Introduce faster approximate sinh/atan functions

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 elastic#41166.

libs/core/src/main/java/org/elasticsearch/common/util/ESSloppyMath.java

@@ -0,0 +1,180 @@
+/*
+ * 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;
+                }
+            }
+        }
+    }
+}

libs/core/src/test/java/org/elasticsearch/common/util/ESSloppyMathTests.java

@@ -0,0 +1,37 @@
+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);
+        }
+    }
+}

server/src/main/java/org/elasticsearch/search/aggregations/bucket/geogrid/GeoTileUtils.java

@@ -20,6 +20,7 @@
 
 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;
@@ -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);
     }