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Introduce faster approximate sinh/atan math functions (#49009)
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This commit introduces a new class called ESSloppyMath
that is meant to reflect the purpose of Lucene's SloppyMath,
but add additional unimplemented faster alternatives to math functions.

The two that are used by geotile-grid a lot are sinh/atan.

In a quick elasticsearch rally benchmark for geotile-grid on Switzerland
data points, this shows a (1.22x) 22% speed-up over using Math's functions.

closes #41166.
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@talevy
talevy committed Nov 14, 2019
1 parent ba5b4f1 commit 3ab2de1
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180 changes: 180 additions & 0 deletions libs/core/src/main/java/org/elasticsearch/common/util/ESSloppyMath.java
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/*
* 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.elasticsearch.common.util;

/*
* =============================================================================
*
* sinh and atan ported from
* https://github.com/yannrichet/jmathplot/blob/f25426e0ab0e68647ad2b75f577c7be050ecac86/src/main/java/org/math/plot/utils/FastMath.java
* (Apache 2.0) Copyright 2012 Jeff Hain, which is partially derived from fdlibm package
*
* =============================================================================
* 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.
* =============================================================================
*/


/**
* Similar to Lucene's SloppyMath, but for additional math functions.
*/
public class ESSloppyMath {

private ESSloppyMath() {}

//--------------------------------------------------------------------------
// RE-USABLE CONSTANTS
//--------------------------------------------------------------------------

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 TWO_POW_N28 = Double.longBitsToDouble(0x3E30000000000000L);
private static final double TWO_POW_66 = Double.longBitsToDouble(0x4410000000000000L);
private static final double LOG_DOUBLE_MAX_VALUE = StrictMath.log(Double.MAX_VALUE);

//--------------------------------------------------------------------------
// CONSTANTS AND TABLES FOR ATAN
//--------------------------------------------------------------------------

// We use the formula atan(-x) = -atan(x)
// ---> we only have to compute atan(x) on [0,+infinity[.
// For values corresponding to angles not close to +-PI/2, we use look-up tables;
// for values corresponding to angles near +-PI/2, we use code derived from fdlibm.

// Supposed to be >= tan(67.7deg), as fdlibm code is supposed to work with values > 2.4375.
private static final double ATAN_MAX_VALUE_FOR_TABS = StrictMath.tan(Math.toRadians(74.0));

private static final int ATAN_TABS_SIZE = 1<<12 + 1;
private static final double ATAN_DELTA = ATAN_MAX_VALUE_FOR_TABS/(ATAN_TABS_SIZE - 1);
private static final double ATAN_INDEXER = 1/ATAN_DELTA;
private static final double[] atanTab = new double[ATAN_TABS_SIZE];
private static final double[] atanDer1DivF1Tab = new double[ATAN_TABS_SIZE];
private static final double[] atanDer2DivF2Tab = new double[ATAN_TABS_SIZE];
private static final double[] atanDer3DivF3Tab = new double[ATAN_TABS_SIZE];
private static final double[] atanDer4DivF4Tab = new double[ATAN_TABS_SIZE];

private static final double ATAN_HI3 = Double.longBitsToDouble(0x3ff921fb54442d18L); // 1.57079632679489655800e+00 atan(inf)hi
private static final double ATAN_LO3 = Double.longBitsToDouble(0x3c91a62633145c07L); // 6.12323399573676603587e-17 atan(inf)lo
private static final double ATAN_AT0 = Double.longBitsToDouble(0x3fd555555555550dL); // 3.33333333333329318027e-01
private static final double ATAN_AT1 = Double.longBitsToDouble(0xbfc999999998ebc4L); // -1.99999999998764832476e-01
private static final double ATAN_AT2 = Double.longBitsToDouble(0x3fc24924920083ffL); // 1.42857142725034663711e-01
private static final double ATAN_AT3 = Double.longBitsToDouble(0xbfbc71c6fe231671L); // -1.11111104054623557880e-01
private static final double ATAN_AT4 = Double.longBitsToDouble(0x3fb745cdc54c206eL); // 9.09088713343650656196e-02
private static final double ATAN_AT5 = Double.longBitsToDouble(0xbfb3b0f2af749a6dL); // -7.69187620504482999495e-02
private static final double ATAN_AT6 = Double.longBitsToDouble(0x3fb10d66a0d03d51L); // 6.66107313738753120669e-02
private static final double ATAN_AT7 = Double.longBitsToDouble(0xbfadde2d52defd9aL); // -5.83357013379057348645e-02
private static final double ATAN_AT8 = Double.longBitsToDouble(0x3fa97b4b24760debL); // 4.97687799461593236017e-02
private static final double ATAN_AT9 = Double.longBitsToDouble(0xbfa2b4442c6a6c2fL); // -3.65315727442169155270e-02
private static final double ATAN_AT10 = Double.longBitsToDouble(0x3f90ad3ae322da11L); // 1.62858201153657823623e-02

static {
// atan
for (int i=0;i<ATAN_TABS_SIZE;i++) {
// x: in [0,ATAN_MAX_VALUE_FOR_TABS].
double x = i * ATAN_DELTA;
double onePlusXSqInv = 1.0/(1+x*x);
double onePlusXSqInv2 = onePlusXSqInv*onePlusXSqInv;
double onePlusXSqInv3 = onePlusXSqInv2*onePlusXSqInv;
double onePlusXSqInv4 = onePlusXSqInv2*onePlusXSqInv2;
atanTab[i] = StrictMath.atan(x);
atanDer1DivF1Tab[i] = onePlusXSqInv;
atanDer2DivF2Tab[i] = (-2*x*onePlusXSqInv2) * ONE_DIV_F2;
atanDer3DivF3Tab[i] = ((-2+6*x*x)*onePlusXSqInv3) * ONE_DIV_F3;
atanDer4DivF4Tab[i] = ((24*x*(1-x*x))*onePlusXSqInv4) * ONE_DIV_F4;
}
}

/**
* A faster and less accurate {@link Math#sinh}
*
* @param value A double value.
* @return Value hyperbolic sine.
*/
public static double sinh(double value) {
// sinh(x) = (exp(x)-exp(-x))/2
double h;
if (value < 0.0) {
value = -value;
h = -0.5;
} else {
h = 0.5;
}
if (value < 22.0) {
if (value < TWO_POW_N28) {
return (h < 0.0) ? -value : value;
} else {
double t = Math.expm1(value);
// Might be more accurate, if value < 1: return h*((t+t)-t*t/(t+1.0)).
return h * (t + t/(t+1.0));
}
} else if (value < LOG_DOUBLE_MAX_VALUE) {
return h * Math.exp(value);
} else {
double t = Math.exp(value*0.5);
return (h*t)*t;
}
}

/**
* A faster and less accurate {@link Math#atan}
*
* @param value A double value.
* @return Value arctangent, in radians, in [-PI/2,PI/2].
*/
public static double atan(double value) {
boolean negateResult;
if (value < 0.0) {
value = -value;
negateResult = true;
} else {
negateResult = false;
}
if (value == 1.0) {
// We want "exact" result for 1.0.
return negateResult ? -Math.PI/4 : Math.PI/4;
} else if (value <= ATAN_MAX_VALUE_FOR_TABS) {
int index = (int)(value * ATAN_INDEXER + 0.5);
double delta = value - index * ATAN_DELTA;
double result = atanTab[index] + delta * (atanDer1DivF1Tab[index] + delta * (atanDer2DivF2Tab[index]
+ delta * (atanDer3DivF3Tab[index] + delta * atanDer4DivF4Tab[index])));
return negateResult ? -result : result;
} else { // value > ATAN_MAX_VALUE_FOR_TABS, or value is NaN
// This part is derived from fdlibm.
if (value < TWO_POW_66) {
double x = -1/value;
double x2 = x*x;
double x4 = x2*x2;
double s1 = x2*(ATAN_AT0+x4*(ATAN_AT2+x4*(ATAN_AT4+x4*(ATAN_AT6+x4*(ATAN_AT8+x4*ATAN_AT10)))));
double s2 = x4*(ATAN_AT1+x4*(ATAN_AT3+x4*(ATAN_AT5+x4*(ATAN_AT7+x4*ATAN_AT9))));
double result = ATAN_HI3-((x*(s1+s2)-ATAN_LO3)-x);
return negateResult ? -result : result;
} else { // value >= 2^66, or value is NaN
if (Double.isNaN(value)) {
return Double.NaN;
} else {
return negateResult ? -Math.PI/2 : Math.PI/2;
}
}
}
}
}
53 changes: 53 additions & 0 deletions libs/core/src/test/java/org/elasticsearch/common/util/ESSloppyMathTests.java
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/*
* 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.elasticsearch.common.util;

import org.elasticsearch.test.ESTestCase;

import static org.elasticsearch.common.util.ESSloppyMath.atan;
import static org.elasticsearch.common.util.ESSloppyMath.sinh;

public class ESSloppyMathTests extends ESTestCase {

// accuracy for atan(x)
static double ATAN_DELTA = 1E-15;
// accuracy for sinh(x)
static double SINH_DELTA = 1E-15; // for small x

public void testAtan() {
assertTrue(Double.isNaN(atan(Double.NaN)));
assertEquals(-Math.PI/2, atan(Double.NEGATIVE_INFINITY), ATAN_DELTA);
assertEquals(Math.PI/2, atan(Double.POSITIVE_INFINITY), ATAN_DELTA);
for (int i = 0; i < 10000; i++) {
assertEquals(StrictMath.atan(i), atan(i), ATAN_DELTA);
assertEquals(StrictMath.atan(-i), atan(-i), ATAN_DELTA);
}
}

public void testSinh() {
assertTrue(Double.isNaN(sinh(Double.NaN)));
assertEquals(Double.NEGATIVE_INFINITY, sinh(Double.NEGATIVE_INFINITY), SINH_DELTA);
assertEquals(Double.POSITIVE_INFINITY, sinh(Double.POSITIVE_INFINITY), SINH_DELTA);
for (int i = 0; i < 10000; i++) {
double d = randomDoubleBetween(-2 * Math.PI, 2 * Math.PI, true);
if (random().nextBoolean()) {
d = -d;
}
assertEquals(StrictMath.sinh(d), sinh(d), SINH_DELTA);
}
}
}
3 changes: 2 additions & 1 deletion server/src/main/java/org/elasticsearch/search/aggregations/bucket/geogrid/GeoTileUtils.java
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import org.elasticsearch.ElasticsearchParseException;
import org.elasticsearch.common.geo.GeoPoint;
import org.elasticsearch.common.util.ESSloppyMath;
import org.elasticsearch.common.xcontent.ObjectParser.ValueType;
import org.elasticsearch.common.xcontent.XContentParser;
import org.elasticsearch.common.xcontent.support.XContentMapValues;
Expand Down Expand Up @@ -210,7 +211,7 @@ private static int validateZXY(int zoom, int xTile, int yTile) {
private static GeoPoint zxyToGeoPoint(int zoom, int xTile, int yTile) {
final int tiles = validateZXY(zoom, xTile, yTile);
final double n = Math.PI - (2.0 * Math.PI * (yTile + 0.5)) / tiles;
final double lat = Math.toDegrees(Math.atan(Math.sinh(n)));
final double lat = Math.toDegrees(ESSloppyMath.atan(ESSloppyMath.sinh(n)));
final double lon = ((xTile + 0.5) / tiles * 360.0) - 180;
return new GeoPoint(lat, lon);
}
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