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/*
* CDDL HEADER START
*
* The contents of this file are subject to the terms of the
* Common Development and Distribution License (the "License").
* You may not use this file except in compliance with the License.
*
* You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
* or http://www.opensolaris.org/os/licensing.
* See the License for the specific language governing permissions
* and limitations under the License.
*
* When distributing Covered Code, include this CDDL HEADER in each
* file and include the License file at usr/src/OPENSOLARIS.LICENSE.
* If applicable, add the following below this CDDL HEADER, with the
* fields enclosed by brackets "[]" replaced with your own identifying
* information: Portions Copyright [yyyy] [name of copyright owner]
*
* CDDL HEADER END
*/
/*
* Copyright 2011 Nexenta Systems, Inc. All rights reserved.
*/
/*
* Copyright 2006 Sun Microsystems, Inc. All rights reserved.
* Use is subject to license terms.
*/
/* INDENT OFF */
/*
* double __k_cexp(double x, int *n);
* Returns the exponential of x in the form of 2**n * y, y=__k_cexp(x,&n).
*
* 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 Remez 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. Return n = k and __k_cexp = 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.
*
* Range and Accuracy:
* When |x| is really big, say |x| > 50000, the accuracy
* is not important because the ultimate result will over or under
* flow. So we will simply replace n = 50000 and r = 0.0. For
* moderate size x, 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.
*/
/* INDENT ON */
#include "libm.h" /* __k_cexp */
#include "complex_wrapper.h" /* HI_WORD/LO_WORD */
/* INDENT OFF */
static const double
one = 1.0,
two128 = 3.40282366920938463463e+38,
halF[2] = {
0.5, -0.5,
},
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 */
/* INDENT ON */
double
__k_cexp(double x, int *n) {
double hi = 0.0L, lo = 0.0L, c, t;
int k, xsb;
unsigned hx, lx;
hx = HI_WORD(x); /* high word of x */
lx = LO_WORD(x); /* low word of x */
xsb = (hx >> 31) & 1; /* sign bit of x */
hx &= 0x7fffffff; /* high word of |x| */
/* filter out non-finite argument */
if (hx >= 0x40e86a00) { /* if |x| > 50000 */
if (hx >= 0x7ff00000) {
*n = 1;
if (((hx & 0xfffff) | lx) != 0)
return (x + x); /* NaN */
else
return ((xsb == 0) ? x : 0.0);
/* exp(+-inf)={inf,0} */
}
*n = (xsb == 0) ? 50000 : -50000;
return (one + ln2LO[1] * ln2LO[1]); /* generate inexact */
}
*n = 0;
/* 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 for t<2**20 */
lo = t * ln2LO[0];
}
x = hi - lo;
*n = k;
} else if (hx < 0x3e300000) { /* when |x|<2**-28 */
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 {
t = one - ((lo - (x * c) / (2.0 - c)) - hi);
if (k > 128) {
t *= two128;
*n = k - 128;
} else if (k > 0) {
HI_WORD(t) += (k << 20);
*n = 0;
}
return (t);
}
}
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