forked from xuos/xiuos
221 lines
6.2 KiB
C
221 lines
6.2 KiB
C
/* Copyright JS Foundation and other contributors, http://js.foundation
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*
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* This file is based on work under the following copyright and permission
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* notice:
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*
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* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
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*
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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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* @(#)e_exp.c 1.6 04/04/22
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*/
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#include "jerry-math-internal.h"
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/* exp(x)
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* Returns the exponential of x.
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*
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* Method:
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* 1. Argument reduction:
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* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
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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.
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*
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* Here r will be represented as r = hi-lo for better
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* accuracy.
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*
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* 2. Approximation of exp(r) by a special rational function on
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* the interval [0,0.34658]:
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* Write
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* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
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* We use a special Remes algorithm on [0,0.34658] to generate
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* a polynomial of degree 5 to approximate R. The maximum error
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* of this polynomial approximation is bounded by 2**-59. In
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* other words,
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* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
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* (where z=r*r, and the values of P1 to P5 are listed below)
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* and
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* | 5 | -59
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* | 2.0+P1*z+...+P5*z - R(z) | <= 2
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* | |
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* The computation of exp(r) thus becomes
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* 2*r
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* exp(r) = 1 + -------
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* R - r
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* r*R1(r)
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* = 1 + r + ----------- (for better accuracy)
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* 2 - R1(r)
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* where
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* 2 4 10
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* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
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*
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* 3. Scale back to obtain exp(x):
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* From step 1, we have
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* exp(x) = 2^k * exp(r)
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*
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* Special cases:
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* exp(INF) is INF, exp(NaN) is NaN;
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* exp(-INF) is 0, and
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* for finite argument, only exp(0)=1 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 exp(x) overflow
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* if x < -7.45133219101941108420e+02 then exp(x) underflow
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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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static const double halF[2] =
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{
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0.5,
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-0.5,
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};
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static const double ln2HI[2] =
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{
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6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
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-6.93147180369123816490e-01, /* 0xbfe62e42, 0xfee00000 */
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};
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static const double ln2LO[2] =
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{
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1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
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-1.90821492927058770002e-10, /* 0xbdea39ef, 0x35793c76 */
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};
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#define one 1.0
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#define huge 1.0e+300
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#define twom1000 9.33263618503218878990e-302 /* 2**-1000=0x01700000,0 */
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#define o_threshold 7.09782712893383973096e+02 /* 0x40862E42, 0xFEFA39EF */
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#define u_threshold -7.45133219101941108420e+02 /* 0xc0874910, 0xD52D3051 */
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#define invln2 1.44269504088896338700e+00 /* 0x3ff71547, 0x652b82fe */
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#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
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#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
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#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
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#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
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#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */
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double
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exp (double x) /* default IEEE double exp */
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{
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double hi, lo, c, t;
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int k = 0, xsb;
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unsigned hx;
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hx = __HI (x); /* high word of x */
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xsb = (hx >> 31) & 1; /* sign bit of x */
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hx &= 0x7fffffff; /* high word of |x| */
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/* filter out non-finite argument */
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if (hx >= 0x40862E42) /* if |x| >= 709.78... */
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{
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if (hx >= 0x7ff00000)
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{
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if (((hx & 0xfffff) | __LO (x)) != 0) /* NaN */
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{
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return x + x;
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}
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else /* exp(+-inf) = {inf,0} */
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{
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return (xsb == 0) ? x : 0.0;
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}
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}
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if (x > o_threshold) /* overflow */
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{
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return huge * huge;
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}
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if (x < u_threshold) /* underflow */
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{
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return twom1000 * twom1000;
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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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{
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if (hx < 0x3FF0A2B2) /* and |x| < 1.5 ln2 */
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{
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hi = x - ln2HI[xsb];
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lo = ln2LO[xsb];
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k = 1 - xsb - xsb;
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}
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else
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{
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k = (int) (invln2 * x + halF[xsb]);
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t = k;
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hi = x - t * ln2HI[0]; /* t * ln2HI is exact here */
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lo = t * ln2LO[0];
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}
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x = hi - lo;
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}
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else if (hx < 0x3e300000) /* when |x| < 2**-28 */
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{
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if (huge + x > one) /* trigger inexact */
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{
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return one + x;
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}
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}
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else
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{
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k = 0;
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}
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double_accessor ret;
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/* x is now in primary range */
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t = x * x;
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c = x - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
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if (k == 0)
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{
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return one - ((x * c) / (c - 2.0) - x);
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}
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else
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{
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ret.dbl = one - ((lo - (x * c) / (2.0 - c)) - hi);
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}
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if (k >= -1021)
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{
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ret.as_int.hi += (((unsigned int) k) << 20); /* add k to y's exponent */
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return ret.dbl;
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}
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else
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{
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ret.as_int.hi += ((k + 1000) << 20); /* add k to y's exponent */
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return ret.dbl * twom1000;
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}
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} /* exp */
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#undef one
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#undef huge
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#undef twom1000
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#undef o_threshold
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#undef u_threshold
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#undef invln2
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#undef P1
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#undef P2
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#undef P3
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#undef P4
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#undef P5
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