# code: 9ferno

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```/* derived from /netlib/fdlibm */

/* @(#)e_log.c 1.3 95/01/18 */
/*
* ====================================================
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/

/* __ieee754_log(x)
* Return the logrithm of x
*
* Method :
*   1. Argument Reduction: find k and f such that
*			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
*      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.
*
* 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.
*
* Accuracy:
*	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
* constants. The decimal values may be used, provided that the
* compiler will convert from decimal to binary accurately enough
* to produce the hexadecimal values shown.
*/

#include "fdlibm.h"

static const double
ln2_hi  =  6.93147180369123816490e-01,	/* 3fe62e42 fee00000 */
ln2_lo  =  1.90821492927058770002e-10,	/* 3dea39ef 35793c76 */
two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */

static double zero   =  0.0;

double __ieee754_log(double x)
{
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 */

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);
}
}
```