| /*- |
| * SPDX-License-Identifier: BSD-3-Clause |
| * |
| * Copyright (c) 1985, 1993 |
| * The Regents of the University of California. All rights reserved. |
| * |
| * Redistribution and use in source and binary forms, with or without |
| * modification, are permitted provided that the following conditions |
| * are met: |
| * 1. Redistributions of source code must retain the above copyright |
| * notice, this list of conditions and the following disclaimer. |
| * 2. Redistributions in binary form must reproduce the above copyright |
| * notice, this list of conditions and the following disclaimer in the |
| * documentation and/or other materials provided with the distribution. |
| * 3. Neither the name of the University nor the names of its contributors |
| * may be used to endorse or promote products derived from this software |
| * without specific prior written permission. |
| * |
| * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND |
| * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
| * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE |
| * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE |
| * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL |
| * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS |
| * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) |
| * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT |
| * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY |
| * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF |
| * SUCH DAMAGE. |
| */ |
| |
| /* @(#)exp.c 8.1 (Berkeley) 6/4/93 */ |
| #include <sys/cdefs.h> |
| __FBSDID("$FreeBSD$"); |
| |
| /* EXP(X) |
| * RETURN THE EXPONENTIAL OF X |
| * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) |
| * CODED IN C BY K.C. NG, 1/19/85; |
| * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. |
| * |
| * Required system supported functions: |
| * ldexp(x,n) |
| * copysign(x,y) |
| * isfinite(x) |
| * |
| * Method: |
| * 1. Argument Reduction: given the input x, find r and integer k such |
| * that |
| * x = k*ln2 + r, |r| <= 0.5*ln2. |
| * r will be represented as r := z+c for better accuracy. |
| * |
| * 2. Compute exp(r) by |
| * |
| * exp(r) = 1 + r + r*R1/(2-R1), |
| * where |
| * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). |
| * |
| * 3. exp(x) = 2^k * exp(r) . |
| * |
| * Special cases: |
| * exp(INF) is INF, exp(NaN) is NaN; |
| * exp(-INF)= 0; |
| * for finite argument, only exp(0)=1 is exact. |
| * |
| * Accuracy: |
| * exp(x) returns the exponential of x nearly rounded. In a test run |
| * with 1,156,000 random arguments on a VAX, the maximum observed |
| * error was 0.869 ulps (units in the last place). |
| */ |
| static const double |
| p1 = 1.6666666666666660e-01, /* 0x3fc55555, 0x55555553 */ |
| p2 = -2.7777777777564776e-03, /* 0xbf66c16c, 0x16c0ac3c */ |
| p3 = 6.6137564717940088e-05, /* 0x3f11566a, 0xb5c2ba0d */ |
| p4 = -1.6534060280704225e-06, /* 0xbebbbd53, 0x273e8fb7 */ |
| p5 = 4.1437773411069054e-08; /* 0x3e663f2a, 0x09c94b6c */ |
| |
| static const double |
| ln2hi = 0x1.62e42fee00000p-1, /* High 32 bits round-down. */ |
| ln2lo = 0x1.a39ef35793c76p-33; /* Next 53 bits round-to-nearst. */ |
| |
| static const double |
| lnhuge = 0x1.6602b15b7ecf2p9, /* (DBL_MAX_EXP + 9) * log(2.) */ |
| lntiny = -0x1.77af8ebeae354p9, /* (DBL_MIN_EXP - 53 - 10) * log(2.) */ |
| invln2 = 0x1.71547652b82fep0; /* 1 / log(2.) */ |
| |
| /* returns exp(r = x + c) for |c| < |x| with no overlap. */ |
| |
| static double |
| __exp__D(double x, double c) |
| { |
| double hi, lo, z; |
| int k; |
| |
| if (x != x) /* x is NaN. */ |
| return(x); |
| |
| if (x <= lnhuge) { |
| if (x >= lntiny) { |
| /* argument reduction: x --> x - k*ln2 */ |
| z = invln2 * x; |
| k = z + copysign(0.5, x); |
| |
| /* |
| * Express (x + c) - k * ln2 as hi - lo. |
| * Let x = hi - lo rounded. |
| */ |
| hi = x - k * ln2hi; /* Exact. */ |
| lo = k * ln2lo - c; |
| x = hi - lo; |
| |
| /* Return 2^k*[1+x+x*c/(2+c)] */ |
| z = x * x; |
| c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 + |
| z * p5)))); |
| c = (x * c) / (2 - c); |
| |
| return (ldexp(1 + (hi - (lo - c)), k)); |
| } else { |
| /* exp(-INF) is 0. exp(-big) underflows to 0. */ |
| return (isfinite(x) ? ldexp(1., -5000) : 0); |
| } |
| } else |
| /* exp(INF) is INF, exp(+big#) overflows to INF */ |
| return (isfinite(x) ? ldexp(1., 5000) : x); |
| } |
