1 /*- 2 * SPDX-License-Identifier: BSD-3-Clause 3 * 4 * Copyright (c) 1985, 1993 5 * The Regents of the University of California. All rights reserved. 6 * 7 * Redistribution and use in source and binary forms, with or without 8 * modification, are permitted provided that the following conditions 9 * are met: 10 * 1. Redistributions of source code must retain the above copyright 11 * notice, this list of conditions and the following disclaimer. 12 * 2. Redistributions in binary form must reproduce the above copyright 13 * notice, this list of conditions and the following disclaimer in the 14 * documentation and/or other materials provided with the distribution. 15 * 3. Neither the name of the University nor the names of its contributors 16 * may be used to endorse or promote products derived from this software 17 * without specific prior written permission. 18 * 19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 22 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 29 * SUCH DAMAGE. 30 */ 31 32 /* EXP(X) 33 * RETURN THE EXPONENTIAL OF X 34 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS) 35 * CODED IN C BY K.C. NG, 1/19/85; 36 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86. 37 * 38 * Required system supported functions: 39 * ldexp(x,n) 40 * copysign(x,y) 41 * isfinite(x) 42 * 43 * Method: 44 * 1. Argument Reduction: given the input x, find r and integer k such 45 * that 46 * x = k*ln2 + r, |r| <= 0.5*ln2. 47 * r will be represented as r := z+c for better accuracy. 48 * 49 * 2. Compute exp(r) by 50 * 51 * exp(r) = 1 + r + r*R1/(2-R1), 52 * where 53 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))). 54 * 55 * 3. exp(x) = 2^k * exp(r) . 56 * 57 * Special cases: 58 * exp(INF) is INF, exp(NaN) is NaN; 59 * exp(-INF)= 0; 60 * for finite argument, only exp(0)=1 is exact. 61 * 62 * Accuracy: 63 * exp(x) returns the exponential of x nearly rounded. In a test run 64 * with 1,156,000 random arguments on a VAX, the maximum observed 65 * error was 0.869 ulps (units in the last place). 66 */ 67 static const double 68 p1 = 1.6666666666666660e-01, /* 0x3fc55555, 0x55555553 */ 69 p2 = -2.7777777777564776e-03, /* 0xbf66c16c, 0x16c0ac3c */ 70 p3 = 6.6137564717940088e-05, /* 0x3f11566a, 0xb5c2ba0d */ 71 p4 = -1.6534060280704225e-06, /* 0xbebbbd53, 0x273e8fb7 */ 72 p5 = 4.1437773411069054e-08; /* 0x3e663f2a, 0x09c94b6c */ 73 74 static const double 75 ln2hi = 0x1.62e42fee00000p-1, /* High 32 bits round-down. */ 76 ln2lo = 0x1.a39ef35793c76p-33; /* Next 53 bits round-to-nearst. */ 77 78 static const double 79 lnhuge = 0x1.6602b15b7ecf2p9, /* (DBL_MAX_EXP + 9) * log(2.) */ 80 lntiny = -0x1.77af8ebeae354p9, /* (DBL_MIN_EXP - 53 - 10) * log(2.) */ 81 invln2 = 0x1.71547652b82fep0; /* 1 / log(2.) */ 82 83 /* returns exp(r = x + c) for |c| < |x| with no overlap. */ 84 85 static double 86 __exp__D(double x, double c) 87 { 88 double hi, lo, z; 89 int k; 90 91 if (x != x) /* x is NaN. */ 92 return(x); 93 94 if (x <= lnhuge) { 95 if (x >= lntiny) { 96 /* argument reduction: x --> x - k*ln2 */ 97 z = invln2 * x; 98 k = z + copysign(0.5, x); 99 100 /* 101 * Express (x + c) - k * ln2 as hi - lo. 102 * Let x = hi - lo rounded. 103 */ 104 hi = x - k * ln2hi; /* Exact. */ 105 lo = k * ln2lo - c; 106 x = hi - lo; 107 108 /* Return 2^k*[1+x+x*c/(2+c)] */ 109 z = x * x; 110 c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 + 111 z * p5)))); 112 c = (x * c) / (2 - c); 113 114 return (ldexp(1 + (hi - (lo - c)), k)); 115 } else { 116 /* exp(-INF) is 0. exp(-big) underflows to 0. */ 117 return (isfinite(x) ? ldexp(1., -5000) : 0); 118 } 119 } else 120 /* exp(INF) is INF, exp(+big#) overflows to INF */ 121 return (isfinite(x) ? ldexp(1., 5000) : x); 122 } 123