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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 com.google.errorprone.annotations.CanIgnoreReturnValue;
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 (long) x - 1;
}
case CEILING:
if (x <= 0.0 || isMathematicalInteger(x)) {
return x;
} else {
return (long) x + 1;
}
case DOWN:
return x;
case UP:
if (isMathematicalInteger(x)) {
return x;
} else {
return (long) x + (x > 0 ? 1 : -1);
}
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}
*
*/
@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
*
* - {@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}
*
*/
@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
*
* - {@code x} is infinite or NaN
*
- {@code x} is not a mathematical integer and {@code mode} is {@link
* RoundingMode#UNNECESSARY}
*
*/
// #roundIntermediate, java.lang.Math.getExponent, com.google.common.math.DoubleUtils
@GwtIncompatible
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) {
if (x > 0.0 && isFinite(x)) {
long significand = getSignificand(x);
return (significand & (significand - 1)) == 0;
}
return false;
}
/**
* 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
}
/**
* 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
*/
@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;
}
private static final double LN_2 = log(2);
/**
* 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)}.
*/
@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}.
*
*
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));
}
}
/**
* Returns the arithmetic mean of
* {@code values}.
*
*
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
* @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
* values.
*/
@Deprecated
// com.google.common.math.DoubleUtils
@GwtIncompatible
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 arithmetic mean of
* {@code values}.
*
*
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
* @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
* values.
*/
@Deprecated
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 arithmetic mean of
* {@code values}.
*
*
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
* @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
* values.
*/
@Deprecated
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 arithmetic mean of
* {@code values}.
*
*
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
* @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
* values.
*/
@Deprecated
// com.google.common.math.DoubleUtils
@GwtIncompatible
public static double mean(Iterable values) {
return mean(values.iterator());
}
/**
* Returns the arithmetic mean of
* {@code values}.
*
*
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
* @deprecated Use {@link Stats#meanOf} instead, noting the less strict handling of non-finite
* values.
*/
@Deprecated
// com.google.common.math.DoubleUtils
@GwtIncompatible
public static double mean(Iterator 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
@CanIgnoreReturnValue
private static double checkFinite(double argument) {
checkArgument(isFinite(argument));
return argument;
}
private DoubleMath() {}
}