src/libm/e_log.c
author Ryan C. Gordon <icculus@icculus.org>
Mon, 21 May 2018 12:00:21 -0400
changeset 11994 8e094f91ddab
parent 11711 8a982ed61896
permissions -rw-r--r--
thread: fixed compiler warnings on non-Linux systems that use pthread.

(static function rtkit_setpriority was unused, moved it in with rest of
__LINUX__ section.)
     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)