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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.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
*/
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.
*/
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
*
* - {@code x} is infinite or NaN
*
- {@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}
*
- {@code x} is not a mathematical integer and {@code mode} is
* {@link RoundingMode#UNNECESSARY}
*
*/
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
*
* - {@code x} is infinite or NaN
*
- {@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}
*
- {@code x} is not a mathematical integer and {@code mode} is
* {@link RoundingMode#UNNECESSARY}
*
*/
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
*
* - {@code x} is infinite or NaN
*
- {@code x} is not a mathematical integer and {@code mode} is
* {@link RoundingMode#UNNECESSARY}
*
*/
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}.
*/
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.
*
* Special cases:
*
* - If {@code x} is NaN or less than zero, the result is NaN.
*
- If {@code x} is positive infinity, the result is positive infinity.
*
- If {@code x} is positive or negative zero, the result is negative infinity.
*
*
* The computed result is within 1 ulp of the exact result.
*
*
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}.
*
*
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
*/
@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.
*
*
This is equivalent to, but not necessarily implemented as, the expression {@code
* !Double.isNaN(x) && !Double.isInfinite(x) && x == Math.rint(x)}.
*/
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 e n!}, or
* {@link Double#POSITIVE_INFINITY} if {@code n! > Double.MAX_VALUE}.
*
*
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.
*
*
Technically speaking, this is equivalent to
* {@code Math.abs(a - b) <= tolerance || Double.valueOf(a).equals(Double.valueOf(b))}.
*
*
Notable special cases include:
*
* - All NaNs are fuzzily equal.
*
- If {@code a == b}, then {@code a} and {@code b} are always fuzzily equal.
*
- Positive and negative zero are always fuzzily equal.
*
- 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}.
*
- With {@link Double#POSITIVE_INFINITY} tolerance, all non-NaN values are fuzzily equal.
*
- With finite tolerance, {@code Double.POSITIVE_INFINITY} and {@code
* Double.NEGATIVE_INFINITY} are fuzzily equal only to themselves.
*
*
* This is reflexive and symmetric, but not transitive, so it is not an
* equivalence relation and not 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.
*
*
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}).
*
*
This is not a total ordering and is not 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));
}
}
private static final class MeanAccumulator {
private long count = 0;
private double mean = 0.0;
void add(double value) {
checkArgument(isFinite(value));
++count;
// Art of Computer Programming vol. 2, Knuth, 4.2.2, (15)
mean += (value - mean) / count;
}
double mean() {
checkArgument(count > 0, "Cannot take mean of 0 values");
return mean;
}
}
/**
* Returns the arithmetic mean of the values. There must be at least one value, and they must all
* be finite.
*/
public static double mean(double... values) {
MeanAccumulator accumulator = new MeanAccumulator();
for (double value : values) {
accumulator.add(value);
}
return accumulator.mean();
}
/**
* Returns the arithmetic mean of the values. There must be at least one value. The values will
* be converted to doubles, which does not cause any loss of precision for ints.
*/
public static double mean(int... values) {
MeanAccumulator accumulator = new MeanAccumulator();
for (int value : values) {
accumulator.add(value);
}
return accumulator.mean();
}
/**
* Returns the arithmetic mean of the values. There must be at least one value. The values will
* be converted to doubles, which causes loss of precision for longs of magnitude over 2^53
* (slightly over 9e15).
*/
public static double mean(long... values) {
MeanAccumulator accumulator = new MeanAccumulator();
for (long value : values) {
accumulator.add(value);
}
return accumulator.mean();
}
/**
* Returns the arithmetic mean of the values. There must be at least one value, and they must all
* be finite. The values will be converted to doubles, which may cause loss of precision for some
* numeric types.
*/
public static double mean(Iterable values) {
MeanAccumulator accumulator = new MeanAccumulator();
for (Number value : values) {
accumulator.add(value.doubleValue());
}
return accumulator.mean();
}
/**
* Returns the arithmetic mean of the values. There must be at least one value, and they must all
* be finite. The values will be converted to doubles, which may cause loss of precision for some
* numeric types.
*/
public static double mean(Iterator values) {
MeanAccumulator accumulator = new MeanAccumulator();
while (values.hasNext()) {
accumulator.add(values.next().doubleValue());
}
return accumulator.mean();
}
private DoubleMath() {}
}