/* * 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); } }
