src/libm/e_log.c
author Sam Lantinga <slouken@libsdl.org>
Sat, 04 Nov 2017 15:53:19 -0700
changeset 11683 48bcba563d9c
parent 6044 35448a5ea044
child 11711 8a982ed61896
permissions -rw-r--r--
Updated math code from the uClibc 0.9.33 release
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/*
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 * ====================================================
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 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 *
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 * Developed at SunPro, a Sun Microsystems, Inc. business.
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 * Permission to use, copy, modify, and distribute this
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 * software is freely granted, provided that this notice
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 * is preserved.
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 * ====================================================
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 */
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/* __ieee754_log(x)
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 * Return the logrithm of x
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 *
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 * Method :
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 *   1. Argument Reduction: find k and f such that
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 *			x = 2^k * (1+f),
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 *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
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 *
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 *   2. Approximation of log(1+f).
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 *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
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 *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
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 *	     	 = 2s + s*R
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 *      We use a special Reme algorithm on [0,0.1716] to generate
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 * 	a polynomial of degree 14 to approximate R The maximum error
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 *	of this polynomial approximation is bounded by 2**-58.45. In
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 *	other words,
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 *		        2      4      6      8      10      12      14
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 *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
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 *  	(the values of Lg1 to Lg7 are listed in the program)
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 *	and
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 *	    |      2          14          |     -58.45
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 *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
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 *	    |                             |
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 *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
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 *	In order to guarantee error in log below 1ulp, we compute log
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 *	by
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 *		log(1+f) = f - s*(f - R)	(if f is not too large)
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 *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
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 *
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 *	3. Finally,  log(x) = k*ln2 + log(1+f).
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 *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
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 *	   Here ln2 is split into two floating point number:
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 *			ln2_hi + ln2_lo,
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 *	   where n*ln2_hi is always exact for |n| < 2000.
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 *
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 * Special cases:
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 *	log(x) is NaN with signal if x < 0 (including -INF) ;
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 *	log(+INF) is +INF; log(0) is -INF with signal;
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 *	log(NaN) is that NaN with no signal.
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 *
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 * Accuracy:
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 *	according to an error analysis, the error is always less than
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 *	1 ulp (unit in the last place).
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 *
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 * Constants:
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 * The hexadecimal values are the intended ones for the following
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 * constants. The decimal values may be used, provided that the
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 * compiler will convert from decimal to binary accurately enough
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 * to produce the hexadecimal values shown.
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 */
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#include "math_libm.h"
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#include "math_private.h"
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static const double
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ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
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ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
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two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
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Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
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Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
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Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
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Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
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Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
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Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
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Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
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static const double zero   =  0.0;
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double attribute_hidden __ieee754_log(double x)
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{
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	double hfsq,f,s,z,R,w,t1,t2,dk;
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	int32_t k,hx,i,j;
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	u_int32_t lx;
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	EXTRACT_WORDS(hx,lx,x);
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	k=0;
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	if (hx < 0x00100000) {			/* x < 2**-1022  */
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	    if (((hx&0x7fffffff)|lx)==0)
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		return -two54/zero;		/* log(+-0)=-inf */
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	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
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	    k -= 54; x *= two54; /* subnormal number, scale up x */
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	    GET_HIGH_WORD(hx,x);
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	}
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	if (hx >= 0x7ff00000) return x+x;
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	k += (hx>>20)-1023;
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	hx &= 0x000fffff;
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	i = (hx+0x95f64)&0x100000;
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	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
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	k += (i>>20);
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	f = x-1.0;
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	if((0x000fffff&(2+hx))<3) {	/* |f| < 2**-20 */
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	    if(f==zero) {if(k==0) return zero;  else {dk=(double)k;
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				 return dk*ln2_hi+dk*ln2_lo;}
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	    }
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	    R = f*f*(0.5-0.33333333333333333*f);
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	    if(k==0) return f-R; else {dk=(double)k;
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	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
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	}
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 	s = f/(2.0+f);
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	dk = (double)k;
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	z = s*s;
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	i = hx-0x6147a;
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	w = z*z;
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	j = 0x6b851-hx;
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	t1= w*(Lg2+w*(Lg4+w*Lg6));
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	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
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	i |= j;
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	R = t2+t1;
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	if(i>0) {
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	    hfsq=0.5*f*f;
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	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
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		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
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	} else {
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	    if(k==0) return f-s*(f-R); else
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		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
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	}
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}
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/*
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 * wrapper log(x)
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 */
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#ifndef _IEEE_LIBM
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double log(double x)
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{
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	double z = __ieee754_log(x);
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	if (_LIB_VERSION == _IEEE_ || isnan(x) || x > 0.0)
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		return z;
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	if (x == 0.0)
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		return __kernel_standard(x, x, 16); /* log(0) */
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	return __kernel_standard(x, x, 17); /* log(x<0) */
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}
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#else
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strong_alias(__ieee754_log, log)
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#endif
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libm_hidden_def(log)