text: /*
   * ====================================================
   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   *
   * Developed at SunPro, 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 "math_libm.h"
  #include "math_private.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 const double zero   =  0.0;
  
  double attribute_hidden __ieee754_log(double x)
  {
  	double hfsq,f,s,z,R,w,t1,t2,dk;
  	int32_t k,hx,i,j;
  	u_int32_t lx;
  
  	EXTRACT_WORDS(hx,lx,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 */
  	    GET_HIGH_WORD(hx,x);
  	}
  	if (hx >= 0x7ff00000) return x+x;
  	k += (hx>>20)-1023;
  	hx &= 0x000fffff;
  	i = (hx+0x95f64)&0x100000;
  	SET_HIGH_WORD(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);
  	}
  }
  
  /*
   * wrapper log(x)
   */
  #ifndef _IEEE_LIBM
  double log(double x)
  {
  	double z = __ieee754_log(x);
  	if (_LIB_VERSION == _IEEE_ || isnan(x) || x > 0.0)
  		return z;
  	if (x == 0.0)
  		return __kernel_standard(x, x, 16); /* log(0) */
  	return __kernel_standard(x, x, 17); /* log(x<0) */
  }
  #else
  strong_alias(__ieee754_log, log)
  #endif
  libm_hidden_def(log)
  