GCC Code Coverage Report
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File: lib/libm/src/s_expm1.c Lines: 50 60 83.3 %
Date: 2017-11-13 Branches: 23 36 63.9 %

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/* @(#)s_expm1.c 5.1 93/09/24 */
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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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/* expm1(x)
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 * Returns exp(x)-1, the exponential of x minus 1.
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 *
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 * Method
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 *   1. Argument reduction:
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 *	Given x, find r and integer k such that
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 *
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 *               x = k*ln2 + r,  |r| <= 0.5*ln2 ~ 0.34658
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 *
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 *      Here a correction term c will be computed to compensate
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 *	the error in r when rounded to a floating-point number.
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 *
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 *   2. Approximating expm1(r) by a special rational function on
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 *	the interval [0,0.34658]:
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 *	Since
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 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
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 *	we define R1(r*r) by
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 *	    r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
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 *	That is,
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 *	    R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
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 *		     = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
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 *		     = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
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 *      We use a special Remes algorithm on [0,0.347] to generate
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 * 	a polynomial of degree 5 in r*r to approximate R1. The
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 *	maximum error of this polynomial approximation is bounded
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 *	by 2**-61. In other words,
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 *	    R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
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 *	where 	Q1  =  -1.6666666666666567384E-2,
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 * 		Q2  =   3.9682539681370365873E-4,
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 * 		Q3  =  -9.9206344733435987357E-6,
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 * 		Q4  =   2.5051361420808517002E-7,
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 * 		Q5  =  -6.2843505682382617102E-9;
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 *  	(where z=r*r, and the values of Q1 to Q5 are listed below)
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 *	with error bounded by
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 *	    |                  5           |     -61
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 *	    | 1.0+Q1*z+...+Q5*z   -  R1(z) | <= 2
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 *	    |                              |
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 *
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 *	expm1(r) = exp(r)-1 is then computed by the following
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 * 	specific way which minimize the accumulation rounding error:
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 *			       2     3
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 *			      r     r    [ 3 - (R1 + R1*r/2)  ]
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 *	      expm1(r) = r + --- + --- * [--------------------]
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 *		              2     2    [ 6 - r*(3 - R1*r/2) ]
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 *
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 *	To compensate the error in the argument reduction, we use
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 *		expm1(r+c) = expm1(r) + c + expm1(r)*c
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 *			   ~ expm1(r) + c + r*c
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 *	Thus c+r*c will be added in as the correction terms for
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 *	expm1(r+c). Now rearrange the term to avoid optimization
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 * 	screw up:
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 *		        (      2                                    2 )
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 *		        ({  ( r    [ R1 -  (3 - R1*r/2) ]  )  }    r  )
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 *	 expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
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 *	                ({  ( 2    [ 6 - r*(3 - R1*r/2) ]  )  }    2  )
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 *                      (                                             )
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 *
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 *		   = r - E
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 *   3. Scale back to obtain expm1(x):
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 *	From step 1, we have
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 *	   expm1(x) = either 2^k*[expm1(r)+1] - 1
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 *		    = or     2^k*[expm1(r) + (1-2^-k)]
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 *   4. Implementation notes:
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 *	(A). To save one multiplication, we scale the coefficient Qi
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 *	     to Qi*2^i, and replace z by (x^2)/2.
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 *	(B). To achieve maximum accuracy, we compute expm1(x) by
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 *	  (i)   if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
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 *	  (ii)  if k=0, return r-E
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 *	  (iii) if k=-1, return 0.5*(r-E)-0.5
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 *        (iv)	if k=1 if r < -0.25, return 2*((r+0.5)- E)
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 *	       	       else	     return  1.0+2.0*(r-E);
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 *	  (v)   if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
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 *	  (vi)  if k <= 20, return 2^k((1-2^-k)-(E-r)), else
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 *	  (vii) return 2^k(1-((E+2^-k)-r))
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 *
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 * Special cases:
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 *	expm1(INF) is INF, expm1(NaN) is NaN;
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 *	expm1(-INF) is -1, and
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 *	for finite argument, only expm1(0)=0 is exact.
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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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 * Misc. info.
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 *	For IEEE double
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 *	    if x >  7.09782712893383973096e+02 then expm1(x) overflow
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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 <float.h>
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#include <math.h>
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#include "math_private.h"
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static const double
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one		= 1.0,
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huge		= 1.0e+300,
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tiny		= 1.0e-300,
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o_threshold	= 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */
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ln2_hi		= 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */
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ln2_lo		= 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */
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invln2		= 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */
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	/* scaled coefficients related to expm1 */
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Q1  =  -3.33333333333331316428e-02, /* BFA11111 111110F4 */
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Q2  =   1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */
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Q3  =  -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */
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Q4  =   4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */
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Q5  =  -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
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double
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expm1(double x)
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{
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	double y,hi,lo,c,t,e,hxs,hfx,r1;
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	int32_t k,xsb;
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	u_int32_t hx;
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	GET_HIGH_WORD(hx,x);
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	xsb = hx&0x80000000;		/* sign bit of x */
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	if(xsb==0) y=x; else y= -x;	/* y = |x| */
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	hx &= 0x7fffffff;		/* high word of |x| */
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    /* filter out huge and non-finite argument */
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	if(hx >= 0x4043687A) {			/* if |x|>=56*ln2 */
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	    if(hx >= 0x40862E42) {		/* if |x|>=709.78... */
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                if(hx>=0x7ff00000) {
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		    u_int32_t low;
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		    GET_LOW_WORD(low,x);
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		    if(((hx&0xfffff)|low)!=0)
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		         return x+x; 	 /* NaN */
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		    else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
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	        }
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	        if(x > o_threshold) return huge*huge; /* overflow */
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	    }
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	    if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
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		if(x+tiny<0.0)		/* raise inexact */
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		return tiny-one;	/* return -1 */
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	    }
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	}
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    /* argument reduction */
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	if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
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	    if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
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		if(xsb==0)
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		    {hi = x - ln2_hi; lo =  ln2_lo;  k =  1;}
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		else
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		    {hi = x + ln2_hi; lo = -ln2_lo;  k = -1;}
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	    } else {
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		k  = invln2*x+((xsb==0)?0.5:-0.5);
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		t  = k;
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		hi = x - t*ln2_hi;	/* t*ln2_hi is exact here */
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		lo = t*ln2_lo;
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	    }
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	    x  = hi - lo;
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	    c  = (hi-x)-lo;
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	}
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	else if(hx < 0x3c900000) {  	/* when |x|<2**-54, return x */
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	    t = huge+x;	/* return x with inexact flags when x!=0 */
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	    return x - (t-(huge+x));
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	}
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	else k = 0;
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    /* x is now in primary range */
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	hfx = 0.5*x;
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	hxs = x*hfx;
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	r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
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	t  = 3.0-r1*hfx;
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	e  = hxs*((r1-t)/(6.0 - x*t));
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	if(k==0) return x - (x*e-hxs);		/* c is 0 */
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	else {
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	    e  = (x*(e-c)-c);
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	    e -= hxs;
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	    if(k== -1) return 0.5*(x-e)-0.5;
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	    if(k==1) {
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	       	if(x < -0.25) return -2.0*(e-(x+0.5));
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	       	else 	      return  one+2.0*(x-e);
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	    }
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	    if (k <= -2 || k>56) {   /* suffice to return exp(x)-1 */
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	        u_int32_t high;
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	        y = one-(e-x);
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		GET_HIGH_WORD(high,y);
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		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
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	        return y-one;
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	    }
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	    t = one;
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	    if(k<20) {
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	        u_int32_t high;
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	        SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k));  /* t=1-2^-k */
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	       	y = t-(e-x);
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		GET_HIGH_WORD(high,y);
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		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
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	   } else {
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	        u_int32_t high;
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		SET_HIGH_WORD(t,((0x3ff-k)<<20));	/* 2^-k */
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	       	y = x-(e+t);
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	       	y += one;
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		GET_HIGH_WORD(high,y);
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		SET_HIGH_WORD(y,high+(k<<20));	/* add k to y's exponent */
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	    }
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	}
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	return y;
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8946
}
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DEF_STD(expm1);
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LDBL_MAYBE_CLONE(expm1);