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/* * Copyright (C) 2011 The Guava Authors * * Licensed 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 com.google.common.math; import static com.google.common.base.Preconditions.checkArgument; import static com.google.common.math.DoubleUtils.IMPLICIT_BIT; import static com.google.common.math.DoubleUtils.SIGNIFICAND_BITS; import static com.google.common.math.DoubleUtils.getSignificand; import static com.google.common.math.DoubleUtils.isFinite; import static com.google.common.math.DoubleUtils.isNormal; import static com.google.common.math.DoubleUtils.scaleNormalize; import static com.google.common.math.MathPreconditions.checkInRange; import static com.google.common.math.MathPreconditions.checkNonNegative; import static com.google.common.math.MathPreconditions.checkRoundingUnnecessary; import static java.lang.Math.abs; import static java.lang.Math.copySign; import static java.lang.Math.getExponent; import static java.lang.Math.log; import static java.lang.Math.rint; import com.google.common.annotations.GwtCompatible; import com.google.common.annotations.GwtIncompatible; import com.google.common.annotations.VisibleForTesting; import com.google.common.primitives.Booleans; import java.math.BigInteger; import java.math.RoundingMode; import java.util.Iterator; /** * A class for arithmetic on doubles that is not covered by {@link java.lang.Math}. * * @author Louis Wasserman * @since 11.0 */ @GwtCompatible(emulated = true) public final class DoubleMath { /* * This method returns a value y such that rounding y DOWN (towards zero) gives the same result * as rounding x according to the specified mode. */ @GwtIncompatible("#isMathematicalInteger, com.google.common.math.DoubleUtils") static double roundIntermediate(double x, RoundingMode mode) { if (!isFinite(x)) { throw new ArithmeticException("input is infinite or NaN"); } switch (mode) { case UNNECESSARY: checkRoundingUnnecessary(isMathematicalInteger(x)); return x; case FLOOR: if (x >= 0.0 || isMathematicalInteger(x)) { return x; } else { return x - 1.0; } case CEILING: if (x <= 0.0 || isMathematicalInteger(x)) { return x; } else { return x + 1.0; } case DOWN: return x; case UP: if (isMathematicalInteger(x)) { return x; } else { return x + Math.copySign(1.0, x); } case HALF_EVEN: return rint(x); case HALF_UP: { double z = rint(x); if (abs(x - z) == 0.5) { return x + copySign(0.5, x); } else { return z; } } case HALF_DOWN: { double z = rint(x); if (abs(x - z) == 0.5) { return x; } else { return z; } } default: throw new AssertionError(); } } /** * Returns the {@code int} value that is equal to {@code x} rounded with the specified rounding * mode, if possible. * * @throws ArithmeticException if * <ul> * <li>{@code x} is infinite or NaN * <li>{@code x}, after being rounded to a mathematical integer using the specified * rounding mode, is either less than {@code Integer.MIN_VALUE} or greater than {@code * Integer.MAX_VALUE} * <li>{@code x} is not a mathematical integer and {@code mode} is * {@link RoundingMode#UNNECESSARY} * </ul> */ @GwtIncompatible("#roundIntermediate") public static int roundToInt(double x, RoundingMode mode) { double z = roundIntermediate(x, mode); checkInRange(z > MIN_INT_AS_DOUBLE - 1.0 & z < MAX_INT_AS_DOUBLE + 1.0); return (int) z; } private static final double MIN_INT_AS_DOUBLE = -0x1p31; private static final double MAX_INT_AS_DOUBLE = 0x1p31 - 1.0; /** * Returns the {@code long} value that is equal to {@code x} rounded with the specified rounding * mode, if possible. * * @throws ArithmeticException if * <ul> * <li>{@code x} is infinite or NaN * <li>{@code x}, after being rounded to a mathematical integer using the specified * rounding mode, is either less than {@code Long.MIN_VALUE} or greater than {@code * Long.MAX_VALUE} * <li>{@code x} is not a mathematical integer and {@code mode} is * {@link RoundingMode#UNNECESSARY} * </ul> */ @GwtIncompatible("#roundIntermediate") public static long roundToLong(double x, RoundingMode mode) { double z = roundIntermediate(x, mode); checkInRange(MIN_LONG_AS_DOUBLE - z < 1.0 & z < MAX_LONG_AS_DOUBLE_PLUS_ONE); return (long) z; } private static final double MIN_LONG_AS_DOUBLE = -0x1p63; /* * We cannot store Long.MAX_VALUE as a double without losing precision. Instead, we store * Long.MAX_VALUE + 1 == -Long.MIN_VALUE, and then offset all comparisons by 1. */ private static final double MAX_LONG_AS_DOUBLE_PLUS_ONE = 0x1p63; /** * Returns the {@code BigInteger} value that is equal to {@code x} rounded with the specified * rounding mode, if possible. * * @throws ArithmeticException if * <ul> * <li>{@code x} is infinite or NaN * <li>{@code x} is not a mathematical integer and {@code mode} is * {@link RoundingMode#UNNECESSARY} * </ul> */ @GwtIncompatible("#roundIntermediate, java.lang.Math.getExponent, " + "com.google.common.math.DoubleUtils") public static BigInteger roundToBigInteger(double x, RoundingMode mode) { x = roundIntermediate(x, mode); if (MIN_LONG_AS_DOUBLE - x < 1.0 & x < MAX_LONG_AS_DOUBLE_PLUS_ONE) { return BigInteger.valueOf((long) x); } int exponent = getExponent(x); long significand = getSignificand(x); BigInteger result = BigInteger.valueOf(significand).shiftLeft(exponent - SIGNIFICAND_BITS); return (x < 0) ? result.negate() : result; } /** * Returns {@code true} if {@code x} is exactly equal to {@code 2^k} for some finite integer * {@code k}. */ @GwtIncompatible("com.google.common.math.DoubleUtils") public static boolean isPowerOfTwo(double x) { return x > 0.0 && isFinite(x) && LongMath.isPowerOfTwo(getSignificand(x)); } /** * Returns the base 2 logarithm of a double value. * * <p>Special cases: * <ul> * <li>If {@code x} is NaN or less than zero, the result is NaN. * <li>If {@code x} is positive infinity, the result is positive infinity. * <li>If {@code x} is positive or negative zero, the result is negative infinity. * </ul> * * <p>The computed result is within 1 ulp of the exact result. * * <p>If the result of this method will be immediately rounded to an {@code int}, * {@link #log2(double, RoundingMode)} is faster. */ public static double log2(double x) { return log(x) / LN_2; // surprisingly within 1 ulp according to tests } private static final double LN_2 = log(2); /** * Returns the base 2 logarithm of a double value, rounded with the specified rounding mode to an * {@code int}. * * <p>Regardless of the rounding mode, this is faster than {@code (int) log2(x)}. * * @throws IllegalArgumentException if {@code x <= 0.0}, {@code x} is NaN, or {@code x} is * infinite */ @GwtIncompatible("java.lang.Math.getExponent, com.google.common.math.DoubleUtils") @SuppressWarnings("fallthrough") public static int log2(double x, RoundingMode mode) { checkArgument(x > 0.0 && isFinite(x), "x must be positive and finite"); int exponent = getExponent(x); if (!isNormal(x)) { return log2(x * IMPLICIT_BIT, mode) - SIGNIFICAND_BITS; // Do the calculation on a normal value. } // x is positive, finite, and normal boolean increment; switch (mode) { case UNNECESSARY: checkRoundingUnnecessary(isPowerOfTwo(x)); // fall through case FLOOR: increment = false; break; case CEILING: increment = !isPowerOfTwo(x); break; case DOWN: increment = exponent < 0 & !isPowerOfTwo(x); break; case UP: increment = exponent >= 0 & !isPowerOfTwo(x); break; case HALF_DOWN: case HALF_EVEN: case HALF_UP: double xScaled = scaleNormalize(x); // sqrt(2) is irrational, and the spec is relative to the "exact numerical result," // so log2(x) is never exactly exponent + 0.5. increment = (xScaled * xScaled) > 2.0; break; default: throw new AssertionError(); } return increment ? exponent + 1 : exponent; } /** * Returns {@code true} if {@code x} represents a mathematical integer. * * <p>This is equivalent to, but not necessarily implemented as, the expression {@code * !Double.isNaN(x) && !Double.isInfinite(x) && x == Math.rint(x)}. */ @GwtIncompatible("java.lang.Math.getExponent, com.google.common.math.DoubleUtils") public static boolean isMathematicalInteger(double x) { return isFinite(x) && (x == 0.0 || SIGNIFICAND_BITS - Long.numberOfTrailingZeros(getSignificand(x)) <= getExponent(x)); } /** * Returns {@code n!}, that is, the product of the first {@code n} positive * integers, {@code 1} if {@code n == 0}, or {@code n!}, or * {@link Double#POSITIVE_INFINITY} if {@code n! > Double.MAX_VALUE}. * * <p>The result is within 1 ulp of the true value. * * @throws IllegalArgumentException if {@code n < 0} */ public static double factorial(int n) { checkNonNegative("n", n); if (n > MAX_FACTORIAL) { return Double.POSITIVE_INFINITY; } else { // Multiplying the last (n & 0xf) values into their own accumulator gives a more accurate // result than multiplying by everySixteenthFactorial[n >> 4] directly. double accum = 1.0; for (int i = 1 + (n & ~0xf); i <= n; i++) { accum *= i; } return accum * everySixteenthFactorial[n >> 4]; } } @VisibleForTesting static final int MAX_FACTORIAL = 170; @VisibleForTesting static final double[] everySixteenthFactorial = { 0x1.0p0, 0x1.30777758p44, 0x1.956ad0aae33a4p117, 0x1.ee69a78d72cb6p202, 0x1.fe478ee34844ap295, 0x1.c619094edabffp394, 0x1.3638dd7bd6347p498, 0x1.7cac197cfe503p605, 0x1.1e5dfc140e1e5p716, 0x1.8ce85fadb707ep829, 0x1.95d5f3d928edep945 }; /** * Returns {@code true} if {@code a} and {@code b} are within {@code tolerance} of each other. * * <p>Technically speaking, this is equivalent to * {@code Math.abs(a - b) <= tolerance || Double.valueOf(a).equals(Double.valueOf(b))}. * * <p>Notable special cases include: * <ul> * <li>All NaNs are fuzzily equal. * <li>If {@code a == b}, then {@code a} and {@code b} are always fuzzily equal. * <li>Positive and negative zero are always fuzzily equal. * <li>If {@code tolerance} is zero, and neither {@code a} nor {@code b} is NaN, then * {@code a} and {@code b} are fuzzily equal if and only if {@code a == b}. * <li>With {@link Double#POSITIVE_INFINITY} tolerance, all non-NaN values are fuzzily equal. * <li>With finite tolerance, {@code Double.POSITIVE_INFINITY} and {@code * Double.NEGATIVE_INFINITY} are fuzzily equal only to themselves. * </li> * * <p>This is reflexive and symmetric, but <em>not</em> transitive, so it is <em>not</em> an * equivalence relation and <em>not</em> suitable for use in {@link Object#equals} * implementations. * * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN * @since 13.0 */ public static boolean fuzzyEquals(double a, double b, double tolerance) { MathPreconditions.checkNonNegative("tolerance", tolerance); return Math.copySign(a - b, 1.0) <= tolerance // copySign(x, 1.0) is a branch-free version of abs(x), but with different NaN semantics || (a == b) // needed to ensure that infinities equal themselves || (Double.isNaN(a) && Double.isNaN(b)); } /** * Compares {@code a} and {@code b} "fuzzily," with a tolerance for nearly-equal values. * * <p>This method is equivalent to * {@code fuzzyEquals(a, b, tolerance) ? 0 : Double.compare(a, b)}. In particular, like * {@link Double#compare(double, double)}, it treats all NaN values as equal and greater than all * other values (including {@link Double#POSITIVE_INFINITY}). * * <p>This is <em>not</em> a total ordering and is <em>not</em> suitable for use in * {@link Comparable#compareTo} implementations. In particular, it is not transitive. * * @throws IllegalArgumentException if {@code tolerance} is {@code < 0} or NaN * @since 13.0 */ public static int fuzzyCompare(double a, double b, double tolerance) { if (fuzzyEquals(a, b, tolerance)) { return 0; } else if (a < b) { return -1; } else if (a > b) { return 1; } else { return Booleans.compare(Double.isNaN(a), Double.isNaN(b)); } } /** * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of * {@code values}. * * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of * the arithmetic mean of the population. * * @param values a nonempty series of values * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value */ @GwtIncompatible("com.google.common.math.DoubleUtils") public static double mean(double... values) { checkArgument(values.length > 0, "Cannot take mean of 0 values"); long count = 1; double mean = checkFinite(values[0]); for (int index = 1; index < values.length; ++index) { checkFinite(values[index]); count++; // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15) mean += (values[index] - mean) / count; } return mean; } /** * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of * {@code values}. * * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of * the arithmetic mean of the population. * * @param values a nonempty series of values * @throws IllegalArgumentException if {@code values} is empty */ public static double mean(int... values) { checkArgument(values.length > 0, "Cannot take mean of 0 values"); // The upper bound on the the length of an array and the bounds on the int values mean that, in // this case only, we can compute the sum as a long without risking overflow or loss of // precision. So we do that, as it's slightly quicker than the Knuth algorithm. long sum = 0; for (int index = 0; index < values.length; ++index) { sum += values[index]; } return (double) sum / values.length; } /** * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of * {@code values}. * * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of * the arithmetic mean of the population. * * @param values a nonempty series of values, which will be converted to {@code double} values * (this may cause loss of precision for longs of magnitude over 2^53 (slightly over 9e15)) * @throws IllegalArgumentException if {@code values} is empty */ public static double mean(long... values) { checkArgument(values.length > 0, "Cannot take mean of 0 values"); long count = 1; double mean = values[0]; for (int index = 1; index < values.length; ++index) { count++; // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15) mean += (values[index] - mean) / count; } return mean; } /** * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of * {@code values}. * * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of * the arithmetic mean of the population. * * @param values a nonempty series of values, which will be converted to {@code double} values * (this may cause loss of precision) * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value */ @GwtIncompatible("com.google.common.math.DoubleUtils") public static double mean(Iterable<? extends Number> values) { return mean(values.iterator()); } /** * Returns the <a href="http://en.wikipedia.org/wiki/Arithmetic_mean">arithmetic mean</a> of * {@code values}. * * <p>If these values are a sample drawn from a population, this is also an unbiased estimator of * the arithmetic mean of the population. * * @param values a nonempty series of values, which will be converted to {@code double} values * (this may cause loss of precision) * @throws IllegalArgumentException if {@code values} is empty or contains any non-finite value */ @GwtIncompatible("com.google.common.math.DoubleUtils") public static double mean(Iterator<? extends Number> values) { checkArgument(values.hasNext(), "Cannot take mean of 0 values"); long count = 1; double mean = checkFinite(values.next().doubleValue()); while (values.hasNext()) { double value = checkFinite(values.next().doubleValue()); count++; // Art of Computer Programming vol. 2, Knuth, 4.2.2, (15) mean += (value - mean) / count; } return mean; } @GwtIncompatible("com.google.common.math.DoubleUtils") private static double checkFinite(double argument) { checkArgument(isFinite(argument)); return argument; } private DoubleMath() { } }