/* Copyright 2016 Samsung Electronics Co., Ltd.
* Copyright 2016 University of Szeged
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
* This file is based on work under the following copyright and permission
* notice:
*
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
*
* @(#)e_exp.c 1.6 04/04/22
*/
#include "jerry-libm-internal.h"
/* 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 Remes 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.
*/
static const double halF[2] =
{
0.5,
-0.5,
};
static const double ln2HI[2] =
{
6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
-6.93147180369123816490e-01, /* 0xbfe62e42, 0xfee00000 */
};
static const double ln2LO[2] =
{
1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
-1.90821492927058770002e-10, /* 0xbdea39ef, 0x35793c76 */
};
#define one 1.0
#define huge 1.0e+300
#define twom1000 9.33263618503218878990e-302 /* 2**-1000=0x01700000,0 */
#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
#define u_threshold -7.45133219101941108420e+02 /* 0xc0874910, 0xD52D3051 */
#define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */
#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */
double
exp (double x) /* default IEEE double exp */
{
double y, hi, lo, c, t;
int k = 0, 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) /* NaN */
{
return x + x;
}
else /* exp(+-inf) = {inf,0} */
{
return (xsb == 0) ? x : 0.0;
}
}
if (x > o_threshold) /* overflow */
{
return huge * huge;
}
if (x < u_threshold) /* underflow */
{
return twom1000 * twom1000;
}
}
/* 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 = (int) (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) /* trigger inexact */
{
return one + x;
}
}
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;
}
} /* exp */
#undef one
#undef huge
#undef twom1000
#undef o_threshold
#undef u_threshold
#undef invln2
#undef P1
#undef P2
#undef P3
#undef P4
#undef P5