# code: 9ferno

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

/* @(#)e_exp.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_exp(x)
* Returns the exponential of x.
*
* Method
*   1. Argument reduction:
*      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
*	Given x, find r and integer k such that
*
*               x = k*ln2 + r,  |r| <= 0.5*ln2.
*
*      Here r will be represented as r = hi-lo for better
*	accuracy.
*
*   2. Approximation of exp(r) by a special rational function on
*	the interval [0,0.34658]:
*	Write
*	    R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
*      We use a special Reme algorithm on [0,0.34658] to generate
* 	a polynomial of degree 5 to approximate R. The maximum error
*	of this polynomial approximation is bounded by 2**-59. In
*	other words,
*	    R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
*  	(where z=r*r, and the values of P1 to P5 are listed below)
*	and
*	    |                  5          |     -59
*	    | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
*	    |                             |
*	The computation of exp(r) thus becomes
*                             2*r
*		exp(r) = 1 + -------
*		              R - r
*                                 r*R1(r)
*		       = 1 + r + ----------- (for better accuracy)
*		                  2 - R1(r)
*	where
*			         2       4             10
*		R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
*
*   3. Scale back to obtain exp(x):
*	From step 1, we have
*	   exp(x) = 2^k * exp(r)
*
* Special cases:
*	exp(INF) is INF, exp(NaN) is NaN;
*	exp(-INF) is 0, and
*	for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
*	according to an error analysis, the error is always less than
*	1 ulp (unit in the last place).
*
* Misc. info.
*	For IEEE double
*	    if x >  7.09782712893383973096e+02 then exp(x) overflow
*	    if x < -7.45133219101941108420e+02 then exp(x) underflow
*
* 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
one	= 1.0,
halF[2]	= {0.5,-0.5,},
Huge	= 1.0e+300,
twom1000= 9.33263618503218878990e-302,     /* 2**-1000=0x01700000,0*/
o_threshold=  7.09782712893383973096e+02,  /* 0x40862E42, 0xFEFA39EF */
u_threshold= -7.45133219101941108420e+02,  /* 0xc0874910, 0xD52D3051 */
ln2HI[2]   ={ 6.93147180369123816490e-01,  /* 0x3fe62e42, 0xfee00000 */
-6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
ln2LO[2]   ={ 1.90821492927058770002e-10,  /* 0x3dea39ef, 0x35793c76 */
-1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
invln2 =  1.44269504088896338700e+00, /* 0x3ff71547, 0x652b82fe */
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
P5   =  4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */

double __ieee754_exp(double x)	/* default IEEE double exp */
{
double y,hi,lo,c,t;
int k,xsb;
unsigned hx;

hx  = __HI(x);	/* high word of x */
xsb = (hx>>31)&1;		/* sign bit of x */
hx &= 0x7fffffff;		/* high word of |x| */

/* filter out non-finite argument */
if(hx >= 0x40862E42) {			/* if |x|>=709.78... */
if(hx>=0x7ff00000) {
if(((hx&0xfffff)|__LO(x))!=0)
return x+x; 		/* NaN */
else return (xsb==0)? x:0.0;	/* exp(+-inf)={inf,0} */
}
if(x > o_threshold) return Huge*Huge; /* overflow */
if(x < u_threshold) return twom1000*twom1000; /* underflow */
}

/* argument reduction */
if(hx > 0x3fd62e42) {		/* if  |x| > 0.5 ln2 */
if(hx < 0x3FF0A2B2) {	/* and |x| < 1.5 ln2 */
hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
} else {
k  = invln2*x+halF[xsb];
t  = k;
hi = x - t*ln2HI[0];	/* t*ln2HI is exact here */
lo = t*ln2LO[0];
}
x  = hi - lo;
}
else if(hx < 0x3e300000)  {	/* when |x|<2**-28 */
if(Huge+x>one) return one+x;/* trigger inexact */
}
else k = 0;

/* x is now in primary range */
t  = x*x;
c  = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
if(k==0) 	return one-((x*c)/(c-2.0)-x);
else 		y = one-((lo-(x*c)/(2.0-c))-hi);
if(k >= -1021) {
__HI(y) += (k<<20);	/* add k to y's exponent */
return y;
} else {
__HI(y) += ((k+1000)<<20);/* add k to y's exponent */
return y*twom1000;
}
}
```