF,+INF) */
|
||||
case 2: return 3.0*pi_o_4+tiny;/*atan(+INF,-INF)*/
|
||||
case 3: return -3.0*pi_o_4-tiny;/*atan(-INF,-INF)*/
|
||||
}
|
||||
} else {
|
||||
switch(m) {
|
||||
case 0: return zero ; /* atan(+...,+INF) */
|
||||
case 1: return -zero ; /* atan(-...,+INF) */
|
||||
case 2: return pi+tiny ; /* atan(+...,-INF) */
|
||||
case 3: return -pi-tiny ; /* atan(-...,-INF) */
|
||||
}
|
||||
}
|
||||
}
|
||||
/* when y is INF */
|
||||
if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
|
||||
/* when y = 0 */
|
||||
if ((iy | ly) == 0)
|
||||
{
|
||||
switch (m)
|
||||
{
|
||||
case 0:
|
||||
case 1:
|
||||
{
|
||||
return y; /* atan(+-0,+anything) = +-0 */
|
||||
}
|
||||
case 2:
|
||||
{
|
||||
return pi + tiny; /* atan(+0,-anything) = pi */
|
||||
}
|
||||
case 3:
|
||||
{
|
||||
return -pi - tiny; /* atan(-0,-anything) = -pi */
|
||||
}
|
||||
}
|
||||
}
|
||||
/* when x = 0 */
|
||||
if ((ix | lx) == 0)
|
||||
{
|
||||
return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
|
||||
}
|
||||
|
||||
/* compute y/x */
|
||||
k = (iy-ix)>>20;
|
||||
if(k > 60) z=pi_o_2+0.5*pi_lo; /* |y/x| > 2**60 */
|
||||
else if(hx<0&&k<-60) z=0.0; /* |y|/x < -2**60 */
|
||||
else z=atan(fabs(y/x)); /* safe to do y/x */
|
||||
switch (m) {
|
||||
case 0: return z ; /* atan(+,+) */
|
||||
case 1: __HI(z) ^= 0x80000000;
|
||||
return z ; /* atan(-,+) */
|
||||
case 2: return pi-(z-pi_lo);/* atan(+,-) */
|
||||
default: /* case 3 */
|
||||
return (z-pi_lo)-pi;/* atan(-,-) */
|
||||
}
|
||||
}
|
||||
/* when x is INF */
|
||||
if (ix == 0x7ff00000)
|
||||
{
|
||||
if (iy == 0x7ff00000)
|
||||
{
|
||||
switch (m)
|
||||
{
|
||||
case 0: /* atan(+INF,+INF) */
|
||||
{
|
||||
return pi_o_4 + tiny;
|
||||
}
|
||||
case 1: /* atan(-INF,+INF) */
|
||||
{
|
||||
return -pi_o_4 - tiny;
|
||||
}
|
||||
case 2: /* atan(+INF,-INF) */
|
||||
{
|
||||
return 3.0 * pi_o_4 + tiny;
|
||||
}
|
||||
case 3: /* atan(-INF,-INF) */
|
||||
{
|
||||
return -3.0 * pi_o_4 - tiny;
|
||||
}
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
switch (m)
|
||||
{
|
||||
case 0: /* atan(+...,+INF) */
|
||||
{
|
||||
return zero;
|
||||
}
|
||||
case 1: /* atan(-...,+INF) */
|
||||
{
|
||||
return -zero;
|
||||
}
|
||||
case 2: /* atan(+...,-INF) */
|
||||
{
|
||||
return pi + tiny;
|
||||
}
|
||||
case 3: /* atan(-...,-INF) */
|
||||
{
|
||||
return -pi - tiny;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
/* when y is INF */
|
||||
if (iy == 0x7ff00000)
|
||||
{
|
||||
return (hy < 0) ? -pi_o_2 - tiny : pi_o_2 + tiny;
|
||||
}
|
||||
|
||||
/* compute y / x */
|
||||
k = (iy - ix) >> 20;
|
||||
if (k > 60) /* |y / x| > 2**60 */
|
||||
{
|
||||
z = pi_o_2 + 0.5 * pi_lo;
|
||||
}
|
||||
else if (hx < 0 && k < -60) /* |y| / x < -2**60 */
|
||||
{
|
||||
z = 0.0;
|
||||
}
|
||||
else /* safe to do y / x */
|
||||
{
|
||||
z = atan (fabs (y / x));
|
||||
}
|
||||
switch (m)
|
||||
{
|
||||
case 0: /* atan(+,+) */
|
||||
{
|
||||
return z;
|
||||
}
|
||||
case 1: /* atan(-,+) */
|
||||
{
|
||||
__HI (z) ^= 0x80000000;
|
||||
return z;
|
||||
}
|
||||
case 2: /* atan(+,-) */
|
||||
{
|
||||
return pi - (z - pi_lo);
|
||||
}
|
||||
/* case 3: */
|
||||
default: /* atan(-,-) */
|
||||
{
|
||||
return (z - pi_lo) - pi;
|
||||
}
|
||||
}
|
||||
} /* atan2 */
|
||||
|
||||
140
third-party/fdlibm/s_ceil.c
vendored
140
third-party/fdlibm/s_ceil.c
vendored
@ -6,64 +6,110 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* ceil(x)
|
||||
/* ceil(x)
|
||||
* Return x rounded toward -inf to integral value
|
||||
*
|
||||
* Method:
|
||||
* Bit twiddling.
|
||||
* Bit twiddling.
|
||||
*
|
||||
* Exception:
|
||||
* Inexact flag raised if x not equal to ceil(x).
|
||||
* Inexact flag raised if x not equal to ceil(x).
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
#define huge 1.0e300
|
||||
|
||||
double ceil(double x)
|
||||
double
|
||||
ceil (double x)
|
||||
{
|
||||
int i0,i1,j0;
|
||||
unsigned i,j;
|
||||
i0 = __HI(x);
|
||||
i1 = __LO(x);
|
||||
j0 = ((i0>>20)&0x7ff)-0x3ff;
|
||||
if(j0<20) {
|
||||
if(j0<0) { /* raise inexact if x != 0 */
|
||||
if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */
|
||||
if(i0<0) {i0=0x80000000;i1=0;}
|
||||
else if((i0|i1)!=0) { i0=0x3ff00000;i1=0;}
|
||||
}
|
||||
} else {
|
||||
i = (0x000fffff)>>j0;
|
||||
if(((i0&i)|i1)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0>0) i0 += (0x00100000)>>j0;
|
||||
i0 &= (~i); i1=0;
|
||||
}
|
||||
}
|
||||
} else if (j0>51) {
|
||||
if(j0==0x400) return x+x; /* inf or NaN */
|
||||
else return x; /* x is integral */
|
||||
} else {
|
||||
i = ((unsigned)(0xffffffff))>>(j0-20);
|
||||
if((i1&i)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0>0) {
|
||||
if(j0==20) i0+=1;
|
||||
else {
|
||||
j = i1 + (1<<(52-j0));
|
||||
if(j<i1) i0+=1; /* got a carry */
|
||||
i1 = j;
|
||||
}
|
||||
}
|
||||
i1 &= (~i);
|
||||
}
|
||||
}
|
||||
__HI(x) = i0;
|
||||
__LO(x) = i1;
|
||||
return x;
|
||||
}
|
||||
int i0, i1, j0;
|
||||
unsigned i, j;
|
||||
|
||||
i0 = __HI (x);
|
||||
i1 = __LO (x);
|
||||
j0 = ((i0 >> 20) & 0x7ff) - 0x3ff;
|
||||
if (j0 < 20)
|
||||
{
|
||||
if (j0 < 0) /* raise inexact if x != 0 */
|
||||
{
|
||||
if (huge + x > 0.0) /* return 0 * sign(x) if |x| < 1 */
|
||||
{
|
||||
if (i0 < 0)
|
||||
{
|
||||
i0 = 0x80000000;
|
||||
i1 = 0;
|
||||
}
|
||||
else if ((i0 | i1) != 0)
|
||||
{
|
||||
i0 = 0x3ff00000;
|
||||
i1 = 0;
|
||||
}
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
i = (0x000fffff) >> j0;
|
||||
if (((i0 & i) | i1) == 0) /* x is integral */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
if (huge + x > 0.0) /* raise inexact flag */
|
||||
{
|
||||
if (i0 > 0)
|
||||
{
|
||||
i0 += (0x00100000) >> j0;
|
||||
}
|
||||
i0 &= (~i);
|
||||
i1 = 0;
|
||||
}
|
||||
}
|
||||
}
|
||||
else if (j0 > 51)
|
||||
{
|
||||
if (j0 == 0x400) /* inf or NaN */
|
||||
{
|
||||
return x + x;
|
||||
}
|
||||
else /* x is integral */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
i = ((unsigned) (0xffffffff)) >> (j0 - 20);
|
||||
if ((i1 & i) == 0) /* x is integral */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
if (huge + x > 0.0) /* raise inexact flag */
|
||||
{
|
||||
if (i0 > 0)
|
||||
{
|
||||
if (j0 == 20)
|
||||
{
|
||||
i0 += 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
j = i1 + (1 << (52 - j0));
|
||||
if (j < i1) /* got a carry */
|
||||
{
|
||||
i0 += 1;
|
||||
}
|
||||
i1 = j;
|
||||
}
|
||||
}
|
||||
i1 &= (~i);
|
||||
}
|
||||
}
|
||||
__HI (x) = i0;
|
||||
__LO (x) = i1;
|
||||
return x;
|
||||
} /* ceil */
|
||||
|
||||
15
third-party/fdlibm/s_copysign.c
vendored
15
third-party/fdlibm/s_copysign.c
vendored
@ -6,21 +6,20 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* copysign(double x, double y)
|
||||
* copysign(x,y) returns a value with the magnitude of x and
|
||||
/* copysign(x,y) returns a value with the magnitude of x and
|
||||
* with the sign bit of y.
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
double copysign(double x, double y)
|
||||
double
|
||||
copysign (double x, double y)
|
||||
{
|
||||
__HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000);
|
||||
return x;
|
||||
}
|
||||
__HI (x) = (__HI (x) & 0x7fffffff) | (__HI (y) & 0x80000000);
|
||||
return x;
|
||||
} /* copysign */
|
||||
|
||||
235
third-party/fdlibm/s_exp.c
vendored
235
third-party/fdlibm/s_exp.c
vendored
@ -5,7 +5,7 @@
|
||||
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
@ -13,58 +13,58 @@
|
||||
/* exp(x)
|
||||
* Returns the exponential of x.
|
||||
*
|
||||
* Method
|
||||
* Method:
|
||||
* 1. Argument reduction:
|
||||
* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
|
||||
* Given x, find r and integer k such that
|
||||
* Given x, find r and integer k such that
|
||||
*
|
||||
* x = k*ln2 + r, |r| <= 0.5*ln2.
|
||||
*
|
||||
* Here r will be represented as r = hi-lo for better
|
||||
* accuracy.
|
||||
* accuracy.
|
||||
*
|
||||
* 2. Approximation of exp(r) by a special rational function on
|
||||
* the interval [0,0.34658]:
|
||||
* Write
|
||||
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
|
||||
* the interval [0,0.34658]:
|
||||
* Write
|
||||
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
|
||||
* We use a special Remes algorithm on [0,0.34658] to generate
|
||||
* a polynomial of degree 5 to approximate R. The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-59. In
|
||||
* other words,
|
||||
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
|
||||
* (where z=r*r, and the values of P1 to P5 are listed below)
|
||||
* and
|
||||
* | 5 | -59
|
||||
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
|
||||
* | |
|
||||
* The computation of exp(r) thus becomes
|
||||
* a polynomial of degree 5 to approximate R. The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-59. In
|
||||
* other words,
|
||||
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
|
||||
* (where z=r*r, and the values of P1 to P5 are listed below)
|
||||
* and
|
||||
* | 5 | -59
|
||||
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
|
||||
* | |
|
||||
* The computation of exp(r) thus becomes
|
||||
* 2*r
|
||||
* exp(r) = 1 + -------
|
||||
* R - r
|
||||
* exp(r) = 1 + -------
|
||||
* R - r
|
||||
* r*R1(r)
|
||||
* = 1 + r + ----------- (for better accuracy)
|
||||
* 2 - R1(r)
|
||||
* where
|
||||
* 2 4 10
|
||||
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
|
||||
* = 1 + r + ----------- (for better accuracy)
|
||||
* 2 - R1(r)
|
||||
* where
|
||||
* 2 4 10
|
||||
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
|
||||
*
|
||||
* 3. Scale back to obtain exp(x):
|
||||
* From step 1, we have
|
||||
* exp(x) = 2^k * exp(r)
|
||||
* From step 1, we have
|
||||
* exp(x) = 2^k * exp(r)
|
||||
*
|
||||
* Special cases:
|
||||
* exp(INF) is INF, exp(NaN) is NaN;
|
||||
* exp(-INF) is 0, and
|
||||
* for finite argument, only exp(0)=1 is exact.
|
||||
* exp(INF) is INF, exp(NaN) is NaN;
|
||||
* exp(-INF) is 0, and
|
||||
* for finite argument, only exp(0)=1 is exact.
|
||||
*
|
||||
* Accuracy:
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
*
|
||||
* Misc. info.
|
||||
* For IEEE double
|
||||
* if x > 7.09782712893383973096e+02 then exp(x) overflow
|
||||
* if x < -7.45133219101941108420e+02 then exp(x) underflow
|
||||
* Misc. info:
|
||||
* For IEEE double
|
||||
* if x > 7.09782712893383973096e+02 then exp(x) overflow
|
||||
* if x < -7.45133219101941108420e+02 then exp(x) underflow
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
@ -75,73 +75,118 @@
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
static const double
|
||||
halF[2] = {0.5,-0.5,},
|
||||
ln2HI[2] = { 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
||||
-6.93147180369123816490e-01,}, /* 0xbfe62e42, 0xfee00000 */
|
||||
ln2LO[2] = { 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
||||
-1.90821492927058770002e-10,}; /* 0xbdea39ef, 0x35793c76 */
|
||||
static const double halF[2] =
|
||||
{
|
||||
0.5,
|
||||
-0.5,
|
||||
};
|
||||
static const double ln2HI[2] =
|
||||
{
|
||||
6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
||||
-6.93147180369123816490e-01, /* 0xbfe62e42, 0xfee00000 */
|
||||
};
|
||||
static const double ln2LO[2] =
|
||||
{
|
||||
1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
||||
-1.90821492927058770002e-10, /* 0xbdea39ef, 0x35793c76 */
|
||||
};
|
||||
|
||||
#define one 1.0
|
||||
#define huge 1.0e+300
|
||||
#define twom1000 9.33263618503218878990e-302 /* 2**-1000=0x01700000,0 */
|
||||
#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
|
||||
#define u_threshold -7.45133219101941108420e+02 /* 0xc0874910, 0xD52D3051 */
|
||||
#define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */
|
||||
#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
|
||||
#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
|
||||
#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
|
||||
#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
|
||||
#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */
|
||||
#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
|
||||
#define u_threshold -7.45133219101941108420e+02 /* 0xc0874910, 0xD52D3051 */
|
||||
#define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */
|
||||
#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
|
||||
#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
|
||||
#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
|
||||
#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
|
||||
#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */
|
||||
|
||||
double exp(double x) /* default IEEE double exp */
|
||||
double
|
||||
exp (double x) /* default IEEE double exp */
|
||||
{
|
||||
double y,hi,lo,c,t;
|
||||
int k = 0,xsb;
|
||||
unsigned hx;
|
||||
double y, hi, lo, c, t;
|
||||
int k = 0, xsb;
|
||||
unsigned hx;
|
||||
|
||||
hx = __HI(x); /* high word of x */
|
||||
xsb = (hx>>31)&1; /* sign bit of x */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
hx = __HI (x); /* high word of x */
|
||||
xsb = (hx >> 31) & 1; /* sign bit of x */
|
||||
hx &= 0x7fffffff; /* high word of |x| */
|
||||
|
||||
/* filter out non-finite argument */
|
||||
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
||||
if(hx>=0x7ff00000) {
|
||||
if(((hx&0xfffff)|__LO(x))!=0)
|
||||
return x+x; /* NaN */
|
||||
else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
|
||||
}
|
||||
if(x > o_threshold) return huge*huge; /* overflow */
|
||||
if(x < u_threshold) return twom1000*twom1000; /* underflow */
|
||||
}
|
||||
/* filter out non-finite argument */
|
||||
if (hx >= 0x40862E42) /* if |x| >= 709.78... */
|
||||
{
|
||||
if (hx >= 0x7ff00000)
|
||||
{
|
||||
if (((hx & 0xfffff) | __LO (x)) != 0) /* NaN */
|
||||
{
|
||||
return x + x;
|
||||
}
|
||||
else /* exp(+-inf) = {inf,0} */
|
||||
{
|
||||
return (xsb == 0) ? x : 0.0;
|
||||
}
|
||||
}
|
||||
if (x > o_threshold) /* overflow */
|
||||
{
|
||||
return huge * huge;
|
||||
}
|
||||
if (x < u_threshold) /* underflow */
|
||||
{
|
||||
return twom1000 * twom1000;
|
||||
}
|
||||
}
|
||||
|
||||
/* argument reduction */
|
||||
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
||||
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
||||
hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
|
||||
} else {
|
||||
k = (int)(invln2*x+halF[xsb]);
|
||||
t = k;
|
||||
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
|
||||
lo = t*ln2LO[0];
|
||||
}
|
||||
x = hi - lo;
|
||||
}
|
||||
else if(hx < 0x3e300000) { /* when |x|<2**-28 */
|
||||
if(huge+x>one) return one+x;/* trigger inexact */
|
||||
}
|
||||
else k = 0;
|
||||
/* argument reduction */
|
||||
if (hx > 0x3fd62e42) /* if |x| > 0.5 ln2 */
|
||||
{
|
||||
if (hx < 0x3FF0A2B2) /* and |x| < 1.5 ln2 */
|
||||
{
|
||||
hi = x - ln2HI[xsb];
|
||||
lo = ln2LO[xsb];
|
||||
k = 1 - xsb - xsb;
|
||||
}
|
||||
else
|
||||
{
|
||||
k = (int) (invln2 * x + halF[xsb]);
|
||||
t = k;
|
||||
hi = x - t * ln2HI[0]; /* t * ln2HI is exact here */
|
||||
lo = t * ln2LO[0];
|
||||
}
|
||||
x = hi - lo;
|
||||
}
|
||||
else if (hx < 0x3e300000) /* when |x| < 2**-28 */
|
||||
{
|
||||
if (huge + x > one) /* trigger inexact */
|
||||
{
|
||||
return one + x;
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
k = 0;
|
||||
}
|
||||
|
||||
/* x is now in primary range */
|
||||
t = x*x;
|
||||
c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
if(k==0) return one-((x*c)/(c-2.0)-x);
|
||||
else y = one-((lo-(x*c)/(2.0-c))-hi);
|
||||
if(k >= -1021) {
|
||||
__HI(y) += (k<<20); /* add k to y's exponent */
|
||||
return y;
|
||||
} else {
|
||||
__HI(y) += ((k+1000)<<20);/* add k to y's exponent */
|
||||
return y*twom1000;
|
||||
}
|
||||
}
|
||||
/* x is now in primary range */
|
||||
t = x * x;
|
||||
c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
|
||||
if (k == 0)
|
||||
{
|
||||
return one - ((x * c) / (c - 2.0) - x);
|
||||
}
|
||||
else
|
||||
{
|
||||
y = one - ((lo - (x * c) / (2.0 - c)) - hi);
|
||||
}
|
||||
if (k >= -1021)
|
||||
{
|
||||
__HI (y) += (k << 20); /* add k to y's exponent */
|
||||
return y;
|
||||
}
|
||||
else
|
||||
{
|
||||
__HI (y) += ((k + 1000) << 20); /* add k to y's exponent */
|
||||
return y * twom1000;
|
||||
}
|
||||
} /* exp */
|
||||
|
||||
14
third-party/fdlibm/s_fabs.c
vendored
14
third-party/fdlibm/s_fabs.c
vendored
@ -6,19 +6,19 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* fabs(x) returns the absolute value of x.
|
||||
/* fabs(x) returns the absolute value of x.
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
double fabs(double x)
|
||||
double
|
||||
fabs (double x)
|
||||
{
|
||||
__HI(x) &= 0x7fffffff;
|
||||
return x;
|
||||
}
|
||||
__HI (x) &= 0x7fffffff;
|
||||
return x;
|
||||
} /* fabs */
|
||||
|
||||
17
third-party/fdlibm/s_finite.c
vendored
17
third-party/fdlibm/s_finite.c
vendored
@ -6,21 +6,22 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* finite(x) returns 1 is x is finite, else 0;
|
||||
/* finite(x) returns 1 is x is finite, else 0;
|
||||
* no branching!
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
int finite(double x)
|
||||
int
|
||||
finite (double x)
|
||||
{
|
||||
int hx;
|
||||
hx = __HI(x);
|
||||
return (unsigned)((hx&0x7fffffff)-0x7ff00000)>>31;
|
||||
}
|
||||
int hx;
|
||||
|
||||
hx = __HI (x);
|
||||
return (unsigned) ((hx & 0x7fffffff) - 0x7ff00000) >> 31;
|
||||
} /* finite */
|
||||
|
||||
140
third-party/fdlibm/s_floor.c
vendored
140
third-party/fdlibm/s_floor.c
vendored
@ -6,65 +6,109 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* floor(x)
|
||||
/* floor(x)
|
||||
* Return x rounded toward -inf to integral value
|
||||
*
|
||||
* Method:
|
||||
* Bit twiddling.
|
||||
* Bit twiddling.
|
||||
*
|
||||
* Exception:
|
||||
* Inexact flag raised if x not equal to floor(x).
|
||||
* Inexact flag raised if x not equal to floor(x).
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
#define huge 1.0e300
|
||||
|
||||
double floor(double x)
|
||||
double
|
||||
floor (double x)
|
||||
{
|
||||
int i0,i1,j0;
|
||||
unsigned i,j;
|
||||
i0 = __HI(x);
|
||||
i1 = __LO(x);
|
||||
j0 = ((i0>>20)&0x7ff)-0x3ff;
|
||||
if(j0<20) {
|
||||
if(j0<0) { /* raise inexact if x != 0 */
|
||||
if(huge+x>0.0) {/* return 0*sign(x) if |x|<1 */
|
||||
if(i0>=0) {i0=i1=0;}
|
||||
else if(((i0&0x7fffffff)|i1)!=0)
|
||||
{ i0=0xbff00000;i1=0;}
|
||||
}
|
||||
} else {
|
||||
i = (0x000fffff)>>j0;
|
||||
if(((i0&i)|i1)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0<0) i0 += (0x00100000)>>j0;
|
||||
i0 &= (~i); i1=0;
|
||||
}
|
||||
}
|
||||
} else if (j0>51) {
|
||||
if(j0==0x400) return x+x; /* inf or NaN */
|
||||
else return x; /* x is integral */
|
||||
} else {
|
||||
i = ((unsigned)(0xffffffff))>>(j0-20);
|
||||
if((i1&i)==0) return x; /* x is integral */
|
||||
if(huge+x>0.0) { /* raise inexact flag */
|
||||
if(i0<0) {
|
||||
if(j0==20) i0+=1;
|
||||
else {
|
||||
j = i1+(1<<(52-j0));
|
||||
if(j<i1) i0 +=1 ; /* got a carry */
|
||||
i1=j;
|
||||
}
|
||||
}
|
||||
i1 &= (~i);
|
||||
}
|
||||
}
|
||||
__HI(x) = i0;
|
||||
__LO(x) = i1;
|
||||
return x;
|
||||
}
|
||||
int i0, i1, j0;
|
||||
unsigned i, j;
|
||||
|
||||
i0 = __HI (x);
|
||||
i1 = __LO (x);
|
||||
j0 = ((i0 >> 20) & 0x7ff) - 0x3ff;
|
||||
if (j0 < 20)
|
||||
{
|
||||
if (j0 < 0) /* raise inexact if x != 0 */
|
||||
{
|
||||
if (huge + x > 0.0) /* return 0 * sign(x) if |x| < 1 */
|
||||
{
|
||||
if (i0 >= 0)
|
||||
{
|
||||
i0 = i1 = 0;
|
||||
}
|
||||
else if (((i0 & 0x7fffffff) | i1) != 0)
|
||||
{
|
||||
i0 = 0xbff00000;
|
||||
i1 = 0;
|
||||
}
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
i = (0x000fffff) >> j0;
|
||||
if (((i0 & i) | i1) == 0) /* x is integral */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
if (huge + x > 0.0) /* raise inexact flag */
|
||||
{
|
||||
if (i0 < 0)
|
||||
{
|
||||
i0 += (0x00100000) >> j0;
|
||||
}
|
||||
i0 &= (~i);
|
||||
i1 = 0;
|
||||
}
|
||||
}
|
||||
}
|
||||
else if (j0 > 51)
|
||||
{
|
||||
if (j0 == 0x400) /* inf or NaN */
|
||||
{
|
||||
return x + x;
|
||||
}
|
||||
else /* x is integral */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
i = ((unsigned) (0xffffffff)) >> (j0 - 20);
|
||||
if ((i1 & i) == 0) /* x is integral */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
if (huge + x > 0.0) /* raise inexact flag */
|
||||
{
|
||||
if (i0 < 0)
|
||||
{
|
||||
if (j0 == 20)
|
||||
{
|
||||
i0 += 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
j = i1 + (1 << (52 - j0));
|
||||
if (j < i1) /* got a carry */
|
||||
{
|
||||
i0 += 1;
|
||||
}
|
||||
i1 = j;
|
||||
}
|
||||
}
|
||||
i1 &= (~i);
|
||||
}
|
||||
}
|
||||
__HI (x) = i0;
|
||||
__LO (x) = i1;
|
||||
return x;
|
||||
} /* floor */
|
||||
|
||||
293
third-party/fdlibm/s_fmod.c
vendored
293
third-party/fdlibm/s_fmod.c
vendored
@ -11,124 +11,209 @@
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* fmod(x,y)
|
||||
/* fmod(x,y)
|
||||
* Return x mod y in exact arithmetic
|
||||
*
|
||||
* Method: shift and subtract
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
static const double
|
||||
Zero[] = {0.0, -0.0,};
|
||||
static const double Zero[] = { 0.0, -0.0, };
|
||||
|
||||
#define one 1.0
|
||||
|
||||
double fmod(double x, double y)
|
||||
double
|
||||
fmod (double x, double y)
|
||||
{
|
||||
int n,hx,hy,hz,ix,iy,sx,i;
|
||||
unsigned lx,ly,lz;
|
||||
int n, hx, hy, hz, ix, iy, sx, i;
|
||||
unsigned lx, ly, lz;
|
||||
|
||||
hx = __HI(x); /* high word of x */
|
||||
lx = __LO(x); /* low word of x */
|
||||
hy = __HI(y); /* high word of y */
|
||||
ly = __LO(y); /* low word of y */
|
||||
sx = hx&0x80000000; /* sign of x */
|
||||
hx ^=sx; /* |x| */
|
||||
hy &= 0x7fffffff; /* |y| */
|
||||
hx = __HI (x); /* high word of x */
|
||||
lx = __LO (x); /* low word of x */
|
||||
hy = __HI (y); /* high word of y */
|
||||
ly = __LO (y); /* low word of y */
|
||||
sx = hx & 0x80000000; /* sign of x */
|
||||
hx ^= sx; /* |x| */
|
||||
hy &= 0x7fffffff; /* |y| */
|
||||
|
||||
/* purge off exception values */
|
||||
if((hy|ly)==0||(hx>=0x7ff00000)|| /* y=0,or x not finite */
|
||||
((hy|((ly|-ly)>>31))>0x7ff00000)) /* or y is NaN */
|
||||
return (x*y)/(x*y);
|
||||
if(hx<=hy) {
|
||||
if((hx<hy)||(lx<ly)) return x; /* |x|<|y| return x */
|
||||
if(lx==ly)
|
||||
return Zero[(unsigned)sx>>31]; /* |x|=|y| return x*0*/
|
||||
}
|
||||
/* purge off exception values */
|
||||
if ((hy | ly) == 0 || (hx >= 0x7ff00000) || /* y = 0, or x not finite */
|
||||
((hy | ((ly | -ly) >> 31)) > 0x7ff00000)) /* or y is NaN */
|
||||
{
|
||||
return (x * y) / (x * y);
|
||||
}
|
||||
if (hx <= hy)
|
||||
{
|
||||
if ((hx < hy) || (lx < ly)) /* |x| < |y| return x */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
if (lx == ly) /* |x| = |y| return x * 0 */
|
||||
{
|
||||
return Zero[(unsigned) sx >> 31];
|
||||
}
|
||||
}
|
||||
|
||||
/* determine ix = ilogb(x) */
|
||||
if(hx<0x00100000) { /* subnormal x */
|
||||
if(hx==0) {
|
||||
for (ix = -1043, i=lx; i>0; i<<=1) ix -=1;
|
||||
} else {
|
||||
for (ix = -1022,i=(hx<<11); i>0; i<<=1) ix -=1;
|
||||
}
|
||||
} else ix = (hx>>20)-1023;
|
||||
/* determine ix = ilogb(x) */
|
||||
if (hx < 0x00100000) /* subnormal x */
|
||||
{
|
||||
if (hx == 0)
|
||||
{
|
||||
for (ix = -1043, i = lx; i > 0; i <<= 1)
|
||||
{
|
||||
ix -= 1;
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
for (ix = -1022, i = (hx << 11); i > 0; i <<= 1)
|
||||
{
|
||||
ix -= 1;
|
||||
}
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
ix = (hx >> 20) - 1023;
|
||||
}
|
||||
|
||||
/* determine iy = ilogb(y) */
|
||||
if(hy<0x00100000) { /* subnormal y */
|
||||
if(hy==0) {
|
||||
for (iy = -1043, i=ly; i>0; i<<=1) iy -=1;
|
||||
} else {
|
||||
for (iy = -1022,i=(hy<<11); i>0; i<<=1) iy -=1;
|
||||
}
|
||||
} else iy = (hy>>20)-1023;
|
||||
/* determine iy = ilogb(y) */
|
||||
if (hy < 0x00100000) /* subnormal y */
|
||||
{
|
||||
if (hy == 0)
|
||||
{
|
||||
for (iy = -1043, i = ly; i > 0; i <<= 1)
|
||||
{
|
||||
iy -= 1;
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
for (iy = -1022, i = (hy << 11); i > 0; i <<= 1)
|
||||
{
|
||||
iy -= 1;
|
||||
}
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
iy = (hy >> 20) - 1023;
|
||||
}
|
||||
|
||||
/* set up {hx,lx}, {hy,ly} and align y to x */
|
||||
if(ix >= -1022)
|
||||
hx = 0x00100000|(0x000fffff&hx);
|
||||
else { /* subnormal x, shift x to normal */
|
||||
n = -1022-ix;
|
||||
if(n<=31) {
|
||||
hx = (hx<<n)|(lx>>(32-n));
|
||||
lx <<= n;
|
||||
} else {
|
||||
hx = lx<<(n-32);
|
||||
lx = 0;
|
||||
}
|
||||
}
|
||||
if(iy >= -1022)
|
||||
hy = 0x00100000|(0x000fffff&hy);
|
||||
else { /* subnormal y, shift y to normal */
|
||||
n = -1022-iy;
|
||||
if(n<=31) {
|
||||
hy = (hy<<n)|(ly>>(32-n));
|
||||
ly <<= n;
|
||||
} else {
|
||||
hy = ly<<(n-32);
|
||||
ly = 0;
|
||||
}
|
||||
}
|
||||
/* set up {hx,lx}, {hy,ly} and align y to x */
|
||||
if (ix >= -1022)
|
||||
{
|
||||
hx = 0x00100000 | (0x000fffff & hx);
|
||||
}
|
||||
else /* subnormal x, shift x to normal */
|
||||
{
|
||||
n = -1022 - ix;
|
||||
if (n <= 31)
|
||||
{
|
||||
hx = (hx << n) | (lx >> (32 - n));
|
||||
lx <<= n;
|
||||
}
|
||||
else
|
||||
{
|
||||
hx = lx << (n - 32);
|
||||
lx = 0;
|
||||
}
|
||||
}
|
||||
if (iy >= -1022)
|
||||
{
|
||||
hy = 0x00100000 | (0x000fffff & hy);
|
||||
}
|
||||
else /* subnormal y, shift y to normal */
|
||||
{
|
||||
n = -1022 - iy;
|
||||
if (n <= 31)
|
||||
{
|
||||
hy = (hy << n) | (ly >> (32 - n));
|
||||
ly <<= n;
|
||||
}
|
||||
else
|
||||
{
|
||||
hy = ly << (n - 32);
|
||||
ly = 0;
|
||||
}
|
||||
}
|
||||
|
||||
/* fix point fmod */
|
||||
n = ix - iy;
|
||||
while(n--) {
|
||||
hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1;
|
||||
if(hz<0){hx = hx+hx+(lx>>31); lx = lx+lx;}
|
||||
else {
|
||||
if((hz|lz)==0) /* return sign(x)*0 */
|
||||
return Zero[(unsigned)sx>>31];
|
||||
hx = hz+hz+(lz>>31); lx = lz+lz;
|
||||
}
|
||||
}
|
||||
hz=hx-hy;lz=lx-ly; if(lx<ly) hz -= 1;
|
||||
if(hz>=0) {hx=hz;lx=lz;}
|
||||
/* fix point fmod */
|
||||
n = ix - iy;
|
||||
while (n--)
|
||||
{
|
||||
hz = hx - hy;
|
||||
lz = lx - ly;
|
||||
if (lx < ly)
|
||||
{
|
||||
hz -= 1;
|
||||
}
|
||||
if (hz < 0)
|
||||
{
|
||||
hx = hx + hx + (lx >> 31);
|
||||
lx = lx + lx;
|
||||
}
|
||||
else
|
||||
{
|
||||
if ((hz | lz) == 0) /* return sign(x) * 0 */
|
||||
{
|
||||
return Zero[(unsigned) sx >> 31];
|
||||
}
|
||||
hx = hz + hz + (lz >> 31);
|
||||
lx = lz + lz;
|
||||
}
|
||||
}
|
||||
hz = hx - hy;
|
||||
lz = lx - ly;
|
||||
if (lx < ly)
|
||||
{
|
||||
hz -= 1;
|
||||
}
|
||||
if (hz >= 0)
|
||||
{
|
||||
hx = hz;
|
||||
lx = lz;
|
||||
}
|
||||
|
||||
/* convert back to floating value and restore the sign */
|
||||
if((hx|lx)==0) /* return sign(x)*0 */
|
||||
return Zero[(unsigned)sx>>31];
|
||||
while(hx<0x00100000) { /* normalize x */
|
||||
hx = hx+hx+(lx>>31); lx = lx+lx;
|
||||
iy -= 1;
|
||||
}
|
||||
if(iy>= -1022) { /* normalize output */
|
||||
hx = ((hx-0x00100000)|((iy+1023)<<20));
|
||||
__HI(x) = hx|sx;
|
||||
__LO(x) = lx;
|
||||
} else { /* subnormal output */
|
||||
n = -1022 - iy;
|
||||
if(n<=20) {
|
||||
lx = (lx>>n)|((unsigned)hx<<(32-n));
|
||||
hx >>= n;
|
||||
} else if (n<=31) {
|
||||
lx = (hx<<(32-n))|(lx>>n); hx = sx;
|
||||
} else {
|
||||
lx = hx>>(n-32); hx = sx;
|
||||
}
|
||||
__HI(x) = hx|sx;
|
||||
__LO(x) = lx;
|
||||
x *= one; /* create necessary signal */
|
||||
}
|
||||
return x; /* exact output */
|
||||
}
|
||||
/* convert back to floating value and restore the sign */
|
||||
if ((hx | lx) == 0) /* return sign(x) * 0 */
|
||||
{
|
||||
return Zero[(unsigned) sx >> 31];
|
||||
}
|
||||
while (hx < 0x00100000) /* normalize x */
|
||||
{
|
||||
hx = hx + hx + (lx >> 31);
|
||||
lx = lx + lx;
|
||||
iy -= 1;
|
||||
}
|
||||
if (iy >= -1022) /* normalize output */
|
||||
{
|
||||
hx = ((hx - 0x00100000) | ((iy + 1023) << 20));
|
||||
__HI (x) = hx | sx;
|
||||
__LO (x) = lx;
|
||||
}
|
||||
else /* subnormal output */
|
||||
{
|
||||
n = -1022 - iy;
|
||||
if (n <= 20)
|
||||
{
|
||||
lx = (lx >> n) | ((unsigned) hx << (32 - n));
|
||||
hx >>= n;
|
||||
}
|
||||
else if (n <= 31)
|
||||
{
|
||||
lx = (hx << (32 - n)) | (lx >> n);
|
||||
hx = sx;
|
||||
}
|
||||
else
|
||||
{
|
||||
lx = hx >> (n - 32);
|
||||
hx = sx;
|
||||
}
|
||||
__HI (x) = hx | sx;
|
||||
__LO (x) = lx;
|
||||
x *= one; /* create necessary signal */
|
||||
}
|
||||
return x; /* exact output */
|
||||
} /* fmod */
|
||||
|
||||
23
third-party/fdlibm/s_isnan.c
vendored
23
third-party/fdlibm/s_isnan.c
vendored
@ -6,24 +6,25 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* isnan(x) returns 1 is x is nan, else 0;
|
||||
/* isnan(x) returns 1 is x is nan, else 0;
|
||||
* no branching!
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
int isnan(double x)
|
||||
int
|
||||
isnan (double x)
|
||||
{
|
||||
int hx,lx;
|
||||
hx = (__HI(x)&0x7fffffff);
|
||||
lx = __LO(x);
|
||||
hx |= (unsigned)(lx|(-lx))>>31;
|
||||
hx = 0x7ff00000 - hx;
|
||||
return ((unsigned)(hx))>>31;
|
||||
}
|
||||
int hx, lx;
|
||||
|
||||
hx = (__HI (x) & 0x7fffffff);
|
||||
lx = __LO (x);
|
||||
hx |= (unsigned) (lx | (-lx)) >> 31;
|
||||
hx = 0x7ff00000 - hx;
|
||||
return ((unsigned) (hx)) >> 31;
|
||||
} /* isnan */
|
||||
|
||||
222
third-party/fdlibm/s_log.c
vendored
222
third-party/fdlibm/s_log.c
vendored
@ -6,7 +6,7 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
@ -16,44 +16,44 @@
|
||||
*
|
||||
* Method :
|
||||
* 1. Argument Reduction: find k and f such that
|
||||
* x = 2^k * (1+f),
|
||||
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
||||
* x = 2^k * (1+f),
|
||||
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
||||
*
|
||||
* 2. Approximation of log(1+f).
|
||||
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
||||
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
||||
* = 2s + s*R
|
||||
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
||||
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
||||
* = 2s + s*R
|
||||
* We use a special Reme algorithm on [0,0.1716] to generate
|
||||
* a polynomial of degree 14 to approximate R The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-58.45. In
|
||||
* other words,
|
||||
* 2 4 6 8 10 12 14
|
||||
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
|
||||
* (the values of Lg1 to Lg7 are listed in the program)
|
||||
* and
|
||||
* | 2 14 | -58.45
|
||||
* | Lg1*s +...+Lg7*s - R(z) | <= 2
|
||||
* | |
|
||||
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
||||
* In order to guarantee error in log below 1ulp, we compute log
|
||||
* by
|
||||
* log(1+f) = f - s*(f - R) (if f is not too large)
|
||||
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
|
||||
*
|
||||
* 3. Finally, log(x) = k*ln2 + log(1+f).
|
||||
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
||||
* Here ln2 is split into two floating point number:
|
||||
* ln2_hi + ln2_lo,
|
||||
* where n*ln2_hi is always exact for |n| < 2000.
|
||||
* a polynomial of degree 14 to approximate R The maximum error
|
||||
* of this polynomial approximation is bounded by 2**-58.45. In
|
||||
* other words,
|
||||
* 2 4 6 8 10 12 14
|
||||
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
|
||||
* (the values of Lg1 to Lg7 are listed in the program)
|
||||
* and
|
||||
* | 2 14 | -58.45
|
||||
* | Lg1*s +...+Lg7*s - R(z) | <= 2
|
||||
* | |
|
||||
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
||||
* In order to guarantee error in log below 1ulp, we compute log
|
||||
* by
|
||||
* log(1+f) = f - s*(f - R) (if f is not too large)
|
||||
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
|
||||
*
|
||||
* 3. Finally, log(x) = k*ln2 + log(1+f).
|
||||
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
||||
* Here ln2 is split into two floating point number:
|
||||
* ln2_hi + ln2_lo,
|
||||
* where n*ln2_hi is always exact for |n| < 2000.
|
||||
*
|
||||
* Special cases:
|
||||
* log(x) is NaN with signal if x < 0 (including -INF) ;
|
||||
* log(+INF) is +INF; log(0) is -INF with signal;
|
||||
* log(NaN) is that NaN with no signal.
|
||||
* log(x) is NaN with signal if x < 0 (including -INF) ;
|
||||
* log(+INF) is +INF; log(0) is -INF with signal;
|
||||
* log(NaN) is that NaN with no signal.
|
||||
*
|
||||
* Accuracy:
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
* according to an error analysis, the error is always less than
|
||||
* 1 ulp (unit in the last place).
|
||||
*
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
@ -65,64 +65,108 @@
|
||||
#include "fdlibm.h"
|
||||
|
||||
#define zero 0.0
|
||||
#define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
|
||||
#define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
|
||||
#define two54 1.80143985094819840000e+16 /* 43500000 00000000 */
|
||||
#define Lg1 6.666666666666735130e-01 /* 3FE55555 55555593 */
|
||||
#define Lg2 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
|
||||
#define Lg3 2.857142874366239149e-01 /* 3FD24924 94229359 */
|
||||
#define Lg4 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
|
||||
#define Lg5 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
|
||||
#define Lg6 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
|
||||
#define Lg7 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
|
||||
#define ln2_hi 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
|
||||
#define ln2_lo 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
|
||||
#define two54 1.80143985094819840000e+16 /* 43500000 00000000 */
|
||||
#define Lg1 6.666666666666735130e-01 /* 3FE55555 55555593 */
|
||||
#define Lg2 3.999999999940941908e-01 /* 3FD99999 9997FA04 */
|
||||
#define Lg3 2.857142874366239149e-01 /* 3FD24924 94229359 */
|
||||
#define Lg4 2.222219843214978396e-01 /* 3FCC71C5 1D8E78AF */
|
||||
#define Lg5 1.818357216161805012e-01 /* 3FC74664 96CB03DE */
|
||||
#define Lg6 1.531383769920937332e-01 /* 3FC39A09 D078C69F */
|
||||
#define Lg7 1.479819860511658591e-01 /* 3FC2F112 DF3E5244 */
|
||||
|
||||
double log(double x)
|
||||
double
|
||||
log (double x)
|
||||
{
|
||||
double hfsq,f,s,z,R,w,t1,t2,dk;
|
||||
int k,hx,i,j;
|
||||
unsigned lx;
|
||||
double hfsq, f, s, z, R, w, t1, t2, dk;
|
||||
int k, hx, i, j;
|
||||
unsigned lx;
|
||||
|
||||
hx = __HI(x); /* high word of x */
|
||||
lx = __LO(x); /* low word of x */
|
||||
hx = __HI (x); /* high word of x */
|
||||
lx = __LO (x); /* low word of x */
|
||||
|
||||
k=0;
|
||||
if (hx < 0x00100000) { /* x < 2**-1022 */
|
||||
if (((hx&0x7fffffff)|lx)==0)
|
||||
return -two54/zero; /* log(+-0)=-inf */
|
||||
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
|
||||
k -= 54; x *= two54; /* subnormal number, scale up x */
|
||||
hx = __HI(x); /* high word of x */
|
||||
}
|
||||
if (hx >= 0x7ff00000) return x+x;
|
||||
k += (hx>>20)-1023;
|
||||
hx &= 0x000fffff;
|
||||
i = (hx+0x95f64)&0x100000;
|
||||
__HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */
|
||||
k += (i>>20);
|
||||
f = x-1.0;
|
||||
if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
|
||||
if(f==zero) if(k==0) return zero; else {dk=(double)k;
|
||||
return dk*ln2_hi+dk*ln2_lo;}
|
||||
R = f*f*(0.5-0.33333333333333333*f);
|
||||
if(k==0) return f-R; else {dk=(double)k;
|
||||
return dk*ln2_hi-((R-dk*ln2_lo)-f);}
|
||||
}
|
||||
s = f/(2.0+f);
|
||||
dk = (double)k;
|
||||
z = s*s;
|
||||
i = hx-0x6147a;
|
||||
w = z*z;
|
||||
j = 0x6b851-hx;
|
||||
t1= w*(Lg2+w*(Lg4+w*Lg6));
|
||||
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
|
||||
i |= j;
|
||||
R = t2+t1;
|
||||
if(i>0) {
|
||||
hfsq=0.5*f*f;
|
||||
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
||||
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
|
||||
} else {
|
||||
if(k==0) return f-s*(f-R); else
|
||||
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
|
||||
}
|
||||
}
|
||||
k = 0;
|
||||
if (hx < 0x00100000) /* x < 2**-1022 */
|
||||
{
|
||||
if (((hx & 0x7fffffff) | lx) == 0) /* log(+-0) = -inf */
|
||||
{
|
||||
return -two54 / zero;
|
||||
}
|
||||
if (hx < 0) /* log(-#) = NaN */
|
||||
{
|
||||
return (x - x) / zero;
|
||||
}
|
||||
k -= 54;
|
||||
x *= two54; /* subnormal number, scale up x */
|
||||
hx = __HI (x); /* high word of x */
|
||||
}
|
||||
if (hx >= 0x7ff00000)
|
||||
{
|
||||
return x + x;
|
||||
}
|
||||
k += (hx >> 20) - 1023;
|
||||
hx &= 0x000fffff;
|
||||
i = (hx + 0x95f64) & 0x100000;
|
||||
__HI (x) = hx | (i ^ 0x3ff00000); /* normalize x or x / 2 */
|
||||
k += (i >> 20);
|
||||
f = x - 1.0;
|
||||
if ((0x000fffff & (2 + hx)) < 3) /* |f| < 2**-20 */
|
||||
{
|
||||
if (f == zero)
|
||||
{
|
||||
if (k == 0)
|
||||
{
|
||||
return zero;
|
||||
}
|
||||
else
|
||||
{
|
||||
dk = (double) k;
|
||||
return dk * ln2_hi + dk * ln2_lo;
|
||||
}
|
||||
}
|
||||
R = f * f * (0.5 - 0.33333333333333333 * f);
|
||||
if (k == 0)
|
||||
{
|
||||
return f - R;
|
||||
}
|
||||
else
|
||||
{
|
||||
dk = (double) k;
|
||||
return dk * ln2_hi - ((R - dk * ln2_lo) - f);
|
||||
}
|
||||
}
|
||||
s = f / (2.0 + f);
|
||||
dk = (double) k;
|
||||
z = s * s;
|
||||
i = hx - 0x6147a;
|
||||
w = z * z;
|
||||
j = 0x6b851 - hx;
|
||||
t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
|
||||
t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
|
||||
i |= j;
|
||||
R = t2 + t1;
|
||||
if (i > 0)
|
||||
{
|
||||
hfsq = 0.5 * f * f;
|
||||
if (k == 0)
|
||||
{
|
||||
return f - (hfsq - s * (hfsq + R));
|
||||
}
|
||||
else
|
||||
{
|
||||
return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) - f);
|
||||
}
|
||||
}
|
||||
else
|
||||
{
|
||||
if (k == 0)
|
||||
{
|
||||
return f - s * (f - R);
|
||||
}
|
||||
else
|
||||
{
|
||||
return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
|
||||
}
|
||||
}
|
||||
} /* log */
|
||||
|
||||
631
third-party/fdlibm/s_pow.c
vendored
631
third-party/fdlibm/s_pow.c
vendored
@ -5,50 +5,50 @@
|
||||
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
|
||||
*
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* pow(x,y) return x**y
|
||||
*
|
||||
* n
|
||||
* n
|
||||
* Method: Let x = 2 * (1+f)
|
||||
* 1. Compute and return log2(x) in two pieces:
|
||||
* log2(x) = w1 + w2,
|
||||
* where w1 has 53-24 = 29 bit trailing zeros.
|
||||
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
|
||||
* arithmetic, where |y'|<=0.5.
|
||||
* 3. Return x**y = 2**n*exp(y'*log2)
|
||||
* 1. Compute and return log2(x) in two pieces:
|
||||
* log2(x) = w1 + w2,
|
||||
* where w1 has 53-24 = 29 bit trailing zeros.
|
||||
* 2. Perform y*log2(x) = n+y' by simulating muti-precision
|
||||
* arithmetic, where |y'|<=0.5.
|
||||
* 3. Return x**y = 2**n*exp(y'*log2)
|
||||
*
|
||||
* Special cases:
|
||||
* 1. (anything) ** 0 is 1
|
||||
* 2. (anything) ** 1 is itself
|
||||
* 3. (anything) ** NAN is NAN
|
||||
* 4. NAN ** (anything except 0) is NAN
|
||||
* 5. +-(|x| > 1) ** +INF is +INF
|
||||
* 6. +-(|x| > 1) ** -INF is +0
|
||||
* 7. +-(|x| < 1) ** +INF is +0
|
||||
* 8. +-(|x| < 1) ** -INF is +INF
|
||||
* 9. +-1 ** +-INF is NAN
|
||||
* 10. +0 ** (+anything except 0, NAN) is +0
|
||||
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
|
||||
* 12. +0 ** (-anything except 0, NAN) is +INF
|
||||
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
|
||||
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
|
||||
* 15. +INF ** (+anything except 0,NAN) is +INF
|
||||
* 16. +INF ** (-anything except 0,NAN) is +0
|
||||
* 17. -INF ** (anything) = -0 ** (-anything)
|
||||
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
||||
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
|
||||
* 1. (anything) ** 0 is 1
|
||||
* 2. (anything) ** 1 is itself
|
||||
* 3. (anything) ** NAN is NAN
|
||||
* 4. NAN ** (anything except 0) is NAN
|
||||
* 5. +-(|x| > 1) ** +INF is +INF
|
||||
* 6. +-(|x| > 1) ** -INF is +0
|
||||
* 7. +-(|x| < 1) ** +INF is +0
|
||||
* 8. +-(|x| < 1) ** -INF is +INF
|
||||
* 9. +-1 ** +-INF is NAN
|
||||
* 10. +0 ** (+anything except 0, NAN) is +0
|
||||
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
|
||||
* 12. +0 ** (-anything except 0, NAN) is +INF
|
||||
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
|
||||
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
|
||||
* 15. +INF ** (+anything except 0,NAN) is +INF
|
||||
* 16. +INF ** (-anything except 0,NAN) is +0
|
||||
* 17. -INF ** (anything) = -0 ** (-anything)
|
||||
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
|
||||
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
|
||||
*
|
||||
* Accuracy:
|
||||
* pow(x,y) returns x**y nearly rounded. In particular
|
||||
* pow(integer,integer)
|
||||
* always returns the correct integer provided it is
|
||||
* representable.
|
||||
* pow(x,y) returns x**y nearly rounded. In particular
|
||||
* pow(integer,integer)
|
||||
* always returns the correct integer provided it is
|
||||
* representable.
|
||||
*
|
||||
* Constants :
|
||||
* Constants:
|
||||
* The hexadecimal values are the intended ones for the following
|
||||
* constants. The decimal values may be used, provided that the
|
||||
* compiler will convert from decimal to binary accurately enough
|
||||
@ -57,242 +57,371 @@
|
||||
|
||||
#include "fdlibm.h"
|
||||
|
||||
static const double
|
||||
one = 1.0,
|
||||
bp[] = {1.0, 1.5,},
|
||||
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
|
||||
dp_l[] = { 0.0, 1.35003920212974897128e-08,}; /* 0x3E4CFDEB, 0x43CFD006 */
|
||||
static const double one = 1.0;
|
||||
static const double bp[] =
|
||||
{
|
||||
1.0,
|
||||
1.5,
|
||||
};
|
||||
static const double dp_h[] =
|
||||
{
|
||||
0.0,
|
||||
5.84962487220764160156e-01, /* 0x3FE2B803, 0x40000000 */
|
||||
};
|
||||
static const double dp_l[] =
|
||||
{
|
||||
0.0,
|
||||
1.35003920212974897128e-08, /* 0x3E4CFDEB, 0x43CFD006 */
|
||||
};
|
||||
|
||||
#define zero 0.0
|
||||
#define two 2.0
|
||||
#define two53 9007199254740992.0 /* 0x43400000, 0x00000000 */
|
||||
#define two53 9007199254740992.0 /* 0x43400000, 0x00000000 */
|
||||
#define huge 1.0e300
|
||||
#define tiny 1.0e-300
|
||||
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
|
||||
#define L1 5.99999999999994648725e-01 /* 0x3FE33333, 0x33333303 */
|
||||
#define L2 4.28571428578550184252e-01 /* 0x3FDB6DB6, 0xDB6FABFF */
|
||||
#define L3 3.33333329818377432918e-01 /* 0x3FD55555, 0x518F264D */
|
||||
#define L4 2.72728123808534006489e-01 /* 0x3FD17460, 0xA91D4101 */
|
||||
#define L5 2.30660745775561754067e-01 /* 0x3FCD864A, 0x93C9DB65 */
|
||||
#define L6 2.06975017800338417784e-01 /* 0x3FCA7E28, 0x4A454EEF */
|
||||
#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
|
||||
#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
|
||||
#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
|
||||
#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
|
||||
#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */
|
||||
#define lg2 6.93147180559945286227e-01 /* 0x3FE62E42, 0xFEFA39EF */
|
||||
#define lg2_h 6.93147182464599609375e-01 /* 0x3FE62E43, 0x00000000 */
|
||||
#define lg2_l -1.90465429995776804525e-09 /* 0xBE205C61, 0x0CA86C39 */
|
||||
/* poly coefs for (3/2) * (log(x) - 2s - 2/3 * s**3 */
|
||||
#define L1 5.99999999999994648725e-01 /* 0x3FE33333, 0x33333303 */
|
||||
#define L2 4.28571428578550184252e-01 /* 0x3FDB6DB6, 0xDB6FABFF */
|
||||
#define L3 3.33333329818377432918e-01 /* 0x3FD55555, 0x518F264D */
|
||||
#define L4 2.72728123808534006489e-01 /* 0x3FD17460, 0xA91D4101 */
|
||||
#define L5 2.30660745775561754067e-01 /* 0x3FCD864A, 0x93C9DB65 */
|
||||
#define L6 2.06975017800338417784e-01 /* 0x3FCA7E28, 0x4A454EEF */
|
||||
#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
|
||||
#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
|
||||
#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
|
||||
#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
|
||||
#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */
|
||||
#define lg2 6.93147180559945286227e-01 /* 0x3FE62E42, 0xFEFA39EF */
|
||||
#define lg2_h 6.93147182464599609375e-01 /* 0x3FE62E43, 0x00000000 */
|
||||
#define lg2_l -1.90465429995776804525e-09 /* 0xBE205C61, 0x0CA86C39 */
|
||||
#define ovt 8.0085662595372944372e-0017 /* -(1024-log2(ovfl+.5ulp)) */
|
||||
#define cp 9.61796693925975554329e-01 /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
|
||||
#define cp_h 9.61796700954437255859e-01 /* 0x3FEEC709, 0xE0000000 =(float)cp */
|
||||
#define cp_l -7.02846165095275826516e-09 /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
|
||||
#define ivln2 1.44269504088896338700e+00 /* 0x3FF71547, 0x652B82FE =1/ln2 */
|
||||
#define ivln2_h 1.44269502162933349609e+00 /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
|
||||
#define ivln2_l 1.92596299112661746887e-08 /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
|
||||
#define cp 9.61796693925975554329e-01 /* 0x3FEEC709, 0xDC3A03FD = 2 / (3 ln2) */
|
||||
#define cp_h 9.61796700954437255859e-01 /* 0x3FEEC709, 0xE0000000 = (float) cp */
|
||||
#define cp_l -7.02846165095275826516e-09 /* 0xBE3E2FE0, 0x145B01F5 = tail of cp_h */
|
||||
#define ivln2 1.44269504088896338700e+00 /* 0x3FF71547, 0x652B82FE = 1 / ln2 */
|
||||
#define ivln2_h 1.44269502162933349609e+00 /* 0x3FF71547, 0x60000000 = 24b 1 / ln2 */
|
||||
#define ivln2_l 1.92596299112661746887e-08 /* 0x3E54AE0B, 0xF85DDF44 = 1 / ln2 tail */
|
||||
|
||||
double pow(double x, double y)
|
||||
double
|
||||
pow (double x, double y)
|
||||
{
|
||||
double z,ax,z_h,z_l,p_h,p_l;
|
||||
double y1,t1,t2,r,s,t,u,v,w;
|
||||
int i0,i1,i,j,k,yisint,n;
|
||||
int hx,hy,ix,iy;
|
||||
unsigned lx,ly;
|
||||
double z, ax, z_h, z_l, p_h, p_l;
|
||||
double y1, t1, t2, r, s, t, u, v, w;
|
||||
int i0, i1, i, j, k, yisint, n;
|
||||
int hx, hy, ix, iy;
|
||||
unsigned lx, ly;
|
||||
|
||||
i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
|
||||
hx = __HI(x); lx = __LO(x);
|
||||
hy = __HI(y); ly = __LO(y);
|
||||
ix = hx&0x7fffffff; iy = hy&0x7fffffff;
|
||||
i0 = ((*(int *) &one) >> 29) ^ 1;
|
||||
i1 = 1 - i0;
|
||||
hx = __HI (x);
|
||||
lx = __LO (x);
|
||||
hy = __HI (y);
|
||||
ly = __LO (y);
|
||||
ix = hx & 0x7fffffff;
|
||||
iy = hy & 0x7fffffff;
|
||||
|
||||
/* y==zero: x**0 = 1 */
|
||||
if((iy|ly)==0) return one;
|
||||
/* y == zero: x**0 = 1 */
|
||||
if ((iy | ly) == 0)
|
||||
{
|
||||
return one;
|
||||
}
|
||||
|
||||
/* +-NaN return x+y */
|
||||
if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
|
||||
iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
|
||||
return x+y;
|
||||
/* +-NaN return x + y */
|
||||
if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0)))
|
||||
{
|
||||
return x + y;
|
||||
}
|
||||
|
||||
/* determine if y is an odd int when x < 0
|
||||
* yisint = 0 ... y is not an integer
|
||||
* yisint = 1 ... y is an odd int
|
||||
* yisint = 2 ... y is an even int
|
||||
*/
|
||||
yisint = 0;
|
||||
if(hx<0) {
|
||||
if(iy>=0x43400000) yisint = 2; /* even integer y */
|
||||
else if(iy>=0x3ff00000) {
|
||||
k = (iy>>20)-0x3ff; /* exponent */
|
||||
if(k>20) {
|
||||
j = ly>>(52-k);
|
||||
if((j<<(52-k))==ly) yisint = 2-(j&1);
|
||||
} else if(ly==0) {
|
||||
j = iy>>(20-k);
|
||||
if((j<<(20-k))==iy) yisint = 2-(j&1);
|
||||
}
|
||||
}
|
||||
}
|
||||
/* determine if y is an odd int when x < 0
|
||||
* yisint = 0 ... y is not an integer
|
||||
* yisint = 1 ... y is an odd int
|
||||
* yisint = 2 ... y is an even int
|
||||
*/
|
||||
yisint = 0;
|
||||
if (hx < 0)
|
||||
{
|
||||
if (iy >= 0x43400000) /* even integer y */
|
||||
{
|
||||
yisint = 2;
|
||||
}
|
||||
else if (iy >= 0x3ff00000)
|
||||
{
|
||||
k = (iy >> 20) - 0x3ff; /* exponent */
|
||||
if (k > 20)
|
||||
{
|
||||
j = ly >> (52 - k);
|
||||
if ((j << (52 - k)) == ly)
|
||||
{
|
||||
yisint = 2 - (j & 1);
|
||||
}
|
||||
}
|
||||
else if (ly == 0)
|
||||
{
|
||||
j = iy >> (20 - k);
|
||||
if ((j << (20 - k)) == iy)
|
||||
{
|
||||
yisint = 2 - (j & 1);
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/* special value of y */
|
||||
if(ly==0) {
|
||||
if (iy==0x7ff00000) { /* y is +-inf */
|
||||
if(((ix-0x3ff00000)|lx)==0)
|
||||
return y - y; /* inf**+-1 is NaN */
|
||||
else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
|
||||
return (hy>=0)? y: zero;
|
||||
else /* (|x|<1)**-,+inf = inf,0 */
|
||||
return (hy<0)?-y: zero;
|
||||
}
|
||||
if(iy==0x3ff00000) { /* y is +-1 */
|
||||
if(hy<0) return one/x; else return x;
|
||||
}
|
||||
if(hy==0x40000000) return x*x; /* y is 2 */
|
||||
if(hy==0x3fe00000) { /* y is 0.5 */
|
||||
if(hx>=0) /* x >= +0 */
|
||||
return sqrt(x);
|
||||
}
|
||||
}
|
||||
/* special value of y */
|
||||
if (ly == 0)
|
||||
{
|
||||
if (iy == 0x7ff00000) /* y is +-inf */
|
||||
{
|
||||
if (((ix - 0x3ff00000) | lx) == 0) /* inf**+-1 is NaN */
|
||||
{
|
||||
return y - y;
|
||||
}
|
||||
else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */
|
||||
{
|
||||
return (hy >= 0) ? y : zero;
|
||||
}
|
||||
else /* (|x|<1)**-,+inf = inf,0 */
|
||||
{
|
||||
return (hy < 0) ? -y : zero;
|
||||
}
|
||||
}
|
||||
if (iy == 0x3ff00000) /* y is +-1 */
|
||||
{
|
||||
if (hy < 0)
|
||||
{
|
||||
return one / x;
|
||||
}
|
||||
else
|
||||
{
|
||||
return x;
|
||||
}
|
||||
}
|
||||
if (hy == 0x40000000) /* y is 2 */
|
||||
{
|
||||
return x * x;
|
||||
}
|
||||
if (hy == 0x3fe00000) /* y is 0.5 */
|
||||
{
|
||||
if (hx >= 0) /* x >= +0 */
|
||||
{
|
||||
return sqrt (x);
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
ax = fabs(x);
|
||||
/* special value of x */
|
||||
if(lx==0) {
|
||||
if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
|
||||
z = ax; /*x is +-0,+-inf,+-1*/
|
||||
if(hy<0) z = one/z; /* z = (1/|x|) */
|
||||
if(hx<0) {
|
||||
if(((ix-0x3ff00000)|yisint)==0) {
|
||||
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
|
||||
} else if(yisint==1)
|
||||
z = -z; /* (x<0)**odd = -(|x|**odd) */
|
||||
}
|
||||
return z;
|
||||
}
|
||||
}
|
||||
ax = fabs (x);
|
||||
/* special value of x */
|
||||
if (lx == 0)
|
||||
{
|
||||
if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000)
|
||||
{
|
||||
z = ax; /* x is +-0,+-inf,+-1 */
|
||||
if (hy < 0)
|
||||
{
|
||||
z = one / z; /* z = (1 / |x|) */
|
||||
}
|
||||
if (hx < 0)
|
||||
{
|
||||
if (((ix - 0x3ff00000) | yisint) == 0)
|
||||
{
|
||||
z = (z - z) / (z - z); /* (-1)**non-int is NaN */
|
||||
}
|
||||
else if (yisint == 1)
|
||||
{
|
||||
z = -z; /* (x<0)**odd = -(|x|**odd) */
|
||||
}
|
||||
}
|
||||
return z;
|
||||
}
|
||||
}
|
||||
|
||||
n = (hx>>31)+1;
|
||||
n = (hx >> 31) + 1;
|
||||
|
||||
/* (x<0)**(non-int) is NaN */
|
||||
if((n|yisint)==0) return (x-x)/(x-x);
|
||||
/* (x<0)**(non-int) is NaN */
|
||||
if ((n | yisint) == 0)
|
||||
{
|
||||
return (x - x) / (x - x);
|
||||
}
|
||||
|
||||
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
|
||||
if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
|
||||
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
|
||||
if ((n | (yisint - 1)) == 0)
|
||||
{
|
||||
s = -one; /* (-ve)**(odd int) */
|
||||
}
|
||||
|
||||
/* |y| is huge */
|
||||
if(iy>0x41e00000) { /* if |y| > 2**31 */
|
||||
if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
|
||||
if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
|
||||
if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
|
||||
}
|
||||
/* over/underflow if x is not close to one */
|
||||
if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
|
||||
if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
|
||||
/* now |1-x| is tiny <= 2**-20, suffice to compute
|
||||
log(x) by x-x^2/2+x^3/3-x^4/4 */
|
||||
t = ax-one; /* t has 20 trailing zeros */
|
||||
w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
|
||||
u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
|
||||
v = t*ivln2_l-w*ivln2;
|
||||
t1 = u+v;
|
||||
__LO(t1) = 0;
|
||||
t2 = v-(t1-u);
|
||||
} else {
|
||||
double ss,s2,s_h,s_l,t_h,t_l;
|
||||
n = 0;
|
||||
/* take care subnormal number */
|
||||
if(ix<0x00100000)
|
||||
{ax *= two53; n -= 53; ix = __HI(ax); }
|
||||
n += ((ix)>>20)-0x3ff;
|
||||
j = ix&0x000fffff;
|
||||
/* determine interval */
|
||||
ix = j|0x3ff00000; /* normalize ix */
|
||||
if(j<=0x3988E) k=0; /* |x|<sqrt(3/2) */
|
||||
else if(j<0xBB67A) k=1; /* |x|<sqrt(3) */
|
||||
else {k=0;n+=1;ix -= 0x00100000;}
|
||||
__HI(ax) = ix;
|
||||
/* |y| is huge */
|
||||
if (iy > 0x41e00000) /* if |y| > 2**31 */
|
||||
{
|
||||
if (iy > 0x43f00000) /* if |y| > 2**64, must o/uflow */
|
||||
{
|
||||
if (ix <= 0x3fefffff)
|
||||
{
|
||||
return (hy < 0) ? huge * huge : tiny * tiny;
|
||||
}
|
||||
if (ix >= 0x3ff00000)
|
||||
{
|
||||
return (hy > 0) ? huge * huge : tiny * tiny;
|
||||
}
|
||||
}
|
||||
/* over/underflow if x is not close to one */
|
||||
if (ix < 0x3fefffff)
|
||||
{
|
||||
return (hy < 0) ? s * huge * huge : s * tiny * tiny;
|
||||
}
|
||||
if (ix > 0x3ff00000)
|
||||
{
|
||||
return (hy > 0) ? s * huge * huge : s * tiny * tiny;
|
||||
}
|
||||
/* now |1 - x| is tiny <= 2**-20, suffice to compute
|
||||
log(x) by x - x^2 / 2 + x^3 / 3 - x^4 / 4 */
|
||||
t = ax - one; /* t has 20 trailing zeros */
|
||||
w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
|
||||
u = ivln2_h * t; /* ivln2_h has 21 sig. bits */
|
||||
v = t * ivln2_l - w * ivln2;
|
||||
t1 = u + v;
|
||||
__LO (t1) = 0;
|
||||
t2 = v - (t1 - u);
|
||||
}
|
||||
else
|
||||
{
|
||||
double ss, s2, s_h, s_l, t_h, t_l;
|
||||
|
||||
/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
|
||||
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
|
||||
v = one/(ax+bp[k]);
|
||||
ss = u*v;
|
||||
s_h = ss;
|
||||
__LO(s_h) = 0;
|
||||
/* t_h=ax+bp[k] High */
|
||||
t_h = zero;
|
||||
__HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18);
|
||||
t_l = ax - (t_h-bp[k]);
|
||||
s_l = v*((u-s_h*t_h)-s_h*t_l);
|
||||
/* compute log(ax) */
|
||||
s2 = ss*ss;
|
||||
r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
|
||||
r += s_l*(s_h+ss);
|
||||
s2 = s_h*s_h;
|
||||
t_h = 3.0+s2+r;
|
||||
__LO(t_h) = 0;
|
||||
t_l = r-((t_h-3.0)-s2);
|
||||
/* u+v = ss*(1+...) */
|
||||
u = s_h*t_h;
|
||||
v = s_l*t_h+t_l*ss;
|
||||
/* 2/(3log2)*(ss+...) */
|
||||
p_h = u+v;
|
||||
__LO(p_h) = 0;
|
||||
p_l = v-(p_h-u);
|
||||
z_h = cp_h*p_h; /* cp_h+cp_l = 2/(3*log2) */
|
||||
z_l = cp_l*p_h+p_l*cp+dp_l[k];
|
||||
/* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
|
||||
t = (double)n;
|
||||
t1 = (((z_h+z_l)+dp_h[k])+t);
|
||||
__LO(t1) = 0;
|
||||
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
|
||||
}
|
||||
n = 0;
|
||||
/* take care subnormal number */
|
||||
if (ix < 0x00100000)
|
||||
{
|
||||
ax *= two53;
|
||||
n -= 53;
|
||||
ix = __HI (ax);
|
||||
}
|
||||
n += ((ix) >> 20) - 0x3ff;
|
||||
j = ix & 0x000fffff;
|
||||
/* determine interval */
|
||||
ix = j | 0x3ff00000; /* normalize ix */
|
||||
if (j <= 0x3988E) /* |x| < sqrt(3/2) */
|
||||
{
|
||||
k = 0;
|
||||
}
|
||||
else if (j < 0xBB67A) /* |x| < sqrt(3) */
|
||||
{
|
||||
k = 1;
|
||||
}
|
||||
else
|
||||
{
|
||||
k = 0;
|
||||
n += 1;
|
||||
ix -= 0x00100000;
|
||||
}
|
||||
__HI (ax) = ix;
|
||||
|
||||
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
|
||||
y1 = y;
|
||||
__LO(y1) = 0;
|
||||
p_l = (y-y1)*t1+y*t2;
|
||||
p_h = y1*t1;
|
||||
z = p_l+p_h;
|
||||
j = __HI(z);
|
||||
i = __LO(z);
|
||||
if (j>=0x40900000) { /* z >= 1024 */
|
||||
if(((j-0x40900000)|i)!=0) /* if z > 1024 */
|
||||
return s*huge*huge; /* overflow */
|
||||
else {
|
||||
if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
|
||||
}
|
||||
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
|
||||
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
|
||||
return s*tiny*tiny; /* underflow */
|
||||
else {
|
||||
if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
|
||||
}
|
||||
}
|
||||
/*
|
||||
* compute 2**(p_h+p_l)
|
||||
*/
|
||||
i = j&0x7fffffff;
|
||||
k = (i>>20)-0x3ff;
|
||||
n = 0;
|
||||
if(i>0x3fe00000) { /* if |z| > 0.5, set n = [z+0.5] */
|
||||
n = j+(0x00100000>>(k+1));
|
||||
k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
|
||||
t = zero;
|
||||
__HI(t) = (n&~(0x000fffff>>k));
|
||||
n = ((n&0x000fffff)|0x00100000)>>(20-k);
|
||||
if(j<0) n = -n;
|
||||
p_h -= t;
|
||||
}
|
||||
t = p_l+p_h;
|
||||
__LO(t) = 0;
|
||||
u = t*lg2_h;
|
||||
v = (p_l-(t-p_h))*lg2+t*lg2_l;
|
||||
z = u+v;
|
||||
w = v-(z-u);
|
||||
t = z*z;
|
||||
t1 = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
|
||||
r = (z*t1)/(t1-two)-(w+z*w);
|
||||
z = one-(r-z);
|
||||
j = __HI(z);
|
||||
j += (n<<20);
|
||||
if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
|
||||
else __HI(z) += (n<<20);
|
||||
return s*z;
|
||||
}
|
||||
/* compute ss = s_h + s_l = (x - 1) / (x + 1) or (x - 1.5) / (x + 1.5) */
|
||||
u = ax - bp[k]; /* bp[0] = 1.0, bp[1] = 1.5 */
|
||||
v = one / (ax + bp[k]);
|
||||
ss = u * v;
|
||||
s_h = ss;
|
||||
__LO (s_h) = 0;
|
||||
/* t_h = ax + bp[k] High */
|
||||
t_h = zero;
|
||||
__HI (t_h) = ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18);
|
||||
t_l = ax - (t_h - bp[k]);
|
||||
s_l = v * ((u - s_h * t_h) - s_h * t_l);
|
||||
/* compute log(ax) */
|
||||
s2 = ss * ss;
|
||||
r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
|
||||
r += s_l * (s_h + ss);
|
||||
s2 = s_h * s_h;
|
||||
t_h = 3.0 + s2 + r;
|
||||
__LO (t_h) = 0;
|
||||
t_l = r - ((t_h - 3.0) - s2);
|
||||
/* u + v = ss * (1 + ...) */
|
||||
u = s_h * t_h;
|
||||
v = s_l * t_h + t_l * ss;
|
||||
/* 2 / (3 * log2) * (ss + ...) */
|
||||
p_h = u + v;
|
||||
__LO (p_h) = 0;
|
||||
p_l = v - (p_h - u);
|
||||
z_h = cp_h * p_h; /* cp_h + cp_l = 2 / (3 * log2) */
|
||||
z_l = cp_l * p_h + p_l * cp + dp_l[k];
|
||||
/* log2(ax) = (ss + ...) * 2 / (3 * log2) = n + dp_h + z_h + z_l */
|
||||
t = (double) n;
|
||||
t1 = (((z_h + z_l) + dp_h[k]) + t);
|
||||
__LO (t1) = 0;
|
||||
t2 = z_l - (((t1 - t) - dp_h[k]) - z_h);
|
||||
}
|
||||
|
||||
/* split up y into y1 + y2 and compute (y1 + y2) * (t1 + t2) */
|
||||
y1 = y;
|
||||
__LO (y1) = 0;
|
||||
p_l = (y - y1) * t1 + y * t2;
|
||||
p_h = y1 * t1;
|
||||
z = p_l + p_h;
|
||||
j = __HI (z);
|
||||
i = __LO (z);
|
||||
if (j >= 0x40900000) /* z >= 1024 */
|
||||
{
|
||||
if (((j - 0x40900000) | i) != 0) /* if z > 1024 */
|
||||
{
|
||||
return s * huge * huge; /* overflow */
|
||||
}
|
||||
else
|
||||
{
|
||||
if (p_l + ovt > z - p_h)
|
||||
{
|
||||
return s * huge * huge; /* overflow */
|
||||
}
|
||||
}
|
||||
}
|
||||
else if ((j & 0x7fffffff) >= 0x4090cc00) /* z <= -1075 */
|
||||
{
|
||||
if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */
|
||||
{
|
||||
return s * tiny * tiny; /* underflow */
|
||||
}
|
||||
else
|
||||
{
|
||||
if (p_l <= z - p_h)
|
||||
{
|
||||
return s * tiny * tiny; /* underflow */
|
||||
}
|
||||
}
|
||||
}
|
||||
/*
|
||||
* compute 2**(p_h + p_l)
|
||||
*/
|
||||
i = j & 0x7fffffff;
|
||||
k = (i >> 20) - 0x3ff;
|
||||
n = 0;
|
||||
if (i > 0x3fe00000) /* if |z| > 0.5, set n = [z + 0.5] */
|
||||
{
|
||||
n = j + (0x00100000 >> (k + 1));
|
||||
k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */
|
||||
t = zero;
|
||||
__HI (t) = (n & ~(0x000fffff >> k));
|
||||
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
|
||||
if (j < 0)
|
||||
{
|
||||
n = -n;
|
||||
}
|
||||
p_h -= t;
|
||||
}
|
||||
t = p_l + p_h;
|
||||
__LO (t) = 0;
|
||||
u = t * lg2_h;
|
||||
v = (p_l - (t - p_h)) * lg2 + t * lg2_l;
|
||||
z = u + v;
|
||||
w = v - (z - u);
|
||||
t = z * z;
|
||||
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
|
||||
r = (z * t1) / (t1 - two) - (w + z * w);
|
||||
z = one - (r - z);
|
||||
j = __HI (z);
|
||||
j += (n << 20);
|
||||
if ((j >> 20) <= 0) /* subnormal output */
|
||||
{
|
||||
z = scalbn (z, n);
|
||||
}
|
||||
else
|
||||
{
|
||||
__HI (z) += (n << 20);
|
||||
}
|
||||
return s * z;
|
||||
} /* pow */
|
||||
|
||||
81
third-party/fdlibm/s_scalbn.c
vendored
81
third-party/fdlibm/s_scalbn.c
vendored
@ -6,14 +6,12 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/*
|
||||
* scalbn (double x, int n)
|
||||
* scalbn(x,n) returns x* 2**n computed by exponent
|
||||
/* scalbn(x,n) returns x* 2**n computed by exponent
|
||||
* manipulation rather than by actually performing an
|
||||
* exponentiation or a multiplication.
|
||||
*/
|
||||
@ -25,29 +23,54 @@
|
||||
#define huge 1.0e+300
|
||||
#define tiny 1.0e-300
|
||||
|
||||
double scalbn (double x, int n)
|
||||
double
|
||||
scalbn (double x, int n)
|
||||
{
|
||||
int k,hx,lx;
|
||||
hx = __HI(x);
|
||||
lx = __LO(x);
|
||||
k = (hx&0x7ff00000)>>20; /* extract exponent */
|
||||
if (k==0) { /* 0 or subnormal x */
|
||||
if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
|
||||
x *= two54;
|
||||
hx = __HI(x);
|
||||
k = ((hx&0x7ff00000)>>20) - 54;
|
||||
if (n< -50000) return tiny*x; /*underflow*/
|
||||
}
|
||||
if (k==0x7ff) return x+x; /* NaN or Inf */
|
||||
k = k+n;
|
||||
if (k > 0x7fe) return huge*copysign(huge,x); /* overflow */
|
||||
if (k > 0) /* normal result */
|
||||
{__HI(x) = (hx&0x800fffff)|(k<<20); return x;}
|
||||
if (k <= -54)
|
||||
if (n > 50000) /* in case integer overflow in n+k */
|
||||
return huge*copysign(huge,x); /*overflow*/
|
||||
else return tiny*copysign(tiny,x); /*underflow*/
|
||||
k += 54; /* subnormal result */
|
||||
__HI(x) = (hx&0x800fffff)|(k<<20);
|
||||
return x*twom54;
|
||||
}
|
||||
int k, hx, lx;
|
||||
|
||||
hx = __HI (x);
|
||||
lx = __LO (x);
|
||||
k = (hx & 0x7ff00000) >> 20; /* extract exponent */
|
||||
if (k == 0) /* 0 or subnormal x */
|
||||
{
|
||||
if ((lx | (hx & 0x7fffffff)) == 0) /* +-0 */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
x *= two54;
|
||||
hx = __HI (x);
|
||||
k = ((hx & 0x7ff00000) >> 20) - 54;
|
||||
if (n < -50000) /*underflow */
|
||||
{
|
||||
return tiny * x;
|
||||
}
|
||||
}
|
||||
if (k == 0x7ff) /* NaN or Inf */
|
||||
{
|
||||
return x + x;
|
||||
}
|
||||
k = k + n;
|
||||
if (k > 0x7fe) /* overflow */
|
||||
{
|
||||
return huge * copysign (huge, x);
|
||||
}
|
||||
if (k > 0) /* normal result */
|
||||
{
|
||||
__HI (x) = (hx & 0x800fffff) | (k << 20);
|
||||
return x;
|
||||
}
|
||||
if (k <= -54)
|
||||
{
|
||||
if (n > 50000) /* in case integer overflow in n + k */
|
||||
{
|
||||
return huge * copysign (huge, x); /*overflow */
|
||||
}
|
||||
else
|
||||
{
|
||||
return tiny * copysign (tiny, x); /*underflow */
|
||||
}
|
||||
}
|
||||
k += 54; /* subnormal result */
|
||||
__HI (x) = (hx & 0x800fffff) | (k << 20);
|
||||
return x * twom54;
|
||||
} /* scalbn */
|
||||
|
||||
708
third-party/fdlibm/s_sqrt.c
vendored
708
third-party/fdlibm/s_sqrt.c
vendored
@ -6,79 +6,80 @@
|
||||
*
|
||||
* Developed at SunSoft, a Sun Microsystems, Inc. business.
|
||||
* Permission to use, copy, modify, and distribute this
|
||||
* software is freely granted, provided that this notice
|
||||
* software is freely granted, provided that this notice
|
||||
* is preserved.
|
||||
* ====================================================
|
||||
*/
|
||||
|
||||
/* sqrt(x)
|
||||
* Return correctly rounded sqrt.
|
||||
*
|
||||
* ------------------------------------------
|
||||
* | Use the hardware sqrt if you have one |
|
||||
* | Use the hardware sqrt if you have one |
|
||||
* ------------------------------------------
|
||||
*
|
||||
* Method:
|
||||
* Bit by bit method using integer arithmetic. (Slow, but portable)
|
||||
* 1. Normalization
|
||||
* Scale x to y in [1,4) with even powers of 2:
|
||||
* find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
|
||||
* sqrt(x) = 2^k * sqrt(y)
|
||||
* Scale x to y in [1,4) with even powers of 2:
|
||||
* find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
|
||||
* sqrt(x) = 2^k * sqrt(y)
|
||||
* 2. Bit by bit computation
|
||||
* Let q = sqrt(y) truncated to i bit after binary point (q = 1),
|
||||
* i 0
|
||||
* Let q = sqrt(y) truncated to i bit after binary point (q = 1),
|
||||
* i 0
|
||||
* i+1 2
|
||||
* s = 2*q , and y = 2 * ( y - q ). (1)
|
||||
* i i i i
|
||||
* s = 2*q , and y = 2 * ( y - q ). (1)
|
||||
* i i i i
|
||||
*
|
||||
* To compute q from q , one checks whether
|
||||
* i+1 i
|
||||
* To compute q from q , one checks whether
|
||||
* i+1 i
|
||||
*
|
||||
* -(i+1) 2
|
||||
* (q + 2 ) <= y. (2)
|
||||
* i
|
||||
* -(i+1)
|
||||
* If (2) is false, then q = q ; otherwise q = q + 2 .
|
||||
* i+1 i i+1 i
|
||||
* -(i+1) 2
|
||||
* (q + 2 ) <= y. (2)
|
||||
* i
|
||||
* -(i+1)
|
||||
* If (2) is false, then q = q ; otherwise q = q + 2 .
|
||||
* i+1 i i+1 i
|
||||
*
|
||||
* With some algebric manipulation, it is not difficult to see
|
||||
* that (2) is equivalent to
|
||||
* With some algebric manipulation, it is not difficult to see
|
||||
* that (2) is equivalent to
|
||||
* -(i+1)
|
||||
* s + 2 <= y (3)
|
||||
* i i
|
||||
* s + 2 <= y (3)
|
||||
* i i
|
||||
*
|
||||
* The advantage of (3) is that s and y can be computed by
|
||||
* i i
|
||||
* the following recurrence formula:
|
||||
* if (3) is false
|
||||
* The advantage of (3) is that s and y can be computed by
|
||||
* i i
|
||||
* the following recurrence formula:
|
||||
* if (3) is false
|
||||
*
|
||||
* s = s , y = y ; (4)
|
||||
* i+1 i i+1 i
|
||||
* s = s , y = y ; (4)
|
||||
* i+1 i i+1 i
|
||||
*
|
||||
* otherwise,
|
||||
* otherwise,
|
||||
* -i -(i+1)
|
||||
* s = s + 2 , y = y - s - 2 (5)
|
||||
* s = s + 2 , y = y - s - 2 (5)
|
||||
* i+1 i i+1 i i
|
||||
*
|
||||
* One may easily use induction to prove (4) and (5).
|
||||
* Note. Since the left hand side of (3) contain only i+2 bits,
|
||||
* it does not necessary to do a full (53-bit) comparison
|
||||
* in (3).
|
||||
* One may easily use induction to prove (4) and (5).
|
||||
* Note. Since the left hand side of (3) contain only i+2 bits,
|
||||
* it does not necessary to do a full (53-bit) comparison
|
||||
* in (3).
|
||||
* 3. Final rounding
|
||||
* After generating the 53 bits result, we compute one more bit.
|
||||
* Together with the remainder, we can decide whether the
|
||||
* result is exact, bigger than 1/2ulp, or less than 1/2ulp
|
||||
* (it will never equal to 1/2ulp).
|
||||
* The rounding mode can be detected by checking whether
|
||||
* huge + tiny is equal to huge, and whether huge - tiny is
|
||||
* equal to huge for some floating point number "huge" and "tiny".
|
||||
* After generating the 53 bits result, we compute one more bit.
|
||||
* Together with the remainder, we can decide whether the
|
||||
* result is exact, bigger than 1/2ulp, or less than 1/2ulp
|
||||
* (it will never equal to 1/2ulp).
|
||||
* The rounding mode can be detected by checking whether
|
||||
* huge + tiny is equal to huge, and whether huge - tiny is
|
||||
* equal to huge for some floating point number "huge" and "tiny".
|
||||
*
|
||||
* Special cases:
|
||||
* sqrt(+-0) = +-0 ... exact
|
||||
* sqrt(inf) = inf
|
||||
* sqrt(-ve) = NaN ... with invalid signal
|
||||
* sqrt(NaN) = NaN ... with invalid signal for signaling NaN
|
||||
* sqrt(+-0) = +-0 ... exact
|
||||
* sqrt(inf) = inf
|
||||
* sqrt(-ve) = NaN ... with invalid signal
|
||||
* sqrt(NaN) = NaN ... with invalid signal for signaling NaN
|
||||
*
|
||||
* Other methods : see the appended file at the end of the program below.
|
||||
*---------------
|
||||
* Other methods: see the appended file at the end of the program below.
|
||||
*/
|
||||
|
||||
#include "fdlibm.h"
|
||||
@ -86,103 +87,143 @@
|
||||
#define one 1.0
|
||||
#define tiny 1.0e-300
|
||||
|
||||
double sqrt(double x)
|
||||
double
|
||||
sqrt (double x)
|
||||
{
|
||||
double z;
|
||||
int sign = (int)0x80000000;
|
||||
unsigned r,t1,s1,ix1,q1;
|
||||
int ix0,s0,q,m,t,i;
|
||||
double z;
|
||||
int sign = (int) 0x80000000;
|
||||
unsigned r, t1, s1, ix1, q1;
|
||||
int ix0, s0, q, m, t, i;
|
||||
|
||||
ix0 = __HI(x); /* high word of x */
|
||||
ix1 = __LO(x); /* low word of x */
|
||||
ix0 = __HI (x); /* high word of x */
|
||||
ix1 = __LO (x); /* low word of x */
|
||||
|
||||
/* take care of Inf and NaN */
|
||||
if((ix0&0x7ff00000)==0x7ff00000) {
|
||||
return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
|
||||
sqrt(-inf)=sNaN */
|
||||
}
|
||||
/* take care of zero */
|
||||
if(ix0<=0) {
|
||||
if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
|
||||
else if(ix0<0)
|
||||
return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
|
||||
}
|
||||
/* normalize x */
|
||||
m = (ix0>>20);
|
||||
if(m==0) { /* subnormal x */
|
||||
while(ix0==0) {
|
||||
m -= 21;
|
||||
ix0 |= (ix1>>11); ix1 <<= 21;
|
||||
}
|
||||
for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
|
||||
m -= i-1;
|
||||
ix0 |= (ix1>>(32-i));
|
||||
ix1 <<= i;
|
||||
}
|
||||
m -= 1023; /* unbias exponent */
|
||||
ix0 = (ix0&0x000fffff)|0x00100000;
|
||||
if(m&1){ /* odd m, double x to make it even */
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
}
|
||||
m >>= 1; /* m = [m/2] */
|
||||
/* take care of Inf and NaN */
|
||||
if ((ix0 & 0x7ff00000) == 0x7ff00000)
|
||||
{
|
||||
return x * x + x; /* sqrt(NaN) = NaN, sqrt(+inf) = +inf, sqrt(-inf) = sNaN */
|
||||
}
|
||||
/* take care of zero */
|
||||
if (ix0 <= 0)
|
||||
{
|
||||
if (((ix0 & (~sign)) | ix1) == 0) /* sqrt(+-0) = +-0 */
|
||||
{
|
||||
return x;
|
||||
}
|
||||
else if (ix0 < 0) /* sqrt(-ve) = sNaN */
|
||||
{
|
||||
return (x - x) / (x - x);
|
||||
}
|
||||
}
|
||||
/* normalize x */
|
||||
m = (ix0 >> 20);
|
||||
if (m == 0) /* subnormal x */
|
||||
{
|
||||
while (ix0 == 0)
|
||||
{
|
||||
m -= 21;
|
||||
ix0 |= (ix1 >> 11);
|
||||
ix1 <<= 21;
|
||||
}
|
||||
for (i = 0; (ix0 & 0x00100000) == 0; i++)
|
||||
{
|
||||
ix0 <<= 1;
|
||||
}
|
||||
m -= i - 1;
|
||||
ix0 |= (ix1 >> (32 - i));
|
||||
ix1 <<= i;
|
||||
}
|
||||
m -= 1023; /* unbias exponent */
|
||||
ix0 = (ix0 & 0x000fffff) | 0x00100000;
|
||||
if (m & 1) /* odd m, double x to make it even */
|
||||
{
|
||||
ix0 += ix0 + ((ix1 & sign) >> 31);
|
||||
ix1 += ix1;
|
||||
}
|
||||
m >>= 1; /* m = [m / 2] */
|
||||
|
||||
/* generate sqrt(x) bit by bit */
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
|
||||
r = 0x00200000; /* r = moving bit from right to left */
|
||||
/* generate sqrt(x) bit by bit */
|
||||
ix0 += ix0 + ((ix1 & sign) >> 31);
|
||||
ix1 += ix1;
|
||||
q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
|
||||
r = 0x00200000; /* r = moving bit from right to left */
|
||||
|
||||
while(r!=0) {
|
||||
t = s0+r;
|
||||
if(t<=ix0) {
|
||||
s0 = t+r;
|
||||
ix0 -= t;
|
||||
q += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
r>>=1;
|
||||
}
|
||||
while (r != 0)
|
||||
{
|
||||
t = s0 + r;
|
||||
if (t <= ix0)
|
||||
{
|
||||
s0 = t + r;
|
||||
ix0 -= t;
|
||||
q += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1 & sign) >> 31);
|
||||
ix1 += ix1;
|
||||
r >>= 1;
|
||||
}
|
||||
|
||||
r = sign;
|
||||
while(r!=0) {
|
||||
t1 = s1+r;
|
||||
t = s0;
|
||||
if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
|
||||
s1 = t1+r;
|
||||
if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
|
||||
ix0 -= t;
|
||||
if (ix1 < t1) ix0 -= 1;
|
||||
ix1 -= t1;
|
||||
q1 += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1&sign)>>31);
|
||||
ix1 += ix1;
|
||||
r>>=1;
|
||||
}
|
||||
r = sign;
|
||||
while (r != 0)
|
||||
{
|
||||
t1 = s1 + r;
|
||||
t = s0;
|
||||
if ((t < ix0) || ((t == ix0) && (t1 <= ix1)))
|
||||
{
|
||||
s1 = t1 + r;
|
||||
if (((t1 & sign) == sign) && (s1 & sign) == 0)
|
||||
{
|
||||
s0 += 1;
|
||||
}
|
||||
ix0 -= t;
|
||||
if (ix1 < t1)
|
||||
{
|
||||
ix0 -= 1;
|
||||
}
|
||||
ix1 -= t1;
|
||||
q1 += r;
|
||||
}
|
||||
ix0 += ix0 + ((ix1 & sign) >> 31);
|
||||
ix1 += ix1;
|
||||
r >>= 1;
|
||||
}
|
||||
|
||||
/* use floating add to find out rounding direction */
|
||||
if((ix0|ix1)!=0) {
|
||||
z = one-tiny; /* trigger inexact flag */
|
||||
if (z>=one) {
|
||||
z = one+tiny;
|
||||
if (q1==(unsigned)0xffffffff) { q1=0; q += 1;}
|
||||
else if (z>one) {
|
||||
if (q1==(unsigned)0xfffffffe) q+=1;
|
||||
q1+=2;
|
||||
} else
|
||||
q1 += (q1&1);
|
||||
}
|
||||
}
|
||||
ix0 = (q>>1)+0x3fe00000;
|
||||
ix1 = q1>>1;
|
||||
if ((q&1)==1) ix1 |= sign;
|
||||
ix0 += (m <<20);
|
||||
__HI(z) = ix0;
|
||||
__LO(z) = ix1;
|
||||
return z;
|
||||
}
|
||||
/* use floating add to find out rounding direction */
|
||||
if ((ix0 | ix1) != 0)
|
||||
{
|
||||
z = one - tiny; /* trigger inexact flag */
|
||||
if (z >= one)
|
||||
{
|
||||
z = one + tiny;
|
||||
if (q1 == (unsigned) 0xffffffff)
|
||||
{
|
||||
q1 = 0;
|
||||
q += 1;
|
||||
}
|
||||
else if (z > one)
|
||||
{
|
||||
if (q1 == (unsigned) 0xfffffffe)
|
||||
{
|
||||
q += 1;
|
||||
}
|
||||
q1 += 2;
|
||||
}
|
||||
else
|
||||
{
|
||||
q1 += (q1 & 1);
|
||||
}
|
||||
}
|
||||
}
|
||||
ix0 = (q >> 1) + 0x3fe00000;
|
||||
ix1 = q1 >> 1;
|
||||
if ((q & 1) == 1)
|
||||
{
|
||||
ix1 |= sign;
|
||||
}
|
||||
ix0 += (m << 20);
|
||||
__HI (z) = ix0;
|
||||
__LO (z) = ix1;
|
||||
return z;
|
||||
} /* sqrt */
|
||||
|
||||
/*
|
||||
Other methods (use floating-point arithmetic)
|
||||
@ -190,253 +231,252 @@ Other methods (use floating-point arithmetic)
|
||||
(This is a copy of a drafted paper by Prof W. Kahan
|
||||
and K.C. Ng, written in May, 1986)
|
||||
|
||||
Two algorithms are given here to implement sqrt(x)
|
||||
(IEEE double precision arithmetic) in software.
|
||||
Both supply sqrt(x) correctly rounded. The first algorithm (in
|
||||
Section A) uses newton iterations and involves four divisions.
|
||||
The second one uses reciproot iterations to avoid division, but
|
||||
requires more multiplications. Both algorithms need the ability
|
||||
to chop results of arithmetic operations instead of round them,
|
||||
and the INEXACT flag to indicate when an arithmetic operation
|
||||
is executed exactly with no roundoff error, all part of the
|
||||
standard (IEEE 754-1985). The ability to perform shift, add,
|
||||
subtract and logical AND operations upon 32-bit words is needed
|
||||
too, though not part of the standard.
|
||||
Two algorithms are given here to implement sqrt(x)
|
||||
(IEEE double precision arithmetic) in software.
|
||||
Both supply sqrt(x) correctly rounded. The first algorithm (in
|
||||
Section A) uses newton iterations and involves four divisions.
|
||||
The second one uses reciproot iterations to avoid division, but
|
||||
requires more multiplications. Both algorithms need the ability
|
||||
to chop results of arithmetic operations instead of round them,
|
||||
and the INEXACT flag to indicate when an arithmetic operation
|
||||
is executed exactly with no roundoff error, all part of the
|
||||
standard (IEEE 754-1985). The ability to perform shift, add,
|
||||
subtract and logical AND operations upon 32-bit words is needed
|
||||
too, though not part of the standard.
|
||||
|
||||
A. sqrt(x) by Newton Iteration
|
||||
|
||||
(1) Initial approximation
|
||||
(1) Initial approximation
|
||||
|
||||
Let x0 and x1 be the leading and the trailing 32-bit words of
|
||||
a floating point number x (in IEEE double format) respectively
|
||||
Let x0 and x1 be the leading and the trailing 32-bit words of
|
||||
a floating point number x (in IEEE double format) respectively
|
||||
|
||||
1 11 52 ...widths
|
||||
------------------------------------------------------
|
||||
x: |s| e | f |
|
||||
------------------------------------------------------
|
||||
msb lsb msb lsb ...order
|
||||
1 11 52 ...widths
|
||||
------------------------------------------------------
|
||||
x: |s| e | f |
|
||||
------------------------------------------------------
|
||||
msb lsb msb lsb ...order
|
||||
|
||||
|
||||
------------------------ ------------------------
|
||||
x0: |s| e | f1 | x1: | f2 |
|
||||
------------------------ ------------------------
|
||||
------------------------ ------------------------
|
||||
x0: |s| e | f1 | x1: | f2 |
|
||||
------------------------ ------------------------
|
||||
|
||||
By performing shifts and subtracts on x0 and x1 (both regarded
|
||||
as integers), we obtain an 8-bit approximation of sqrt(x) as
|
||||
follows.
|
||||
By performing shifts and subtracts on x0 and x1 (both regarded
|
||||
as integers), we obtain an 8-bit approximation of sqrt(x) as
|
||||
follows.
|
||||
|
||||
k := (x0>>1) + 0x1ff80000;
|
||||
y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
|
||||
Here k is a 32-bit integer and T1[] is an integer array containing
|
||||
correction terms. Now magically the floating value of y (y's
|
||||
leading 32-bit word is y0, the value of its trailing word is 0)
|
||||
approximates sqrt(x) to almost 8-bit.
|
||||
k := (x0>>1) + 0x1ff80000;
|
||||
y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
|
||||
Here k is a 32-bit integer and T1[] is an integer array containing
|
||||
correction terms. Now magically the floating value of y (y's
|
||||
leading 32-bit word is y0, the value of its trailing word is 0)
|
||||
approximates sqrt(x) to almost 8-bit.
|
||||
|
||||
Value of T1:
|
||||
static int T1[32]= {
|
||||
0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
|
||||
29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
|
||||
83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
|
||||
16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
|
||||
Value of T1:
|
||||
static int T1[32]= {
|
||||
0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
|
||||
29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
|
||||
83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
|
||||
16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
|
||||
|
||||
(2) Iterative refinement
|
||||
(2) Iterative refinement
|
||||
|
||||
Apply Heron's rule three times to y, we have y approximates
|
||||
sqrt(x) to within 1 ulp (Unit in the Last Place):
|
||||
Apply Heron's rule three times to y, we have y approximates
|
||||
sqrt(x) to within 1 ulp (Unit in the Last Place):
|
||||
|
||||
y := (y+x/y)/2 ... almost 17 sig. bits
|
||||
y := (y+x/y)/2 ... almost 35 sig. bits
|
||||
y := y-(y-x/y)/2 ... within 1 ulp
|
||||
y := (y+x/y)/2 ... almost 17 sig. bits
|
||||
y := (y+x/y)/2 ... almost 35 sig. bits
|
||||
y := y-(y-x/y)/2 ... within 1 ulp
|
||||
|
||||
|
||||
Remark 1.
|
||||
Another way to improve y to within 1 ulp is:
|
||||
Remark 1.
|
||||
Another way to improve y to within 1 ulp is:
|
||||
|
||||
y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
|
||||
y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
|
||||
y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
|
||||
y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
|
||||
|
||||
2
|
||||
(x-y )*y
|
||||
y := y + 2* ---------- ...within 1 ulp
|
||||
2
|
||||
3y + x
|
||||
2
|
||||
(x-y )*y
|
||||
y := y + 2* ---------- ...within 1 ulp
|
||||
2
|
||||
3y + x
|
||||
|
||||
|
||||
This formula has one division fewer than the one above; however,
|
||||
it requires more multiplications and additions. Also x must be
|
||||
scaled in advance to avoid spurious overflow in evaluating the
|
||||
expression 3y*y+x. Hence it is not recommended uless division
|
||||
is slow. If division is very slow, then one should use the
|
||||
reciproot algorithm given in section B.
|
||||
This formula has one division fewer than the one above; however,
|
||||
it requires more multiplications and additions. Also x must be
|
||||
scaled in advance to avoid spurious overflow in evaluating the
|
||||
expression 3y*y+x. Hence it is not recommended uless division
|
||||
is slow. If division is very slow, then one should use the
|
||||
reciproot algorithm given in section B.
|
||||
|
||||
(3) Final adjustment
|
||||
|
||||
By twiddling y's last bit it is possible to force y to be
|
||||
correctly rounded according to the prevailing rounding mode
|
||||
as follows. Let r and i be copies of the rounding mode and
|
||||
inexact flag before entering the square root program. Also we
|
||||
use the expression y+-ulp for the next representable floating
|
||||
numbers (up and down) of y. Note that y+-ulp = either fixed
|
||||
point y+-1, or multiply y by nextafter(1,+-inf) in chopped
|
||||
mode.
|
||||
By twiddling y's last bit it is possible to force y to be
|
||||
correctly rounded according to the prevailing rounding mode
|
||||
as follows. Let r and i be copies of the rounding mode and
|
||||
inexact flag before entering the square root program. Also we
|
||||
use the expression y+-ulp for the next representable floating
|
||||
numbers (up and down) of y. Note that y+-ulp = either fixed
|
||||
point y+-1, or multiply y by nextafter(1,+-inf) in chopped
|
||||
mode.
|
||||
|
||||
I := FALSE; ... reset INEXACT flag I
|
||||
R := RZ; ... set rounding mode to round-toward-zero
|
||||
z := x/y; ... chopped quotient, possibly inexact
|
||||
If(not I) then { ... if the quotient is exact
|
||||
if(z=y) {
|
||||
I := i; ... restore inexact flag
|
||||
R := r; ... restore rounded mode
|
||||
return sqrt(x):=y.
|
||||
} else {
|
||||
z := z - ulp; ... special rounding
|
||||
}
|
||||
}
|
||||
i := TRUE; ... sqrt(x) is inexact
|
||||
If (r=RN) then z=z+ulp ... rounded-to-nearest
|
||||
If (r=RP) then { ... round-toward-+inf
|
||||
y = y+ulp; z=z+ulp;
|
||||
}
|
||||
y := y+z; ... chopped sum
|
||||
y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
|
||||
I := i; ... restore inexact flag
|
||||
R := r; ... restore rounded mode
|
||||
return sqrt(x):=y.
|
||||
I := FALSE; ... reset INEXACT flag I
|
||||
R := RZ; ... set rounding mode to round-toward-zero
|
||||
z := x/y; ... chopped quotient, possibly inexact
|
||||
If(not I) then { ... if the quotient is exact
|
||||
if(z=y) {
|
||||
I := i; ... restore inexact flag
|
||||
R := r; ... restore rounded mode
|
||||
return sqrt(x):=y.
|
||||
} else {
|
||||
z := z - ulp; ... special rounding
|
||||
}
|
||||
}
|
||||
i := TRUE; ... sqrt(x) is inexact
|
||||
If (r=RN) then z=z+ulp ... rounded-to-nearest
|
||||
If (r=RP) then { ... round-toward-+inf
|
||||
y = y+ulp; z=z+ulp;
|
||||
}
|
||||
y := y+z; ... chopped sum
|
||||
y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
|
||||
I := i; ... restore inexact flag
|
||||
R := r; ... restore rounded mode
|
||||
return sqrt(x):=y.
|
||||
|
||||
(4) Special cases
|
||||
(4) Special cases
|
||||
|
||||
Square root of +inf, +-0, or NaN is itself;
|
||||
Square root of a negative number is NaN with invalid signal.
|
||||
Square root of +inf, +-0, or NaN is itself;
|
||||
Square root of a negative number is NaN with invalid signal.
|
||||
|
||||
|
||||
B. sqrt(x) by Reciproot Iteration
|
||||
|
||||
(1) Initial approximation
|
||||
(1) Initial approximation
|
||||
|
||||
Let x0 and x1 be the leading and the trailing 32-bit words of
|
||||
a floating point number x (in IEEE double format) respectively
|
||||
(see section A). By performing shifs and subtracts on x0 and y0,
|
||||
we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
|
||||
Let x0 and x1 be the leading and the trailing 32-bit words of
|
||||
a floating point number x (in IEEE double format) respectively
|
||||
(see section A). By performing shifs and subtracts on x0 and y0,
|
||||
we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
|
||||
|
||||
k := 0x5fe80000 - (x0>>1);
|
||||
y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
|
||||
k := 0x5fe80000 - (x0>>1);
|
||||
y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
|
||||
|
||||
Here k is a 32-bit integer and T2[] is an integer array
|
||||
containing correction terms. Now magically the floating
|
||||
value of y (y's leading 32-bit word is y0, the value of
|
||||
its trailing word y1 is set to zero) approximates 1/sqrt(x)
|
||||
to almost 7.8-bit.
|
||||
Here k is a 32-bit integer and T2[] is an integer array
|
||||
containing correction terms. Now magically the floating
|
||||
value of y (y's leading 32-bit word is y0, the value of
|
||||
its trailing word y1 is set to zero) approximates 1/sqrt(x)
|
||||
to almost 7.8-bit.
|
||||
|
||||
Value of T2:
|
||||
static int T2[64]= {
|
||||
0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
|
||||
0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
|
||||
0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
|
||||
0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
|
||||
0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
|
||||
0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
|
||||
0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
|
||||
0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
|
||||
Value of T2:
|
||||
static int T2[64]= {
|
||||
0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
|
||||
0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
|
||||
0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
|
||||
0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
|
||||
0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
|
||||
0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
|
||||
0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
|
||||
0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
|
||||
|
||||
(2) Iterative refinement
|
||||
(2) Iterative refinement
|
||||
|
||||
Apply Reciproot iteration three times to y and multiply the
|
||||
result by x to get an approximation z that matches sqrt(x)
|
||||
to about 1 ulp. To be exact, we will have
|
||||
-1ulp < sqrt(x)-z<1.0625ulp.
|
||||
Apply Reciproot iteration three times to y and multiply the
|
||||
result by x to get an approximation z that matches sqrt(x)
|
||||
to about 1 ulp. To be exact, we will have
|
||||
-1ulp < sqrt(x)-z<1.0625ulp.
|
||||
|
||||
... set rounding mode to Round-to-nearest
|
||||
y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
|
||||
y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
|
||||
... special arrangement for better accuracy
|
||||
z := x*y ... 29 bits to sqrt(x), with z*y<1
|
||||
z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
|
||||
... set rounding mode to Round-to-nearest
|
||||
y := y*(1.5-0.5*x*y*y) ... almost 15 sig. bits to 1/sqrt(x)
|
||||
y := y*((1.5-2^-30)+0.5*x*y*y)... about 29 sig. bits to 1/sqrt(x)
|
||||
... special arrangement for better accuracy
|
||||
z := x*y ... 29 bits to sqrt(x), with z*y<1
|
||||
z := z + 0.5*z*(1-z*y) ... about 1 ulp to sqrt(x)
|
||||
|
||||
Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
|
||||
(a) the term z*y in the final iteration is always less than 1;
|
||||
(b) the error in the final result is biased upward so that
|
||||
-1 ulp < sqrt(x) - z < 1.0625 ulp
|
||||
instead of |sqrt(x)-z|<1.03125ulp.
|
||||
Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
|
||||
(a) the term z*y in the final iteration is always less than 1;
|
||||
(b) the error in the final result is biased upward so that
|
||||
-1 ulp < sqrt(x) - z < 1.0625 ulp
|
||||
instead of |sqrt(x)-z|<1.03125ulp.
|
||||
|
||||
(3) Final adjustment
|
||||
(3) Final adjustment
|
||||
|
||||
By twiddling y's last bit it is possible to force y to be
|
||||
correctly rounded according to the prevailing rounding mode
|
||||
as follows. Let r and i be copies of the rounding mode and
|
||||
inexact flag before entering the square root program. Also we
|
||||
use the expression y+-ulp for the next representable floating
|
||||
numbers (up and down) of y. Note that y+-ulp = either fixed
|
||||
point y+-1, or multiply y by nextafter(1,+-inf) in chopped
|
||||
mode.
|
||||
By twiddling y's last bit it is possible to force y to be
|
||||
correctly rounded according to the prevailing rounding mode
|
||||
as follows. Let r and i be copies of the rounding mode and
|
||||
inexact flag before entering the square root program. Also we
|
||||
use the expression y+-ulp for the next representable floating
|
||||
numbers (up and down) of y. Note that y+-ulp = either fixed
|
||||
point y+-1, or multiply y by nextafter(1,+-inf) in chopped
|
||||
mode.
|
||||
|
||||
R := RZ; ... set rounding mode to round-toward-zero
|
||||
switch(r) {
|
||||
case RN: ... round-to-nearest
|
||||
if(x<= z*(z-ulp)...chopped) z = z - ulp; else
|
||||
if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
|
||||
break;
|
||||
case RZ:case RM: ... round-to-zero or round-to--inf
|
||||
R:=RP; ... reset rounding mod to round-to-+inf
|
||||
if(x<z*z ... rounded up) z = z - ulp; else
|
||||
if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
|
||||
break;
|
||||
case RP: ... round-to-+inf
|
||||
if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
|
||||
if(x>z*z ...chopped) z = z+ulp;
|
||||
break;
|
||||
}
|
||||
R := RZ; ... set rounding mode to round-toward-zero
|
||||
switch(r) {
|
||||
case RN: ... round-to-nearest
|
||||
if(x<= z*(z-ulp)...chopped) z = z - ulp; else
|
||||
if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
|
||||
break;
|
||||
case RZ:case RM: ... round-to-zero or round-to--inf
|
||||
R:=RP; ... reset rounding mod to round-to-+inf
|
||||
if(x<z*z ... rounded up) z = z - ulp; else
|
||||
if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
|
||||
break;
|
||||
case RP: ... round-to-+inf
|
||||
if(x>(z+ulp)*(z+ulp)...chopped) z = z+2*ulp; else
|
||||
if(x>z*z ...chopped) z = z+ulp;
|
||||
break;
|
||||
}
|
||||
|
||||
Remark 3. The above comparisons can be done in fixed point. For
|
||||
example, to compare x and w=z*z chopped, it suffices to compare
|
||||
x1 and w1 (the trailing parts of x and w), regarding them as
|
||||
two's complement integers.
|
||||
Remark 3. The above comparisons can be done in fixed point. For
|
||||
example, to compare x and w=z*z chopped, it suffices to compare
|
||||
x1 and w1 (the trailing parts of x and w), regarding them as
|
||||
two's complement integers.
|
||||
|
||||
...Is z an exact square root?
|
||||
To determine whether z is an exact square root of x, let z1 be the
|
||||
trailing part of z, and also let x0 and x1 be the leading and
|
||||
trailing parts of x.
|
||||
...Is z an exact square root?
|
||||
To determine whether z is an exact square root of x, let z1 be the
|
||||
trailing part of z, and also let x0 and x1 be the leading and
|
||||
trailing parts of x.
|
||||
|
||||
If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
|
||||
I := 1; ... Raise Inexact flag: z is not exact
|
||||
else {
|
||||
j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
|
||||
k := z1 >> 26; ... get z's 25-th and 26-th
|
||||
fraction bits
|
||||
I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
|
||||
}
|
||||
R:= r ... restore rounded mode
|
||||
return sqrt(x):=z.
|
||||
If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
|
||||
I := 1; ... Raise Inexact flag: z is not exact
|
||||
else {
|
||||
j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
|
||||
k := z1 >> 26; ... get z's 25-th and 26-th
|
||||
fraction bits
|
||||
I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
|
||||
}
|
||||
R:= r ... restore rounded mode
|
||||
return sqrt(x):=z.
|
||||
|
||||
If multiplication is cheaper then the foregoing red tape, the
|
||||
Inexact flag can be evaluated by
|
||||
If multiplication is cheaper then the foregoing red tape, the
|
||||
Inexact flag can be evaluated by
|
||||
|
||||
I := i;
|
||||
I := (z*z!=x) or I.
|
||||
I := i;
|
||||
I := (z*z!=x) or I.
|
||||
|
||||
Note that z*z can overwrite I; this value must be sensed if it is
|
||||
True.
|
||||
Note that z*z can overwrite I; this value must be sensed if it is
|
||||
True.
|
||||
|
||||
Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
|
||||
zero.
|
||||
Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
|
||||
zero.
|
||||
|
||||
--------------------
|
||||
z1: | f2 |
|
||||
--------------------
|
||||
bit 31 bit 0
|
||||
--------------------
|
||||
z1: | f2 |
|
||||
--------------------
|
||||
bit 31 bit 0
|
||||
|
||||
Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
|
||||
or even of logb(x) have the following relations:
|
||||
Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
|
||||
or even of logb(x) have the following relations:
|
||||
|
||||
-------------------------------------------------
|
||||
bit 27,26 of z1 bit 1,0 of x1 logb(x)
|
||||
-------------------------------------------------
|
||||
00 00 odd and even
|
||||
01 01 even
|
||||
10 10 odd
|
||||
10 00 even
|
||||
11 01 even
|
||||
-------------------------------------------------
|
||||
|
||||
(4) Special cases (see (4) of Section A).
|
||||
-------------------------------------------------
|
||||
bit 27,26 of z1 bit 1,0 of x1 logb(x)
|
||||
-------------------------------------------------
|
||||
00 00 odd and even
|
||||
01 01 even
|
||||
10 10 odd
|
||||
10 00 even
|
||||
11 01 even
|
||||
-------------------------------------------------
|
||||
|
||||
(4) Special cases (see (4) of Section A).
|
||||
*/
|
||||
|
||||
1383
third-party/fdlibm/s_trig.c
vendored
1383
third-party/fdlibm/s_trig.c
vendored
File diff suppressed because it is too large
Load Diff
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x
Reference in New Issue
Block a user