src/libm/e_log.c
author Ryan C. Gordon <icculus@icculus.org>
Thu, 21 Apr 2016 03:16:44 -0400
changeset 11729 d1ce8396c356
parent 11711 8a982ed61896
permissions -rw-r--r--
Initial shot at a renderer target for Apple's Metal API.

This isn't complete, but is enough to run testsprite2. It's currently
Mac-only; with a little work to figure out how to properly glue in a Metal
layer to a UIView, this will likely work on iOS, too.

This is only wired up to the configure script right now, and disabled by
default. CMake and Xcode still need their bits filled in as appropriate.
     1 /*
     2  * ====================================================
     3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     4  *
     5  * Developed at SunPro, a Sun Microsystems, Inc. business.
     6  * Permission to use, copy, modify, and distribute this
     7  * software is freely granted, provided that this notice
     8  * is preserved.
     9  * ====================================================
    10  */
    11 
    12 #if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
    13 /* C4723: potential divide by zero. */
    14 #pragma warning ( disable : 4723 )
    15 #endif
    16 
    17 /* __ieee754_log(x)
    18  * Return the logrithm of x
    19  *
    20  * Method :
    21  *   1. Argument Reduction: find k and f such that
    22  *			x = 2^k * (1+f),
    23  *	   where  sqrt(2)/2 < 1+f < sqrt(2) .
    24  *
    25  *   2. Approximation of log(1+f).
    26  *	Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
    27  *		 = 2s + 2/3 s**3 + 2/5 s**5 + .....,
    28  *	     	 = 2s + s*R
    29  *      We use a special Reme algorithm on [0,0.1716] to generate
    30  * 	a polynomial of degree 14 to approximate R The maximum error
    31  *	of this polynomial approximation is bounded by 2**-58.45. In
    32  *	other words,
    33  *		        2      4      6      8      10      12      14
    34  *	    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
    35  *  	(the values of Lg1 to Lg7 are listed in the program)
    36  *	and
    37  *	    |      2          14          |     -58.45
    38  *	    | Lg1*s +...+Lg7*s    -  R(z) | <= 2
    39  *	    |                             |
    40  *	Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
    41  *	In order to guarantee error in log below 1ulp, we compute log
    42  *	by
    43  *		log(1+f) = f - s*(f - R)	(if f is not too large)
    44  *		log(1+f) = f - (hfsq - s*(hfsq+R)).	(better accuracy)
    45  *
    46  *	3. Finally,  log(x) = k*ln2 + log(1+f).
    47  *			    = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
    48  *	   Here ln2 is split into two floating point number:
    49  *			ln2_hi + ln2_lo,
    50  *	   where n*ln2_hi is always exact for |n| < 2000.
    51  *
    52  * Special cases:
    53  *	log(x) is NaN with signal if x < 0 (including -INF) ;
    54  *	log(+INF) is +INF; log(0) is -INF with signal;
    55  *	log(NaN) is that NaN with no signal.
    56  *
    57  * Accuracy:
    58  *	according to an error analysis, the error is always less than
    59  *	1 ulp (unit in the last place).
    60  *
    61  * Constants:
    62  * The hexadecimal values are the intended ones for the following
    63  * constants. The decimal values may be used, provided that the
    64  * compiler will convert from decimal to binary accurately enough
    65  * to produce the hexadecimal values shown.
    66  */
    67 
    68 #include "math_libm.h"
    69 #include "math_private.h"
    70 
    71 static const double
    72 ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
    73 ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
    74 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
    75 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
    76 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
    77 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
    78 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
    79 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
    80 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
    81 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
    82 
    83 static const double zero   =  0.0;
    84 
    85 double attribute_hidden __ieee754_log(double x)
    86 {
    87 	double hfsq,f,s,z,R,w,t1,t2,dk;
    88 	int32_t k,hx,i,j;
    89 	u_int32_t lx;
    90 
    91 	EXTRACT_WORDS(hx,lx,x);
    92 
    93 	k=0;
    94 	if (hx < 0x00100000) {			/* x < 2**-1022  */
    95 	    if (((hx&0x7fffffff)|lx)==0)
    96 		return -two54/zero;		/* log(+-0)=-inf */
    97 	    if (hx<0) return (x-x)/zero;	/* log(-#) = NaN */
    98 	    k -= 54; x *= two54; /* subnormal number, scale up x */
    99 	    GET_HIGH_WORD(hx,x);
   100 	}
   101 	if (hx >= 0x7ff00000) return x+x;
   102 	k += (hx>>20)-1023;
   103 	hx &= 0x000fffff;
   104 	i = (hx+0x95f64)&0x100000;
   105 	SET_HIGH_WORD(x,hx|(i^0x3ff00000));	/* normalize x or x/2 */
   106 	k += (i>>20);
   107 	f = x-1.0;
   108 	if((0x000fffff&(2+hx))<3) {	/* |f| < 2**-20 */
   109 	    if(f==zero) {if(k==0) return zero;  else {dk=(double)k;
   110 				 return dk*ln2_hi+dk*ln2_lo;}
   111 	    }
   112 	    R = f*f*(0.5-0.33333333333333333*f);
   113 	    if(k==0) return f-R; else {dk=(double)k;
   114 	    	     return dk*ln2_hi-((R-dk*ln2_lo)-f);}
   115 	}
   116  	s = f/(2.0+f);
   117 	dk = (double)k;
   118 	z = s*s;
   119 	i = hx-0x6147a;
   120 	w = z*z;
   121 	j = 0x6b851-hx;
   122 	t1= w*(Lg2+w*(Lg4+w*Lg6));
   123 	t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
   124 	i |= j;
   125 	R = t2+t1;
   126 	if(i>0) {
   127 	    hfsq=0.5*f*f;
   128 	    if(k==0) return f-(hfsq-s*(hfsq+R)); else
   129 		     return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
   130 	} else {
   131 	    if(k==0) return f-s*(f-R); else
   132 		     return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
   133 	}
   134 }
   135 
   136 /*
   137  * wrapper log(x)
   138  */
   139 #ifndef _IEEE_LIBM
   140 double log(double x)
   141 {
   142 	double z = __ieee754_log(x);
   143 	if (_LIB_VERSION == _IEEE_ || isnan(x) || x > 0.0)
   144 		return z;
   145 	if (x == 0.0)
   146 		return __kernel_standard(x, x, 16); /* log(0) */
   147 	return __kernel_standard(x, x, 17); /* log(x<0) */
   148 }
   149 #else
   150 strong_alias(__ieee754_log, log)
   151 #endif
   152 libm_hidden_def(log)