2015-10-07 01:39:26 +00:00
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/*
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2023-01-30 20:33:01 +00:00
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* Copyright (c) 1998, 2023, Oracle and/or its affiliates. All rights reserved.
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2015-10-07 01:39:26 +00:00
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* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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*
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* This code is free software; you can redistribute it and/or modify it
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* under the terms of the GNU General Public License version 2 only, as
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* published by the Free Software Foundation. Oracle designates this
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* particular file as subject to the "Classpath" exception as provided
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* by Oracle in the LICENSE file that accompanied this code.
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*
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* This code is distributed in the hope that it will be useful, but WITHOUT
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* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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* version 2 for more details (a copy is included in the LICENSE file that
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* accompanied this code).
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*
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* You should have received a copy of the GNU General Public License version
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* 2 along with this work; if not, write to the Free Software Foundation,
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* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
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*
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* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
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* or visit www.oracle.com if you need additional information or have any
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* questions.
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*/
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/**
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* A transliteration of the "Freely Distributable Math Library"
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* algorithms from C into Java. That is, this port of the algorithms
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* is as close to the C originals as possible while still being
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* readable legal Java.
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*/
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public class FdlibmTranslit {
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private FdlibmTranslit() {
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throw new UnsupportedOperationException("No FdLibmTranslit instances for you.");
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}
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/**
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* Return the low-order 32 bits of the double argument as an int.
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*/
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private static int __LO(double x) {
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long transducer = Double.doubleToRawLongBits(x);
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return (int)transducer;
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}
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/**
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* Return a double with its low-order bits of the second argument
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* and the high-order bits of the first argument..
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*/
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private static double __LO(double x, int low) {
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long transX = Double.doubleToRawLongBits(x);
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2016-12-17 05:43:29 +00:00
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return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
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(low & 0x0000_0000_FFFF_FFFFL));
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2015-10-07 01:39:26 +00:00
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}
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/**
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* Return the high-order 32 bits of the double argument as an int.
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*/
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private static int __HI(double x) {
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long transducer = Double.doubleToRawLongBits(x);
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return (int)(transducer >> 32);
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}
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/**
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* Return a double with its high-order bits of the second argument
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* and the low-order bits of the first argument..
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*/
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private static double __HI(double x, int high) {
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long transX = Double.doubleToRawLongBits(x);
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2016-12-17 05:43:29 +00:00
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return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
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( ((long)high)) << 32 );
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2015-10-07 01:39:26 +00:00
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}
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2023-02-15 22:16:30 +00:00
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public static double asin(double x) {
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return Asin.compute(x);
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}
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public static double acos(double x) {
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return Acos.compute(x);
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}
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public static double atan(double x) {
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return Atan.compute(x);
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}
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2015-10-07 01:39:26 +00:00
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public static double hypot(double x, double y) {
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return Hypot.compute(x, y);
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}
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2015-10-14 23:17:08 +00:00
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2023-01-30 20:33:01 +00:00
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public static double cbrt(double x) {
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return Cbrt.compute(x);
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}
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2023-02-11 02:15:46 +00:00
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public static double log(double x) {
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return Log.compute(x);
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}
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2023-01-30 20:33:01 +00:00
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public static double log10(double x) {
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return Log10.compute(x);
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}
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2023-02-02 20:36:34 +00:00
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public static double log1p(double x) {
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return Log1p.compute(x);
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}
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2023-02-17 03:22:06 +00:00
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public static double exp(double x) {
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return Exp.compute(x);
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}
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2023-02-04 00:48:26 +00:00
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public static double expm1(double x) {
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return Expm1.compute(x);
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}
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2023-02-17 03:22:06 +00:00
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public static double sinh(double x) {
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return Sinh.compute(x);
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}
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public static double cosh(double x) {
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return Cosh.compute(x);
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}
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public static double tanh(double x) {
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return Tanh.compute(x);
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}
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2023-02-15 22:16:30 +00:00
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/** Returns the arcsine of x.
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*
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* Method :
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* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
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* we approximate asin(x) on [0,0.5] by
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* asin(x) = x + x*x^2*R(x^2)
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* where
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* R(x^2) is a rational approximation of (asin(x)-x)/x^3
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* and its remez error is bounded by
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* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
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*
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* For x in [0.5,1]
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* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
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* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
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* then for x>0.98
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
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* For x<=0.98, let pio4_hi = pio2_hi/2, then
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* f = hi part of s;
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* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
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* and
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* asin(x) = pi/2 - 2*(s+s*z*R(z))
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* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
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* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
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*
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* Special cases:
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
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*
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*/
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static class Asin {
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private static final double
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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huge = 1.000e+300,
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pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
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pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
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pio4_hi = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
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/* coefficient for R(x^2) */
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pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
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pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
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pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
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pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
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pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
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pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
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qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
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qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
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qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
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qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
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static double compute(double x) {
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double t=0,w,p,q,c,r,s;
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int hx,ix;
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hx = __HI(x);
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ix = hx&0x7fffffff;
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if(ix>= 0x3ff00000) { /* |x|>= 1 */
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if(((ix-0x3ff00000)|__LO(x))==0)
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/* asin(1)=+-pi/2 with inexact */
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return x*pio2_hi+x*pio2_lo;
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return (x-x)/(x-x); /* asin(|x|>1) is NaN */
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} else if (ix<0x3fe00000) { /* |x|<0.5 */
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if(ix<0x3e400000) { /* if |x| < 2**-27 */
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if(huge+x>one) return x;/* return x with inexact if x!=0*/
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} else
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t = x*x;
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p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
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q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
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w = p/q;
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return x+x*w;
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}
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/* 1> |x|>= 0.5 */
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w = one-Math.abs(x);
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t = w*0.5;
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p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
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q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
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s = Math.sqrt(t);
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if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
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w = p/q;
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t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
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} else {
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w = s;
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// __LO(w) = 0;
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w = __LO(w, 0);
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c = (t-w*w)/(s+w);
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r = p/q;
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p = 2.0*s*r-(pio2_lo-2.0*c);
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q = pio4_hi-2.0*w;
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t = pio4_hi-(p-q);
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}
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if(hx>0) return t; else return -t;
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}
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}
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/** Returns the arccosine of x.
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* Method :
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* acos(x) = pi/2 - asin(x)
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* acos(-x) = pi/2 + asin(x)
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* For |x|<=0.5
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* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
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* For x>0.5
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* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
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* = 2asin(sqrt((1-x)/2))
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* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
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* = 2f + (2c + 2s*z*R(z))
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* where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
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* for f so that f+c ~ sqrt(z).
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* For x<-0.5
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* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
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* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
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*
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* Special cases:
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* if x is NaN, return x itself;
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* if |x|>1, return NaN with invalid signal.
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*
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* Function needed: sqrt
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*/
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static class Acos {
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private static final double
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one= 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
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pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
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pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
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pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
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pS0 = 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
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pS1 = -3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
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pS2 = 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
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pS3 = -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
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pS4 = 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
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pS5 = 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
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qS1 = -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
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qS2 = 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
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qS3 = -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
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qS4 = 7.70381505559019352791e-02; /* 0x3FB3B8C5, 0xB12E9282 */
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static double compute(double x) {
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double z,p,q,r,w,s,c,df;
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int hx,ix;
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hx = __HI(x);
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ix = hx&0x7fffffff;
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if(ix>=0x3ff00000) { /* |x| >= 1 */
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if(((ix-0x3ff00000)|__LO(x))==0) { /* |x|==1 */
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if(hx>0) return 0.0; /* acos(1) = 0 */
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else return pi+2.0*pio2_lo; /* acos(-1)= pi */
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}
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return (x-x)/(x-x); /* acos(|x|>1) is NaN */
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}
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if(ix<0x3fe00000) { /* |x| < 0.5 */
|
|
|
|
|
if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
|
|
|
|
|
z = x*x;
|
|
|
|
|
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
|
|
|
|
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
|
|
|
|
r = p/q;
|
|
|
|
|
return pio2_hi - (x - (pio2_lo-x*r));
|
|
|
|
|
} else if (hx<0) { /* x < -0.5 */
|
|
|
|
|
z = (one+x)*0.5;
|
|
|
|
|
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
|
|
|
|
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
|
|
|
|
s = Math.sqrt(z);
|
|
|
|
|
r = p/q;
|
|
|
|
|
w = r*s-pio2_lo;
|
|
|
|
|
return pi - 2.0*(s+w);
|
|
|
|
|
} else { /* x > 0.5 */
|
|
|
|
|
z = (one-x)*0.5;
|
|
|
|
|
s = Math.sqrt(z);
|
|
|
|
|
df = s;
|
|
|
|
|
// __LO(df) = 0;
|
|
|
|
|
df = __LO(df, 0);
|
|
|
|
|
c = (z-df*df)/(s+df);
|
|
|
|
|
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
|
|
|
|
|
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
|
|
|
|
|
r = p/q;
|
|
|
|
|
w = r*s+c;
|
|
|
|
|
return 2.0*(df+w);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* Returns the arctangent of x.
|
|
|
|
|
* Method
|
|
|
|
|
* 1. Reduce x to positive by atan(x) = -atan(-x).
|
|
|
|
|
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
|
|
|
|
|
* is further reduced to one of the following intervals and the
|
|
|
|
|
* arctangent of t is evaluated by the corresponding formula:
|
|
|
|
|
*
|
|
|
|
|
* [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
|
|
|
|
|
* [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
|
|
|
|
|
* [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
|
|
|
|
|
* [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
|
|
|
|
|
* [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
|
|
|
|
|
*
|
|
|
|
|
* 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.
|
|
|
|
|
*/
|
|
|
|
|
static class Atan {
|
|
|
|
|
private static final double atanhi[] = {
|
|
|
|
|
4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
|
|
|
|
|
7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
|
|
|
|
|
9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
|
|
|
|
|
1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
private static final double atanlo[] = {
|
|
|
|
|
2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
|
|
|
|
|
3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
|
|
|
|
|
1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
|
|
|
|
|
6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
private static final double aT[] = {
|
|
|
|
|
3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
|
|
|
|
|
-1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
|
|
|
|
|
1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
|
|
|
|
|
-1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
|
|
|
|
|
9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
|
|
|
|
|
-7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
|
|
|
|
|
6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
|
|
|
|
|
-5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
|
|
|
|
|
4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
|
|
|
|
|
-3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
|
|
|
|
|
1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
|
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
private static final double
|
|
|
|
|
one = 1.0,
|
|
|
|
|
huge = 1.0e300;
|
|
|
|
|
|
|
|
|
|
static double compute(double x) {
|
|
|
|
|
double w,s1,s2,z;
|
|
|
|
|
int ix,hx,id;
|
|
|
|
|
|
|
|
|
|
hx = __HI(x);
|
|
|
|
|
ix = hx&0x7fffffff;
|
|
|
|
|
if(ix>=0x44100000) { /* if |x| >= 2^66 */
|
|
|
|
|
if(ix>0x7ff00000||
|
|
|
|
|
(ix==0x7ff00000&&(__LO(x)!=0)))
|
|
|
|
|
return x+x; /* NaN */
|
|
|
|
|
if(hx>0) return atanhi[3]+atanlo[3];
|
|
|
|
|
else return -atanhi[3]-atanlo[3];
|
|
|
|
|
} if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
|
|
|
|
|
if (ix < 0x3e200000) { /* |x| < 2^-29 */
|
|
|
|
|
if(huge+x>one) return x; /* raise inexact */
|
|
|
|
|
}
|
|
|
|
|
id = -1;
|
|
|
|
|
} else {
|
|
|
|
|
x = Math.abs(x);
|
|
|
|
|
if (ix < 0x3ff30000) { /* |x| < 1.1875 */
|
|
|
|
|
if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
|
|
|
|
|
id = 0; x = (2.0*x-one)/(2.0+x);
|
|
|
|
|
} else { /* 11/16<=|x|< 19/16 */
|
|
|
|
|
id = 1; x = (x-one)/(x+one);
|
|
|
|
|
}
|
|
|
|
|
} else {
|
|
|
|
|
if (ix < 0x40038000) { /* |x| < 2.4375 */
|
|
|
|
|
id = 2; x = (x-1.5)/(one+1.5*x);
|
|
|
|
|
} else { /* 2.4375 <= |x| < 2^66 */
|
|
|
|
|
id = 3; x = -1.0/x;
|
|
|
|
|
}
|
|
|
|
|
}}
|
|
|
|
|
/* end of argument reduction */
|
|
|
|
|
z = x*x;
|
|
|
|
|
w = z*z;
|
|
|
|
|
/* break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly */
|
|
|
|
|
s1 = z*(aT[0]+w*(aT[2]+w*(aT[4]+w*(aT[6]+w*(aT[8]+w*aT[10])))));
|
|
|
|
|
s2 = w*(aT[1]+w*(aT[3]+w*(aT[5]+w*(aT[7]+w*aT[9]))));
|
|
|
|
|
if (id<0) return x - x*(s1+s2);
|
|
|
|
|
else {
|
|
|
|
|
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
|
|
|
|
|
return (hx<0)? -z:z;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
2015-10-14 23:17:08 +00:00
|
|
|
/**
|
|
|
|
|
* cbrt(x)
|
|
|
|
|
* Return cube root of x
|
|
|
|
|
*/
|
|
|
|
|
public static class Cbrt {
|
|
|
|
|
// unsigned
|
|
|
|
|
private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */
|
|
|
|
|
private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
|
|
|
|
|
|
|
|
|
|
private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
|
|
|
|
|
private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */
|
|
|
|
|
private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
|
|
|
|
|
private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
|
|
|
|
|
private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
|
|
|
|
|
|
2023-01-30 20:33:01 +00:00
|
|
|
public static double compute(double x) {
|
2015-10-14 23:17:08 +00:00
|
|
|
int hx;
|
|
|
|
|
double r, s, t=0.0, w;
|
|
|
|
|
int sign; // unsigned
|
|
|
|
|
|
|
|
|
|
hx = __HI(x); // high word of x
|
|
|
|
|
sign = hx & 0x80000000; // sign= sign(x)
|
|
|
|
|
hx ^= sign;
|
|
|
|
|
if (hx >= 0x7ff00000)
|
|
|
|
|
return (x+x); // cbrt(NaN,INF) is itself
|
|
|
|
|
if ((hx | __LO(x)) == 0)
|
|
|
|
|
return(x); // cbrt(0) is itself
|
|
|
|
|
|
|
|
|
|
x = __HI(x, hx); // x <- |x|
|
|
|
|
|
// rough cbrt to 5 bits
|
|
|
|
|
if (hx < 0x00100000) { // subnormal number
|
|
|
|
|
t = __HI(t, 0x43500000); // set t= 2**54
|
|
|
|
|
t *= x;
|
|
|
|
|
t = __HI(t, __HI(t)/3+B2);
|
|
|
|
|
} else {
|
|
|
|
|
t = __HI(t, hx/3+B1);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
// new cbrt to 23 bits, may be implemented in single precision
|
|
|
|
|
r = t * t/x;
|
|
|
|
|
s = C + r*t;
|
|
|
|
|
t *= G + F/(s + E + D/s);
|
|
|
|
|
|
|
|
|
|
// chopped to 20 bits and make it larger than cbrt(x)
|
|
|
|
|
t = __LO(t, 0);
|
|
|
|
|
t = __HI(t, __HI(t)+0x00000001);
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
// one step newton iteration to 53 bits with error less than 0.667 ulps
|
|
|
|
|
s = t * t; // t*t is exact
|
|
|
|
|
r = x / s;
|
|
|
|
|
w = t + t;
|
|
|
|
|
r= (r - t)/(w + r); // r-s is exact
|
|
|
|
|
t= t + t*r;
|
|
|
|
|
|
|
|
|
|
// retore the sign bit
|
|
|
|
|
t = __HI(t, __HI(t) | sign);
|
|
|
|
|
return(t);
|
|
|
|
|
}
|
|
|
|
|
}
|
2015-10-07 01:39:26 +00:00
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* hypot(x,y)
|
|
|
|
|
*
|
|
|
|
|
* Method :
|
|
|
|
|
* If (assume round-to-nearest) z = x*x + y*y
|
|
|
|
|
* has error less than sqrt(2)/2 ulp, than
|
|
|
|
|
* sqrt(z) has error less than 1 ulp (exercise).
|
|
|
|
|
*
|
|
|
|
|
* So, compute sqrt(x*x + y*y) with some care as
|
|
|
|
|
* follows to get the error below 1 ulp:
|
|
|
|
|
*
|
|
|
|
|
* Assume x > y > 0;
|
|
|
|
|
* (if possible, set rounding to round-to-nearest)
|
|
|
|
|
* 1. if x > 2y use
|
|
|
|
|
* x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
|
|
|
|
|
* where x1 = x with lower 32 bits cleared, x2 = x - x1; else
|
|
|
|
|
* 2. if x <= 2y use
|
|
|
|
|
* t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
|
|
|
|
|
* where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
|
|
|
|
|
* y1= y with lower 32 bits chopped, y2 = y - y1.
|
|
|
|
|
*
|
|
|
|
|
* NOTE: scaling may be necessary if some argument is too
|
|
|
|
|
* large or too tiny
|
|
|
|
|
*
|
|
|
|
|
* Special cases:
|
|
|
|
|
* hypot(x,y) is INF if x or y is +INF or -INF; else
|
|
|
|
|
* hypot(x,y) is NAN if x or y is NAN.
|
|
|
|
|
*
|
|
|
|
|
* Accuracy:
|
|
|
|
|
* hypot(x,y) returns sqrt(x^2 + y^2) with error less
|
|
|
|
|
* than 1 ulps (units in the last place)
|
|
|
|
|
*/
|
|
|
|
|
static class Hypot {
|
|
|
|
|
public static double compute(double x, double y) {
|
|
|
|
|
double a = x;
|
|
|
|
|
double b = y;
|
|
|
|
|
double t1, t2, y1, y2, w;
|
|
|
|
|
int j, k, ha, hb;
|
|
|
|
|
|
|
|
|
|
ha = __HI(x) & 0x7fffffff; // high word of x
|
|
|
|
|
hb = __HI(y) & 0x7fffffff; // high word of y
|
|
|
|
|
if(hb > ha) {
|
|
|
|
|
a = y;
|
|
|
|
|
b = x;
|
|
|
|
|
j = ha;
|
|
|
|
|
ha = hb;
|
|
|
|
|
hb = j;
|
|
|
|
|
} else {
|
|
|
|
|
a = x;
|
|
|
|
|
b = y;
|
|
|
|
|
}
|
|
|
|
|
a = __HI(a, ha); // a <- |a|
|
|
|
|
|
b = __HI(b, hb); // b <- |b|
|
|
|
|
|
if ((ha - hb) > 0x3c00000) {
|
|
|
|
|
return a + b; // x / y > 2**60
|
|
|
|
|
}
|
|
|
|
|
k=0;
|
|
|
|
|
if (ha > 0x5f300000) { // a>2**500
|
|
|
|
|
if (ha >= 0x7ff00000) { // Inf or NaN
|
|
|
|
|
w = a + b; // for sNaN
|
|
|
|
|
if (((ha & 0xfffff) | __LO(a)) == 0)
|
|
|
|
|
w = a;
|
|
|
|
|
if (((hb ^ 0x7ff00000) | __LO(b)) == 0)
|
|
|
|
|
w = b;
|
|
|
|
|
return w;
|
|
|
|
|
}
|
|
|
|
|
// scale a and b by 2**-600
|
|
|
|
|
ha -= 0x25800000;
|
|
|
|
|
hb -= 0x25800000;
|
|
|
|
|
k += 600;
|
|
|
|
|
a = __HI(a, ha);
|
|
|
|
|
b = __HI(b, hb);
|
|
|
|
|
}
|
|
|
|
|
if (hb < 0x20b00000) { // b < 2**-500
|
|
|
|
|
if (hb <= 0x000fffff) { // subnormal b or 0 */
|
|
|
|
|
if ((hb | (__LO(b))) == 0)
|
|
|
|
|
return a;
|
|
|
|
|
t1 = 0;
|
|
|
|
|
t1 = __HI(t1, 0x7fd00000); // t1=2^1022
|
|
|
|
|
b *= t1;
|
|
|
|
|
a *= t1;
|
|
|
|
|
k -= 1022;
|
|
|
|
|
} else { // scale a and b by 2^600
|
|
|
|
|
ha += 0x25800000; // a *= 2^600
|
|
|
|
|
hb += 0x25800000; // b *= 2^600
|
|
|
|
|
k -= 600;
|
|
|
|
|
a = __HI(a, ha);
|
|
|
|
|
b = __HI(b, hb);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
// medium size a and b
|
|
|
|
|
w = a - b;
|
|
|
|
|
if (w > b) {
|
|
|
|
|
t1 = 0;
|
|
|
|
|
t1 = __HI(t1, ha);
|
|
|
|
|
t2 = a - t1;
|
|
|
|
|
w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1)));
|
|
|
|
|
} else {
|
|
|
|
|
a = a + a;
|
|
|
|
|
y1 = 0;
|
|
|
|
|
y1 = __HI(y1, hb);
|
|
|
|
|
y2 = b - y1;
|
|
|
|
|
t1 = 0;
|
|
|
|
|
t1 = __HI(t1, ha + 0x00100000);
|
|
|
|
|
t2 = a - t1;
|
|
|
|
|
w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
|
|
|
|
|
}
|
|
|
|
|
if (k != 0) {
|
|
|
|
|
t1 = 1.0;
|
|
|
|
|
int t1_hi = __HI(t1);
|
|
|
|
|
t1_hi += (k << 20);
|
|
|
|
|
t1 = __HI(t1, t1_hi);
|
|
|
|
|
return t1 * w;
|
|
|
|
|
} else
|
|
|
|
|
return w;
|
|
|
|
|
}
|
|
|
|
|
}
|
2016-12-17 05:43:29 +00:00
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Returns the exponential of x.
|
|
|
|
|
*
|
|
|
|
|
* 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 Reme 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. Scale back to obtain exp(x):
|
|
|
|
|
* From step 1, we have
|
|
|
|
|
* exp(x) = 2^k * 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.
|
|
|
|
|
*
|
|
|
|
|
* Accuracy:
|
|
|
|
|
* according to an error analysis, the error is always less than
|
|
|
|
|
* 1 ulp (unit in the last place).
|
|
|
|
|
*
|
|
|
|
|
* Misc. info.
|
|
|
|
|
* For IEEE double
|
|
|
|
|
* if x > 7.09782712893383973096e+02 then exp(x) overflow
|
|
|
|
|
* if x < -7.45133219101941108420e+02 then exp(x) underflow
|
|
|
|
|
*
|
|
|
|
|
* 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.
|
|
|
|
|
*/
|
2023-02-17 03:22:06 +00:00
|
|
|
private static final class Exp {
|
2016-12-17 05:43:29 +00:00
|
|
|
private static final double one = 1.0;
|
|
|
|
|
private static final double[] halF = {0.5,-0.5,};
|
|
|
|
|
private static final double huge = 1.0e+300;
|
|
|
|
|
private static final double twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/
|
|
|
|
|
private static final double o_threshold= 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
|
|
|
|
|
private static final double u_threshold= -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
|
|
|
|
|
private static final double[] ln2HI ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
|
|
|
|
|
-6.93147180369123816490e-01}; /* 0xbfe62e42, 0xfee00000 */
|
|
|
|
|
private static final double[] ln2LO ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
|
|
|
|
|
-1.90821492927058770002e-10,}; /* 0xbdea39ef, 0x35793c76 */
|
|
|
|
|
private static final double invln2 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */
|
|
|
|
|
private static final double P1 = 1.66666666666666019037e-01; /* 0x3FC55555, 0x5555553E */
|
|
|
|
|
private static final double P2 = -2.77777777770155933842e-03; /* 0xBF66C16C, 0x16BEBD93 */
|
|
|
|
|
private static final double P3 = 6.61375632143793436117e-05; /* 0x3F11566A, 0xAF25DE2C */
|
|
|
|
|
private static final double P4 = -1.65339022054652515390e-06; /* 0xBEBBBD41, 0xC5D26BF1 */
|
|
|
|
|
private static final double P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
|
|
|
|
|
|
2023-02-17 03:22:06 +00:00
|
|
|
static double compute(double x) {
|
2016-12-17 05:43:29 +00:00
|
|
|
double y,hi=0,lo=0,c,t;
|
|
|
|
|
int k=0,xsb;
|
|
|
|
|
/*unsigned*/ int hx;
|
|
|
|
|
|
|
|
|
|
hx = __HI(x); /* high word of x */
|
|
|
|
|
xsb = (hx>>31)&1; /* sign bit of x */
|
|
|
|
|
hx &= 0x7fffffff; /* high word of |x| */
|
|
|
|
|
|
|
|
|
|
/* filter out non-finite argument */
|
|
|
|
|
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
|
|
|
|
if(hx>=0x7ff00000) {
|
|
|
|
|
if(((hx&0xfffff)|__LO(x))!=0)
|
|
|
|
|
return x+x; /* NaN */
|
|
|
|
|
else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
|
|
|
|
|
}
|
|
|
|
|
if(x > o_threshold) return huge*huge; /* overflow */
|
|
|
|
|
if(x < u_threshold) return twom1000*twom1000; /* underflow */
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* 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 here */
|
|
|
|
|
lo = t*ln2LO[0];
|
|
|
|
|
}
|
|
|
|
|
x = hi - lo;
|
|
|
|
|
}
|
|
|
|
|
else if(hx < 0x3e300000) { /* when |x|<2**-28 */
|
|
|
|
|
if(huge+x>one) return one+x;/* trigger inexact */
|
|
|
|
|
}
|
|
|
|
|
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 y = one-((lo-(x*c)/(2.0-c))-hi);
|
|
|
|
|
if(k >= -1021) {
|
|
|
|
|
y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */
|
|
|
|
|
return y;
|
|
|
|
|
} else {
|
|
|
|
|
y = __HI(y, __HI(y) + ((k+1000)<<20));/* add k to y's exponent */
|
|
|
|
|
return y*twom1000;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
2023-01-30 20:33:01 +00:00
|
|
|
|
2023-02-11 02:15:46 +00:00
|
|
|
/**
|
|
|
|
|
* Return the logarithm of x
|
|
|
|
|
*
|
|
|
|
|
* Method :
|
|
|
|
|
* 1. Argument Reduction: find k and f such that
|
|
|
|
|
* x = 2^k * (1+f),
|
|
|
|
|
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
|
|
|
|
*
|
|
|
|
|
* 2. Approximation of log(1+f).
|
|
|
|
|
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
|
|
|
|
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
|
|
|
|
* = 2s + s*R
|
|
|
|
|
* We use a special Reme algorithm on [0,0.1716] to generate
|
|
|
|
|
* a polynomial of degree 14 to approximate R The maximum error
|
|
|
|
|
* of this polynomial approximation is bounded by 2**-58.45. In
|
|
|
|
|
* other words,
|
|
|
|
|
* 2 4 6 8 10 12 14
|
|
|
|
|
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
|
|
|
|
|
* (the values of Lg1 to Lg7 are listed in the program)
|
|
|
|
|
* and
|
|
|
|
|
* | 2 14 | -58.45
|
|
|
|
|
* | Lg1*s +...+Lg7*s - R(z) | <= 2
|
|
|
|
|
* | |
|
|
|
|
|
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
|
|
|
|
* In order to guarantee error in log below 1ulp, we compute log
|
|
|
|
|
* by
|
|
|
|
|
* log(1+f) = f - s*(f - R) (if f is not too large)
|
|
|
|
|
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
|
|
|
|
|
*
|
|
|
|
|
* 3. Finally, log(x) = k*ln2 + log(1+f).
|
|
|
|
|
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
|
|
|
|
* Here ln2 is split into two floating point number:
|
|
|
|
|
* ln2_hi + ln2_lo,
|
|
|
|
|
* where n*ln2_hi is always exact for |n| < 2000.
|
|
|
|
|
*
|
|
|
|
|
* Special cases:
|
|
|
|
|
* log(x) is NaN with signal if x < 0 (including -INF) ;
|
|
|
|
|
* log(+INF) is +INF; log(0) is -INF with signal;
|
|
|
|
|
* log(NaN) is that NaN with no signal.
|
|
|
|
|
*
|
|
|
|
|
* Accuracy:
|
|
|
|
|
* 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.
|
|
|
|
|
*/
|
|
|
|
|
private static final class Log {
|
|
|
|
|
private static final double
|
|
|
|
|
ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
|
|
|
|
|
ln2_lo = 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
|
|
|
|
|
two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
|
|
|
|
|
Lg1 = 6.666666666666735130e-01, /* 3FE55555 55555593 */
|
|
|
|
|
Lg2 = 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
|
|
|
|
|
Lg3 = 2.857142874366239149e-01, /* 3FD24924 94229359 */
|
|
|
|
|
Lg4 = 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
|
|
|
|
|
Lg5 = 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
|
|
|
|
|
Lg6 = 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
|
|
|
|
|
Lg7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
|
|
|
|
|
|
|
|
|
|
private static double zero = 0.0;
|
|
|
|
|
|
|
|
|
|
static double compute(double x) {
|
|
|
|
|
double hfsq,f,s,z,R,w,t1,t2,dk;
|
|
|
|
|
int k,hx,i,j;
|
|
|
|
|
/*unsigned*/ int lx;
|
|
|
|
|
|
|
|
|
|
hx = __HI(x); /* high word of x */
|
|
|
|
|
lx = __LO(x); /* low word of x */
|
|
|
|
|
|
|
|
|
|
k=0;
|
|
|
|
|
if (hx < 0x00100000) { /* x < 2**-1022 */
|
|
|
|
|
if (((hx&0x7fffffff)|lx)==0)
|
|
|
|
|
return -two54/zero; /* log(+-0)=-inf */
|
|
|
|
|
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
|
|
|
|
|
k -= 54; x *= two54; /* subnormal number, scale up x */
|
|
|
|
|
hx = __HI(x); /* high word of x */
|
|
|
|
|
}
|
|
|
|
|
if (hx >= 0x7ff00000) return x+x;
|
|
|
|
|
k += (hx>>20)-1023;
|
|
|
|
|
hx &= 0x000fffff;
|
|
|
|
|
i = (hx+0x95f64)&0x100000;
|
|
|
|
|
// __HI(x) = hx|(i^0x3ff00000); /* normalize x or x/2 */
|
|
|
|
|
x =__HI(x, hx|(i^0x3ff00000)); /* normalize x or x/2 */
|
|
|
|
|
k += (i>>20);
|
|
|
|
|
f = x-1.0;
|
|
|
|
|
if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
|
|
|
|
|
if(f==zero) {
|
|
|
|
|
if (k==0) return zero;
|
|
|
|
|
else {dk=(double)k; return dk*ln2_hi+dk*ln2_lo;}
|
|
|
|
|
}
|
|
|
|
|
R = f*f*(0.5-0.33333333333333333*f);
|
|
|
|
|
if(k==0) return f-R; else {dk=(double)k;
|
|
|
|
|
return dk*ln2_hi-((R-dk*ln2_lo)-f);}
|
|
|
|
|
}
|
|
|
|
|
s = f/(2.0+f);
|
|
|
|
|
dk = (double)k;
|
|
|
|
|
z = s*s;
|
|
|
|
|
i = hx-0x6147a;
|
|
|
|
|
w = z*z;
|
|
|
|
|
j = 0x6b851-hx;
|
|
|
|
|
t1= w*(Lg2+w*(Lg4+w*Lg6));
|
|
|
|
|
t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
|
|
|
|
|
i |= j;
|
|
|
|
|
R = t2+t1;
|
|
|
|
|
if(i>0) {
|
|
|
|
|
hfsq=0.5*f*f;
|
|
|
|
|
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
|
|
|
|
return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
|
|
|
|
|
} else {
|
|
|
|
|
if(k==0) return f-s*(f-R); else
|
|
|
|
|
return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
2023-01-30 20:33:01 +00:00
|
|
|
/**
|
|
|
|
|
* Return the base 10 logarithm of x
|
|
|
|
|
*
|
|
|
|
|
* Method :
|
|
|
|
|
* Let log10_2hi = leading 40 bits of log10(2) and
|
|
|
|
|
* log10_2lo = log10(2) - log10_2hi,
|
|
|
|
|
* ivln10 = 1/log(10) rounded.
|
|
|
|
|
* Then
|
|
|
|
|
* n = ilogb(x),
|
|
|
|
|
* if(n<0) n = n+1;
|
|
|
|
|
* x = scalbn(x,-n);
|
|
|
|
|
* log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
|
|
|
|
|
*
|
|
|
|
|
* Note 1:
|
|
|
|
|
* To guarantee log10(10**n)=n, where 10**n is normal, the rounding
|
|
|
|
|
* mode must set to Round-to-Nearest.
|
|
|
|
|
* Note 2:
|
|
|
|
|
* [1/log(10)] rounded to 53 bits has error .198 ulps;
|
|
|
|
|
* log10 is monotonic at all binary break points.
|
|
|
|
|
*
|
|
|
|
|
* Special cases:
|
|
|
|
|
* log10(x) is NaN with signal if x < 0;
|
|
|
|
|
* log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
|
|
|
|
|
* log10(NaN) is that NaN with no signal;
|
|
|
|
|
* log10(10**N) = N for N=0,1,...,22.
|
|
|
|
|
*
|
|
|
|
|
* 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.
|
|
|
|
|
*/
|
|
|
|
|
static class Log10 {
|
|
|
|
|
private static double two54 = 1.80143985094819840000e+16; /* 0x43500000, 0x00000000 */
|
|
|
|
|
private static double ivln10 = 4.34294481903251816668e-01; /* 0x3FDBCB7B, 0x1526E50E */
|
|
|
|
|
|
|
|
|
|
private static double log10_2hi = 3.01029995663611771306e-01; /* 0x3FD34413, 0x509F6000 */
|
|
|
|
|
private static double log10_2lo = 3.69423907715893078616e-13; /* 0x3D59FEF3, 0x11F12B36 */
|
|
|
|
|
|
|
|
|
|
private static double zero = 0.0;
|
|
|
|
|
|
|
|
|
|
public static double compute(double x) {
|
|
|
|
|
double y,z;
|
|
|
|
|
int i,k,hx;
|
|
|
|
|
/*unsigned*/ int lx;
|
|
|
|
|
|
|
|
|
|
hx = __HI(x); /* high word of x */
|
|
|
|
|
lx = __LO(x); /* low word of x */
|
|
|
|
|
|
|
|
|
|
k=0;
|
|
|
|
|
if (hx < 0x00100000) { /* x < 2**-1022 */
|
|
|
|
|
if (((hx&0x7fffffff)|lx)==0)
|
|
|
|
|
return -two54/zero; /* log(+-0)=-inf */
|
|
|
|
|
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
|
|
|
|
|
k -= 54; x *= two54; /* subnormal number, scale up x */
|
|
|
|
|
hx = __HI(x); /* high word of x */
|
|
|
|
|
}
|
|
|
|
|
if (hx >= 0x7ff00000) return x+x;
|
|
|
|
|
k += (hx>>20)-1023;
|
|
|
|
|
i = (k&0x80000000)>>>31; // unsigned shift
|
|
|
|
|
hx = (hx&0x000fffff)|((0x3ff-i)<<20);
|
|
|
|
|
y = (double)(k+i);
|
|
|
|
|
x = __HI(x, hx); //original: __HI(x) = hx;
|
2023-02-11 02:15:46 +00:00
|
|
|
z = y*log10_2lo + ivln10*log(x);
|
2023-01-30 20:33:01 +00:00
|
|
|
return z+y*log10_2hi;
|
|
|
|
|
}
|
|
|
|
|
}
|
2023-02-02 20:36:34 +00:00
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Returns the natural logarithm of the sum of the argument and 1.
|
|
|
|
|
*
|
|
|
|
|
* Method :
|
|
|
|
|
* 1. Argument Reduction: find k and f such that
|
|
|
|
|
* 1+x = 2^k * (1+f),
|
|
|
|
|
* where sqrt(2)/2 < 1+f < sqrt(2) .
|
|
|
|
|
*
|
|
|
|
|
* Note. If k=0, then f=x is exact. However, if k!=0, then f
|
|
|
|
|
* may not be representable exactly. In that case, a correction
|
|
|
|
|
* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
|
|
|
|
|
* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
|
|
|
|
|
* and add back the correction term c/u.
|
|
|
|
|
* (Note: when x > 2**53, one can simply return log(x))
|
|
|
|
|
*
|
|
|
|
|
* 2. Approximation of log1p(f).
|
|
|
|
|
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
|
|
|
|
|
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
|
|
|
|
|
* = 2s + s*R
|
|
|
|
|
* We use a special Reme algorithm on [0,0.1716] to generate
|
|
|
|
|
* a polynomial of degree 14 to approximate R The maximum error
|
|
|
|
|
* of this polynomial approximation is bounded by 2**-58.45. In
|
|
|
|
|
* other words,
|
|
|
|
|
* 2 4 6 8 10 12 14
|
|
|
|
|
* R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
|
|
|
|
|
* (the values of Lp1 to Lp7 are listed in the program)
|
|
|
|
|
* and
|
|
|
|
|
* | 2 14 | -58.45
|
|
|
|
|
* | Lp1*s +...+Lp7*s - R(z) | <= 2
|
|
|
|
|
* | |
|
|
|
|
|
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
|
|
|
|
|
* In order to guarantee error in log below 1ulp, we compute log
|
|
|
|
|
* by
|
|
|
|
|
* log1p(f) = f - (hfsq - s*(hfsq+R)).
|
|
|
|
|
*
|
|
|
|
|
* 3. Finally, log1p(x) = k*ln2 + log1p(f).
|
|
|
|
|
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
|
|
|
|
|
* Here ln2 is split into two floating point number:
|
|
|
|
|
* ln2_hi + ln2_lo,
|
|
|
|
|
* where n*ln2_hi is always exact for |n| < 2000.
|
|
|
|
|
*
|
|
|
|
|
* Special cases:
|
|
|
|
|
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
|
|
|
|
|
* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
|
|
|
|
|
* log1p(NaN) is that NaN with no signal.
|
|
|
|
|
*
|
|
|
|
|
* Accuracy:
|
|
|
|
|
* 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.
|
|
|
|
|
*
|
|
|
|
|
* Note: Assuming log() return accurate answer, the following
|
|
|
|
|
* algorithm can be used to compute log1p(x) to within a few ULP:
|
|
|
|
|
*
|
|
|
|
|
* u = 1+x;
|
|
|
|
|
* if(u==1.0) return x ; else
|
|
|
|
|
* return log(u)*(x/(u-1.0));
|
|
|
|
|
*
|
|
|
|
|
* See HP-15C Advanced Functions Handbook, p.193.
|
|
|
|
|
*/
|
|
|
|
|
static class Log1p {
|
|
|
|
|
private static double ln2_hi = 6.93147180369123816490e-01; /* 3fe62e42 fee00000 */
|
|
|
|
|
private static double ln2_lo = 1.90821492927058770002e-10; /* 3dea39ef 35793c76 */
|
|
|
|
|
private static double two54 = 1.80143985094819840000e+16; /* 43500000 00000000 */
|
|
|
|
|
private static double Lp1 = 6.666666666666735130e-01; /* 3FE55555 55555593 */
|
|
|
|
|
private static double Lp2 = 3.999999999940941908e-01; /* 3FD99999 9997FA04 */
|
|
|
|
|
private static double Lp3 = 2.857142874366239149e-01; /* 3FD24924 94229359 */
|
|
|
|
|
private static double Lp4 = 2.222219843214978396e-01; /* 3FCC71C5 1D8E78AF */
|
|
|
|
|
private static double Lp5 = 1.818357216161805012e-01; /* 3FC74664 96CB03DE */
|
|
|
|
|
private static double Lp6 = 1.531383769920937332e-01; /* 3FC39A09 D078C69F */
|
|
|
|
|
private static double Lp7 = 1.479819860511658591e-01; /* 3FC2F112 DF3E5244 */
|
|
|
|
|
private static double zero = 0.0;
|
|
|
|
|
|
|
|
|
|
public static double compute(double x) {
|
|
|
|
|
double hfsq,f=0,c=0,s,z,R,u;
|
|
|
|
|
int k,hx,hu=0,ax;
|
|
|
|
|
|
|
|
|
|
hx = __HI(x); /* high word of x */
|
|
|
|
|
ax = hx&0x7fffffff;
|
|
|
|
|
|
|
|
|
|
k = 1;
|
|
|
|
|
if (hx < 0x3FDA827A) { /* x < 0.41422 */
|
|
|
|
|
if(ax>=0x3ff00000) { /* x <= -1.0 */
|
|
|
|
|
/*
|
|
|
|
|
* Added redundant test against hx to work around VC++
|
|
|
|
|
* code generation problem.
|
|
|
|
|
*/
|
|
|
|
|
if(x==-1.0 && (hx==0xbff00000)) /* log1p(-1)=-inf */
|
|
|
|
|
return -two54/zero;
|
|
|
|
|
else
|
|
|
|
|
return (x-x)/(x-x); /* log1p(x<-1)=NaN */
|
|
|
|
|
}
|
|
|
|
|
if(ax<0x3e200000) { /* |x| < 2**-29 */
|
|
|
|
|
if(two54+x>zero /* raise inexact */
|
|
|
|
|
&&ax<0x3c900000) /* |x| < 2**-54 */
|
|
|
|
|
return x;
|
|
|
|
|
else
|
|
|
|
|
return x - x*x*0.5;
|
|
|
|
|
}
|
|
|
|
|
if(hx>0||hx<=((int)0xbfd2bec3)) {
|
|
|
|
|
k=0;f=x;hu=1;} /* -0.2929<x<0.41422 */
|
|
|
|
|
}
|
|
|
|
|
if (hx >= 0x7ff00000) return x+x;
|
|
|
|
|
if(k!=0) {
|
|
|
|
|
if(hx<0x43400000) {
|
|
|
|
|
u = 1.0+x;
|
|
|
|
|
hu = __HI(u); /* high word of u */
|
|
|
|
|
k = (hu>>20)-1023;
|
|
|
|
|
c = (k>0)? 1.0-(u-x):x-(u-1.0);/* correction term */
|
|
|
|
|
c /= u;
|
|
|
|
|
} else {
|
|
|
|
|
u = x;
|
|
|
|
|
hu = __HI(u); /* high word of u */
|
|
|
|
|
k = (hu>>20)-1023;
|
|
|
|
|
c = 0;
|
|
|
|
|
}
|
|
|
|
|
hu &= 0x000fffff;
|
|
|
|
|
if(hu<0x6a09e) {
|
|
|
|
|
u = __HI(u, hu|0x3ff00000); /* normalize u */
|
|
|
|
|
} else {
|
|
|
|
|
k += 1;
|
|
|
|
|
u = __HI(u, hu|0x3fe00000); /* normalize u/2 */
|
|
|
|
|
hu = (0x00100000-hu)>>2;
|
|
|
|
|
}
|
|
|
|
|
f = u-1.0;
|
|
|
|
|
}
|
|
|
|
|
hfsq=0.5*f*f;
|
|
|
|
|
if(hu==0) { /* |f| < 2**-20 */
|
|
|
|
|
if(f==zero) { if(k==0) return zero;
|
|
|
|
|
else {c += k*ln2_lo; return k*ln2_hi+c;}}
|
|
|
|
|
R = hfsq*(1.0-0.66666666666666666*f);
|
|
|
|
|
if(k==0) return f-R; else
|
|
|
|
|
return k*ln2_hi-((R-(k*ln2_lo+c))-f);
|
|
|
|
|
}
|
|
|
|
|
s = f/(2.0+f);
|
|
|
|
|
z = s*s;
|
|
|
|
|
R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
|
|
|
|
|
if(k==0) return f-(hfsq-s*(hfsq+R)); else
|
|
|
|
|
return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
|
|
|
|
|
}
|
|
|
|
|
}
|
2023-02-04 00:48:26 +00:00
|
|
|
|
|
|
|
|
/* expm1(x)
|
|
|
|
|
* Returns exp(x)-1, the exponential of x minus 1.
|
|
|
|
|
*
|
|
|
|
|
* Method
|
|
|
|
|
* 1. Argument reduction:
|
|
|
|
|
* Given x, find r and integer k such that
|
|
|
|
|
*
|
|
|
|
|
* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
|
|
|
|
|
*
|
|
|
|
|
* Here a correction term c will be computed to compensate
|
|
|
|
|
* the error in r when rounded to a floating-point number.
|
|
|
|
|
*
|
|
|
|
|
* 2. Approximating expm1(r) by a special rational function on
|
|
|
|
|
* the interval [0,0.34658]:
|
|
|
|
|
* Since
|
|
|
|
|
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
|
|
|
|
|
* we define R1(r*r) by
|
|
|
|
|
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
|
|
|
|
|
* That is,
|
|
|
|
|
* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
|
|
|
|
|
* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
|
|
|
|
|
* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
|
|
|
|
|
* We use a special Reme algorithm on [0,0.347] to generate
|
|
|
|
|
* a polynomial of degree 5 in r*r to approximate R1. The
|
|
|
|
|
* maximum error of this polynomial approximation is bounded
|
|
|
|
|
* by 2**-61. In other words,
|
|
|
|
|
* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
|
|
|
|
|
* where Q1 = -1.6666666666666567384E-2,
|
|
|
|
|
* Q2 = 3.9682539681370365873E-4,
|
|
|
|
|
* Q3 = -9.9206344733435987357E-6,
|
|
|
|
|
* Q4 = 2.5051361420808517002E-7,
|
|
|
|
|
* Q5 = -6.2843505682382617102E-9;
|
|
|
|
|
* (where z=r*r, and the values of Q1 to Q5 are listed below)
|
|
|
|
|
* with error bounded by
|
|
|
|
|
* | 5 | -61
|
|
|
|
|
* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
|
|
|
|
|
* | |
|
|
|
|
|
*
|
|
|
|
|
* expm1(r) = exp(r)-1 is then computed by the following
|
|
|
|
|
* specific way which minimize the accumulation rounding error:
|
|
|
|
|
* 2 3
|
|
|
|
|
* r r [ 3 - (R1 + R1*r/2) ]
|
|
|
|
|
* expm1(r) = r + --- + --- * [--------------------]
|
|
|
|
|
* 2 2 [ 6 - r*(3 - R1*r/2) ]
|
|
|
|
|
*
|
|
|
|
|
* To compensate the error in the argument reduction, we use
|
|
|
|
|
* expm1(r+c) = expm1(r) + c + expm1(r)*c
|
|
|
|
|
* ~ expm1(r) + c + r*c
|
|
|
|
|
* Thus c+r*c will be added in as the correction terms for
|
|
|
|
|
* expm1(r+c). Now rearrange the term to avoid optimization
|
|
|
|
|
* screw up:
|
|
|
|
|
* ( 2 2 )
|
|
|
|
|
* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
|
|
|
|
|
* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
|
|
|
|
|
* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
|
|
|
|
|
* ( )
|
|
|
|
|
*
|
|
|
|
|
* = r - E
|
|
|
|
|
* 3. Scale back to obtain expm1(x):
|
|
|
|
|
* From step 1, we have
|
|
|
|
|
* expm1(x) = either 2^k*[expm1(r)+1] - 1
|
|
|
|
|
* = or 2^k*[expm1(r) + (1-2^-k)]
|
|
|
|
|
* 4. Implementation notes:
|
|
|
|
|
* (A). To save one multiplication, we scale the coefficient Qi
|
|
|
|
|
* to Qi*2^i, and replace z by (x^2)/2.
|
|
|
|
|
* (B). To achieve maximum accuracy, we compute expm1(x) by
|
|
|
|
|
* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
|
|
|
|
|
* (ii) if k=0, return r-E
|
|
|
|
|
* (iii) if k=-1, return 0.5*(r-E)-0.5
|
|
|
|
|
* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
|
|
|
|
|
* else return 1.0+2.0*(r-E);
|
|
|
|
|
* (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
|
|
|
|
|
* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
|
|
|
|
|
* (vii) return 2^k(1-((E+2^-k)-r))
|
|
|
|
|
*
|
|
|
|
|
* Special cases:
|
|
|
|
|
* expm1(INF) is INF, expm1(NaN) is NaN;
|
|
|
|
|
* expm1(-INF) is -1, and
|
|
|
|
|
* for finite argument, only expm1(0)=0 is exact.
|
|
|
|
|
*
|
|
|
|
|
* Accuracy:
|
|
|
|
|
* according to an error analysis, the error is always less than
|
|
|
|
|
* 1 ulp (unit in the last place).
|
|
|
|
|
*
|
|
|
|
|
* Misc. info.
|
|
|
|
|
* For IEEE double
|
|
|
|
|
* if x > 7.09782712893383973096e+02 then expm1(x) overflow
|
|
|
|
|
*
|
|
|
|
|
* 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.
|
|
|
|
|
*/
|
|
|
|
|
static class Expm1 {
|
|
|
|
|
private static final double one = 1.0;
|
|
|
|
|
private static final double huge = 1.0e+300;
|
|
|
|
|
private static final double tiny = 1.0e-300;
|
|
|
|
|
private static final double o_threshold = 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
|
|
|
|
|
private static final double ln2_hi = 6.93147180369123816490e-01; /* 0x3fe62e42, 0xfee00000 */
|
|
|
|
|
private static final double ln2_lo = 1.90821492927058770002e-10; /* 0x3dea39ef, 0x35793c76 */
|
|
|
|
|
private static final double invln2 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */
|
|
|
|
|
/* scaled coefficients related to expm1 */
|
|
|
|
|
private static final double Q1 = -3.33333333333331316428e-02; /* BFA11111 111110F4 */
|
|
|
|
|
private static final double Q2 = 1.58730158725481460165e-03; /* 3F5A01A0 19FE5585 */
|
|
|
|
|
private static final double Q3 = -7.93650757867487942473e-05; /* BF14CE19 9EAADBB7 */
|
|
|
|
|
private static final double Q4 = 4.00821782732936239552e-06; /* 3ED0CFCA 86E65239 */
|
|
|
|
|
private static final double Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */
|
|
|
|
|
|
|
|
|
|
static double compute(double x) {
|
|
|
|
|
double y,hi,lo,c=0,t,e,hxs,hfx,r1;
|
|
|
|
|
int k,xsb;
|
|
|
|
|
/*unsigned*/ int hx;
|
|
|
|
|
|
|
|
|
|
hx = __HI(x); /* high word of x */
|
|
|
|
|
xsb = hx&0x80000000; /* sign bit of x */
|
|
|
|
|
if(xsb==0) y=x; else y= -x; /* y = |x| */
|
|
|
|
|
hx &= 0x7fffffff; /* high word of |x| */
|
|
|
|
|
|
|
|
|
|
/* filter out huge and non-finite argument */
|
|
|
|
|
if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */
|
|
|
|
|
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
|
|
|
|
|
if(hx>=0x7ff00000) {
|
|
|
|
|
if(((hx&0xfffff)|__LO(x))!=0)
|
|
|
|
|
return x+x; /* NaN */
|
|
|
|
|
else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */
|
|
|
|
|
}
|
|
|
|
|
if(x > o_threshold) return huge*huge; /* overflow */
|
|
|
|
|
}
|
|
|
|
|
if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */
|
|
|
|
|
if(x+tiny<0.0) /* raise inexact */
|
|
|
|
|
return tiny-one; /* return -1 */
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* argument reduction */
|
|
|
|
|
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
|
|
|
|
|
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
|
|
|
|
|
if(xsb==0)
|
|
|
|
|
{hi = x - ln2_hi; lo = ln2_lo; k = 1;}
|
|
|
|
|
else
|
|
|
|
|
{hi = x + ln2_hi; lo = -ln2_lo; k = -1;}
|
|
|
|
|
} else {
|
|
|
|
|
k = (int)(invln2*x+((xsb==0)?0.5:-0.5));
|
|
|
|
|
t = k;
|
|
|
|
|
hi = x - t*ln2_hi; /* t*ln2_hi is exact here */
|
|
|
|
|
lo = t*ln2_lo;
|
|
|
|
|
}
|
|
|
|
|
x = hi - lo;
|
|
|
|
|
c = (hi-x)-lo;
|
|
|
|
|
}
|
|
|
|
|
else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */
|
|
|
|
|
t = huge+x; /* return x with inexact flags when x!=0 */
|
|
|
|
|
return x - (t-(huge+x));
|
|
|
|
|
}
|
|
|
|
|
else k = 0;
|
|
|
|
|
|
|
|
|
|
/* x is now in primary range */
|
|
|
|
|
hfx = 0.5*x;
|
|
|
|
|
hxs = x*hfx;
|
|
|
|
|
r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5))));
|
|
|
|
|
t = 3.0-r1*hfx;
|
|
|
|
|
e = hxs*((r1-t)/(6.0 - x*t));
|
|
|
|
|
if(k==0) return x - (x*e-hxs); /* c is 0 */
|
|
|
|
|
else {
|
|
|
|
|
e = (x*(e-c)-c);
|
|
|
|
|
e -= hxs;
|
|
|
|
|
if(k== -1) return 0.5*(x-e)-0.5;
|
|
|
|
|
if(k==1) {
|
|
|
|
|
if(x < -0.25) return -2.0*(e-(x+0.5));
|
|
|
|
|
else return one+2.0*(x-e);
|
|
|
|
|
}
|
|
|
|
|
if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */
|
|
|
|
|
y = one-(e-x);
|
|
|
|
|
y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */
|
|
|
|
|
return y-one;
|
|
|
|
|
}
|
|
|
|
|
t = one;
|
|
|
|
|
if(k<20) {
|
|
|
|
|
t = __HI(t, 0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */
|
|
|
|
|
y = t-(e-x);
|
|
|
|
|
y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */
|
|
|
|
|
} else {
|
|
|
|
|
t = __HI(t, ((0x3ff-k)<<20)); /* 2^-k */
|
|
|
|
|
y = x-(e+t);
|
|
|
|
|
y += one;
|
|
|
|
|
y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
return y;
|
|
|
|
|
}
|
|
|
|
|
}
|
2023-02-17 03:22:06 +00:00
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Method :
|
|
|
|
|
* mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
|
|
|
|
|
* 1. Replace x by |x| (sinh(-x) = -sinh(x)).
|
|
|
|
|
* 2.
|
|
|
|
|
* E + E/(E+1)
|
|
|
|
|
* 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x)
|
|
|
|
|
* 2
|
|
|
|
|
*
|
|
|
|
|
* 22 <= x <= lnovft : sinh(x) := exp(x)/2
|
|
|
|
|
* lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2)
|
|
|
|
|
* ln2ovft < x : sinh(x) := x*shuge (overflow)
|
|
|
|
|
*
|
|
|
|
|
* Special cases:
|
|
|
|
|
* sinh(x) is |x| if x is +INF, -INF, or NaN.
|
|
|
|
|
* only sinh(0)=0 is exact for finite x.
|
|
|
|
|
*/
|
|
|
|
|
private static final class Sinh {
|
|
|
|
|
private static final double one = 1.0, shuge = 1.0e307;
|
|
|
|
|
|
|
|
|
|
static double compute(double x) {
|
|
|
|
|
double t,w,h;
|
|
|
|
|
int ix,jx;
|
|
|
|
|
/* unsigned */ int lx;
|
|
|
|
|
|
|
|
|
|
/* High word of |x|. */
|
|
|
|
|
jx = __HI(x);
|
|
|
|
|
ix = jx&0x7fffffff;
|
|
|
|
|
|
|
|
|
|
/* x is INF or NaN */
|
|
|
|
|
if(ix>=0x7ff00000) return x+x;
|
|
|
|
|
|
|
|
|
|
h = 0.5;
|
|
|
|
|
if (jx<0) h = -h;
|
|
|
|
|
/* |x| in [0,22], return sign(x)*0.5*(E+E/(E+1))) */
|
|
|
|
|
if (ix < 0x40360000) { /* |x|<22 */
|
|
|
|
|
if (ix<0x3e300000) /* |x|<2**-28 */
|
|
|
|
|
if(shuge+x>one) return x;/* sinh(tiny) = tiny with inexact */
|
|
|
|
|
t = FdlibmTranslit.expm1(Math.abs(x));
|
|
|
|
|
if(ix<0x3ff00000) return h*(2.0*t-t*t/(t+one));
|
|
|
|
|
return h*(t+t/(t+one));
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* |x| in [22, log(maxdouble)] return 0.5*exp(|x|) */
|
|
|
|
|
if (ix < 0x40862E42) return h*FdlibmTranslit.exp(Math.abs(x));
|
|
|
|
|
|
|
|
|
|
/* |x| in [log(maxdouble), overflowthresold] */
|
|
|
|
|
// Note: the original FDLIBM sources use
|
|
|
|
|
// lx = *( (((*(unsigned*)&one)>>29)) + (unsigned*)&x);
|
|
|
|
|
// to set lx to the low-order 32 bits of x. The expression
|
|
|
|
|
// in question is an alternate way to implement the
|
|
|
|
|
// functionality of the C FDLIBM __LO macro and the
|
|
|
|
|
// expression is coded to work on both big-edian and
|
|
|
|
|
// little-endian machines. However, this port will instead
|
|
|
|
|
// use the __LO method call to represent this
|
|
|
|
|
// functionality.
|
|
|
|
|
lx = __LO(x);
|
|
|
|
|
if (ix<0x408633CE || ((ix==0x408633ce)&&(Long.compareUnsigned(lx, 0x8fb9f87d) <= 0 ))) {
|
|
|
|
|
w = exp(0.5*Math.abs(x));
|
|
|
|
|
t = h*w;
|
|
|
|
|
return t*w;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* |x| > overflowthresold, sinh(x) overflow */
|
|
|
|
|
return x*shuge;
|
|
|
|
|
}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/**
|
|
|
|
|
* Method :
|
|
|
|
|
* mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
|
|
|
|
|
* 1. Replace x by |x| (cosh(x) = cosh(-x)).
|
|
|
|
|
* 2.
|
|
|
|
|
* [ exp(x) - 1 ]^2
|
|
|
|
|
* 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
|
|
|
|
|
* 2*exp(x)
|
|
|
|
|
*
|
|
|
|
|
* exp(x) + 1/exp(x)
|
|
|
|
|
* ln2/2 <= x <= 22 : cosh(x) := -------------------
|
|
|
|
|
* 2
|
|
|
|
|
* 22 <= x <= lnovft : cosh(x) := exp(x)/2
|
|
|
|
|
* lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
|
|
|
|
|
* ln2ovft < x : cosh(x) := huge*huge (overflow)
|
|
|
|
|
*
|
|
|
|
|
* Special cases:
|
|
|
|
|
* cosh(x) is |x| if x is +INF, -INF, or NaN.
|
|
|
|
|
* only cosh(0)=1 is exact for finite x.
|
|
|
|
|
*/
|
|
|
|
|
private static final class Cosh {
|
|
|
|
|
private static final double one = 1.0, half=0.5, huge = 1.0e300;
|
|
|
|
|
static double compute(double x) {
|
|
|
|
|
double t,w;
|
|
|
|
|
int ix;
|
|
|
|
|
/*unsigned*/ int lx;
|
|
|
|
|
|
|
|
|
|
/* High word of |x|. */
|
|
|
|
|
ix = __HI(x);
|
|
|
|
|
ix &= 0x7fffffff;
|
|
|
|
|
|
|
|
|
|
/* x is INF or NaN */
|
|
|
|
|
if(ix>=0x7ff00000) return x*x;
|
|
|
|
|
|
|
|
|
|
/* |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|)) */
|
|
|
|
|
if(ix<0x3fd62e43) {
|
|
|
|
|
t = expm1(Math.abs(x));
|
|
|
|
|
w = one+t;
|
|
|
|
|
if (ix<0x3c800000) return w; /* cosh(tiny) = 1 */
|
|
|
|
|
return one+(t*t)/(w+w);
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* |x| in [0.5*ln2,22], return (exp(|x|)+1/exp(|x|)/2; */
|
|
|
|
|
if (ix < 0x40360000) {
|
|
|
|
|
t = exp(Math.abs(x));
|
|
|
|
|
return half*t+half/t;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* |x| in [22, log(maxdouble)] return half*exp(|x|) */
|
|
|
|
|
if (ix < 0x40862E42) return half*exp(Math.abs(x));
|
|
|
|
|
|
|
|
|
|
/* |x| in [log(maxdouble), overflowthresold] */
|
|
|
|
|
// See note above in the sinh implementation for how this
|
|
|
|
|
// transliteration port uses __LO(x) in the line below
|
|
|
|
|
// that differs from the idiom used in the original FDLIBM.
|
|
|
|
|
lx = __LO(x);
|
|
|
|
|
if (ix<0x408633CE ||
|
|
|
|
|
((ix==0x408633ce)&&(Integer.compareUnsigned(lx, 0x8fb9f87d) <= 0))) {
|
|
|
|
|
w = exp(half*Math.abs(x));
|
|
|
|
|
t = half*w;
|
|
|
|
|
return t*w;
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
/* |x| > overflowthresold, cosh(x) overflow */
|
|
|
|
|
return huge*huge;
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}
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}
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/**
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* Return the Hyperbolic Tangent of x
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*
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* Method :
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* x -x
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* e - e
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* 0. tanh(x) is defined to be -----------
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* x -x
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* e + e
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* 1. reduce x to non-negative by tanh(-x) = -tanh(x).
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* 2. 0 <= x <= 2**-55 : tanh(x) := x*(one+x)
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* -t
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* 2**-55 < x <= 1 : tanh(x) := -----; t = expm1(-2x)
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* t + 2
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* 2
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* 1 <= x <= 22.0 : tanh(x) := 1- ----- ; t=expm1(2x)
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* t + 2
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* 22.0 < x <= INF : tanh(x) := 1.
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*
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* Special cases:
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* tanh(NaN) is NaN;
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* only tanh(0)=0 is exact for finite argument.
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*/
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private static final class Tanh {
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private static final double one=1.0, two=2.0, tiny = 1.0e-300;
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static double compute(double x) {
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double t,z;
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int jx,ix;
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/* High word of |x|. */
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jx = __HI(x);
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ix = jx&0x7fffffff;
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/* x is INF or NaN */
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if(ix>=0x7ff00000) {
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if (jx>=0) return one/x+one; /* tanh(+-inf)=+-1 */
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else return one/x-one; /* tanh(NaN) = NaN */
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}
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/* |x| < 22 */
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if (ix < 0x40360000) { /* |x|<22 */
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if (ix<0x3c800000) /* |x|<2**-55 */
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|
return x*(one+x); /* tanh(small) = small */
|
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|
|
if (ix>=0x3ff00000) { /* |x|>=1 */
|
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|
|
|
t = expm1(two*Math.abs(x));
|
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|
|
|
z = one - two/(t+two);
|
|
|
|
|
} else {
|
|
|
|
|
t = expm1(-two*Math.abs(x));
|
|
|
|
|
z= -t/(t+two);
|
|
|
|
|
}
|
|
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|
|
/* |x| > 22, return +-1 */
|
|
|
|
|
} else {
|
|
|
|
|
z = one - tiny; /* raised inexact flag */
|
|
|
|
|
}
|
|
|
|
|
return (jx>=0)? z: -z;
|
|
|
|
|
}
|
|
|
|
|
}
|
2015-10-07 01:39:26 +00:00
|
|
|
}
|
