Java tutorial
/* * Licensed to the Apache Software Foundation (ASF) under one or more * contributor license agreements. See the NOTICE file distributed with * this work for additional information regarding copyright ownership. * The ASF licenses this file to You under the Apache License, Version 2.0 * (the "License"); you may not use this file except in compliance with * the License. You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ package org.apache.lucene.util; /* some code derived from jodk: http://code.google.com/p/jodk/ (apache 2.0) * asin() derived from fdlibm: http://www.netlib.org/fdlibm/e_asin.c (public domain): * ============================================================================= * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ============================================================================= */ /** Math functions that trade off accuracy for speed. */ public class SloppyMath { /** * Returns the Haversine distance in meters between two points * specified in decimal degrees (latitude/longitude). This works correctly * even if the dateline is between the two points. * <p> * Error is at most 4E-1 (40cm) from the actual haversine distance, but is typically * much smaller for reasonable distances: around 1E-5 (0.01mm) for distances less than * 1000km. * * @param lat1 Latitude of the first point. * @param lon1 Longitude of the first point. * @param lat2 Latitude of the second point. * @param lon2 Longitude of the second point. * @return distance in meters. */ public static double haversinMeters(double lat1, double lon1, double lat2, double lon2) { return haversinMeters(haversinSortKey(lat1, lon1, lat2, lon2)); } /** * Returns the Haversine distance in meters between two points * given the previous result from {@link #haversinSortKey(double, double, double, double)} * @return distance in meters. */ public static double haversinMeters(double sortKey) { return TO_METERS * 2 * asin(Math.min(1, Math.sqrt(sortKey * 0.5))); } /** * Returns the Haversine distance in kilometers between two points * specified in decimal degrees (latitude/longitude). This works correctly * even if the dateline is between the two points. * * @param lat1 Latitude of the first point. * @param lon1 Longitude of the first point. * @param lat2 Latitude of the second point. * @param lon2 Longitude of the second point. * @return distance in kilometers. * @deprecated Use {@link #haversinMeters(double, double, double, double) instead} */ @Deprecated public static double haversinKilometers(double lat1, double lon1, double lat2, double lon2) { double h = haversinSortKey(lat1, lon1, lat2, lon2); return TO_KILOMETERS * 2 * asin(Math.min(1, Math.sqrt(h * 0.5))); } /** * Returns a sort key for distance. This is less expensive to compute than * {@link #haversinMeters(double, double, double, double)}, but it always compares the same. * This can be converted into an actual distance with {@link #haversinMeters(double)}, which * effectively does the second half of the computation. */ public static double haversinSortKey(double lat1, double lon1, double lat2, double lon2) { double x1 = lat1 * TO_RADIANS; double x2 = lat2 * TO_RADIANS; double h1 = 1 - cos(x1 - x2); double h2 = 1 - cos((lon1 - lon2) * TO_RADIANS); double h = h1 + cos(x1) * cos(x2) * h2; // clobber crazy precision so subsequent rounding does not create ties. return Double.longBitsToDouble(Double.doubleToRawLongBits(h) & 0xFFFFFFFFFFFFFFF8L); } /** * Returns the trigonometric cosine of an angle. * <p> * Error is around 1E-15. * <p> * Special cases: * <ul> * <li>If the argument is {@code NaN} or an infinity, then the result is {@code NaN}. * </ul> * @param a an angle, in radians. * @return the cosine of the argument. * @see Math#cos(double) */ public static double cos(double a) { if (a < 0.0) { a = -a; } if (a > SIN_COS_MAX_VALUE_FOR_INT_MODULO) { return Math.cos(a); } // index: possibly outside tables range. int index = (int) (a * SIN_COS_INDEXER + 0.5); double delta = (a - index * SIN_COS_DELTA_HI) - index * SIN_COS_DELTA_LO; // Making sure index is within tables range. // Last value of each table is the same than first, so we ignore it (tabs size minus one) for modulo. index &= (SIN_COS_TABS_SIZE - 2); // index % (SIN_COS_TABS_SIZE-1) double indexCos = cosTab[index]; double indexSin = sinTab[index]; return indexCos + delta * (-indexSin + delta * (-indexCos * ONE_DIV_F2 + delta * (indexSin * ONE_DIV_F3 + delta * indexCos * ONE_DIV_F4))); } /** * Returns the arc sine of a value. * <p> * The returned angle is in the range <i>-pi</i>/2 through <i>pi</i>/2. * Error is around 1E-7. * <p> * Special cases: * <ul> * <li>If the argument is {@code NaN} or its absolute value is greater than 1, then the result is {@code NaN}. * </ul> * @param a the value whose arc sine is to be returned. * @return arc sine of the argument * @see Math#asin(double) */ // because asin(-x) = -asin(x), asin(x) only needs to be computed on [0,1]. // ---> we only have to compute asin(x) on [0,1]. // For values not close to +-1, we use look-up tables; // for values near +-1, we use code derived from fdlibm. public static double asin(double a) { boolean negateResult; if (a < 0.0) { a = -a; negateResult = true; } else { negateResult = false; } if (a <= ASIN_MAX_VALUE_FOR_TABS) { int index = (int) (a * ASIN_INDEXER + 0.5); double delta = a - index * ASIN_DELTA; double result = asinTab[index] + delta * (asinDer1DivF1Tab[index] + delta * (asinDer2DivF2Tab[index] + delta * (asinDer3DivF3Tab[index] + delta * asinDer4DivF4Tab[index]))); return negateResult ? -result : result; } else { // value > ASIN_MAX_VALUE_FOR_TABS, or value is NaN // This part is derived from fdlibm. if (a < 1.0) { double t = (1.0 - a) * 0.5; double p = t * (ASIN_PS0 + t * (ASIN_PS1 + t * (ASIN_PS2 + t * (ASIN_PS3 + t * (ASIN_PS4 + t * ASIN_PS5))))); double q = 1.0 + t * (ASIN_QS1 + t * (ASIN_QS2 + t * (ASIN_QS3 + t * ASIN_QS4))); double s = Math.sqrt(t); double z = s + s * (p / q); double result = ASIN_PIO2_HI - ((z + z) - ASIN_PIO2_LO); return negateResult ? -result : result; } else { // value >= 1.0, or value is NaN if (a == 1.0) { return negateResult ? -Math.PI / 2 : Math.PI / 2; } else { return Double.NaN; } } } } /** * Convert to degrees. * @param radians radians to convert to degrees * @return degrees */ public static double toDegrees(final double radians) { return radians * TO_DEGREES; } /** * Convert to radians. * @param degrees degrees to convert to radians * @return radians */ public static double toRadians(final double degrees) { return degrees * TO_RADIANS; } // haversin // TODO: remove these for java 9, they fixed Math.toDegrees()/toRadians() to work just like this. public static final double TO_RADIANS = Math.PI / 180D; public static final double TO_DEGREES = 180D / Math.PI; // Earth's mean radius, in meters and kilometers; see http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf private static final double TO_METERS = 6_371_008.7714D; // equatorial radius private static final double TO_KILOMETERS = 6_371.0087714D; // equatorial radius // cos/asin private static final double ONE_DIV_F2 = 1 / 2.0; private static final double ONE_DIV_F3 = 1 / 6.0; private static final double ONE_DIV_F4 = 1 / 24.0; private static final double PIO2_HI = Double.longBitsToDouble(0x3FF921FB54400000L); // 1.57079632673412561417e+00 first 33 bits of pi/2 private static final double PIO2_LO = Double.longBitsToDouble(0x3DD0B4611A626331L); // 6.07710050650619224932e-11 pi/2 - PIO2_HI private static final double TWOPI_HI = 4 * PIO2_HI; private static final double TWOPI_LO = 4 * PIO2_LO; private static final int SIN_COS_TABS_SIZE = (1 << 11) + 1; private static final double SIN_COS_DELTA_HI = TWOPI_HI / (SIN_COS_TABS_SIZE - 1); private static final double SIN_COS_DELTA_LO = TWOPI_LO / (SIN_COS_TABS_SIZE - 1); private static final double SIN_COS_INDEXER = 1 / (SIN_COS_DELTA_HI + SIN_COS_DELTA_LO); private static final double[] sinTab = new double[SIN_COS_TABS_SIZE]; private static final double[] cosTab = new double[SIN_COS_TABS_SIZE]; // Max abs value for fast modulo, above which we use regular angle normalization. // This value must be < (Integer.MAX_VALUE / SIN_COS_INDEXER), to stay in range of int type. // The higher it is, the higher the error, but also the faster it is for lower values. // If you set it to ((Integer.MAX_VALUE / SIN_COS_INDEXER) * 0.99), worse accuracy on double range is about 1e-10. static final double SIN_COS_MAX_VALUE_FOR_INT_MODULO = ((Integer.MAX_VALUE >> 9) / SIN_COS_INDEXER) * 0.99; // Supposed to be >= sin(77.2deg), as fdlibm code is supposed to work with values > 0.975, // but seems to work well enough as long as value >= sin(25deg). private static final double ASIN_MAX_VALUE_FOR_TABS = StrictMath.sin(toRadians(73.0)); private static final int ASIN_TABS_SIZE = (1 << 13) + 1; private static final double ASIN_DELTA = ASIN_MAX_VALUE_FOR_TABS / (ASIN_TABS_SIZE - 1); private static final double ASIN_INDEXER = 1 / ASIN_DELTA; private static final double[] asinTab = new double[ASIN_TABS_SIZE]; private static final double[] asinDer1DivF1Tab = new double[ASIN_TABS_SIZE]; private static final double[] asinDer2DivF2Tab = new double[ASIN_TABS_SIZE]; private static final double[] asinDer3DivF3Tab = new double[ASIN_TABS_SIZE]; private static final double[] asinDer4DivF4Tab = new double[ASIN_TABS_SIZE]; private static final double ASIN_PIO2_HI = Double.longBitsToDouble(0x3FF921FB54442D18L); // 1.57079632679489655800e+00 private static final double ASIN_PIO2_LO = Double.longBitsToDouble(0x3C91A62633145C07L); // 6.12323399573676603587e-17 private static final double ASIN_PS0 = Double.longBitsToDouble(0x3fc5555555555555L); // 1.66666666666666657415e-01 private static final double ASIN_PS1 = Double.longBitsToDouble(0xbfd4d61203eb6f7dL); // -3.25565818622400915405e-01 private static final double ASIN_PS2 = Double.longBitsToDouble(0x3fc9c1550e884455L); // 2.01212532134862925881e-01 private static final double ASIN_PS3 = Double.longBitsToDouble(0xbfa48228b5688f3bL); // -4.00555345006794114027e-02 private static final double ASIN_PS4 = Double.longBitsToDouble(0x3f49efe07501b288L); // 7.91534994289814532176e-04 private static final double ASIN_PS5 = Double.longBitsToDouble(0x3f023de10dfdf709L); // 3.47933107596021167570e-05 private static final double ASIN_QS1 = Double.longBitsToDouble(0xc0033a271c8a2d4bL); // -2.40339491173441421878e+00 private static final double ASIN_QS2 = Double.longBitsToDouble(0x40002ae59c598ac8L); // 2.02094576023350569471e+00 private static final double ASIN_QS3 = Double.longBitsToDouble(0xbfe6066c1b8d0159L); // -6.88283971605453293030e-01 private static final double ASIN_QS4 = Double.longBitsToDouble(0x3fb3b8c5b12e9282L); // 7.70381505559019352791e-02 /** Initializes look-up tables. */ static { // sin and cos final int SIN_COS_PI_INDEX = (SIN_COS_TABS_SIZE - 1) / 2; final int SIN_COS_PI_MUL_2_INDEX = 2 * SIN_COS_PI_INDEX; final int SIN_COS_PI_MUL_0_5_INDEX = SIN_COS_PI_INDEX / 2; final int SIN_COS_PI_MUL_1_5_INDEX = 3 * SIN_COS_PI_INDEX / 2; for (int i = 0; i < SIN_COS_TABS_SIZE; i++) { // angle: in [0,2*PI]. double angle = i * SIN_COS_DELTA_HI + i * SIN_COS_DELTA_LO; double sinAngle = StrictMath.sin(angle); double cosAngle = StrictMath.cos(angle); // For indexes corresponding to null cosine or sine, we make sure the value is zero // and not an epsilon. This allows for a much better accuracy for results close to zero. if (i == SIN_COS_PI_INDEX) { sinAngle = 0.0; } else if (i == SIN_COS_PI_MUL_2_INDEX) { sinAngle = 0.0; } else if (i == SIN_COS_PI_MUL_0_5_INDEX) { cosAngle = 0.0; } else if (i == SIN_COS_PI_MUL_1_5_INDEX) { cosAngle = 0.0; } sinTab[i] = sinAngle; cosTab[i] = cosAngle; } // asin for (int i = 0; i < ASIN_TABS_SIZE; i++) { // x: in [0,ASIN_MAX_VALUE_FOR_TABS]. double x = i * ASIN_DELTA; asinTab[i] = StrictMath.asin(x); double oneMinusXSqInv = 1.0 / (1 - x * x); double oneMinusXSqInv0_5 = StrictMath.sqrt(oneMinusXSqInv); double oneMinusXSqInv1_5 = oneMinusXSqInv0_5 * oneMinusXSqInv; double oneMinusXSqInv2_5 = oneMinusXSqInv1_5 * oneMinusXSqInv; double oneMinusXSqInv3_5 = oneMinusXSqInv2_5 * oneMinusXSqInv; asinDer1DivF1Tab[i] = oneMinusXSqInv0_5; asinDer2DivF2Tab[i] = (x * oneMinusXSqInv1_5) * ONE_DIV_F2; asinDer3DivF3Tab[i] = ((1 + 2 * x * x) * oneMinusXSqInv2_5) * ONE_DIV_F3; asinDer4DivF4Tab[i] = ((5 + 2 * x * (2 + x * (5 - 2 * x))) * oneMinusXSqInv3_5) * ONE_DIV_F4; } } }