601 lines
18 KiB
Java
601 lines
18 KiB
Java
/*
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* Copyright (C) 2011 The Guava Authors
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package com.google.common.math;
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import static com.google.common.base.Preconditions.checkArgument;
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import static com.google.common.base.Preconditions.checkNotNull;
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import static com.google.common.math.MathPreconditions.checkNoOverflow;
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import static com.google.common.math.MathPreconditions.checkNonNegative;
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import static com.google.common.math.MathPreconditions.checkPositive;
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import static com.google.common.math.MathPreconditions.checkRoundingUnnecessary;
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import static java.lang.Math.abs;
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import static java.lang.Math.min;
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import static java.math.RoundingMode.HALF_EVEN;
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import static java.math.RoundingMode.HALF_UP;
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import java.math.BigInteger;
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import java.math.RoundingMode;
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import com.google.common.annotations.GwtCompatible;
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import com.google.common.annotations.GwtIncompatible;
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import com.google.common.annotations.VisibleForTesting;
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/**
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* A class for arithmetic on values of type {@code int}. Where possible, methods
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* are defined and named analogously to their {@code BigInteger} counterparts.
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*
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* <p>
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* The implementations of many methods in this class are based on material from
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* Henry S. Warren, Jr.'s <i>Hacker's Delight</i>, (Addison Wesley, 2002).
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*
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* <p>
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* Similar functionality for {@code long} and for {@link BigInteger} can be
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* found in {@link LongMath} and {@link BigIntegerMath} respectively. For other
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* common operations on {@code int} values, see
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* {@link com.google.common.primitives.Ints}.
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*
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* @author Louis Wasserman
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* @since 11.0
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*/
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@GwtCompatible(emulated = true)
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public final class IntMath {
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// NOTE: Whenever both tests are cheap and functional, it's faster to use &, |
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// instead of &&, ||
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/**
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* Returns {@code true} if {@code x} represents a power of two.
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*
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* <p>
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* This differs from {@code Integer.bitCount(x) == 1}, because
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* {@code Integer.bitCount(Integer.MIN_VALUE) == 1}, but
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* {@link Integer#MIN_VALUE} is not a power of two.
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*/
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public static boolean isPowerOfTwo(int x) {
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return x > 0 & (x & (x - 1)) == 0;
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}
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/**
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* Returns 1 if {@code x < y} as unsigned integers, and 0 otherwise. Assumes
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* that x - y fits into a signed int. The implementation is branch-free, and
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* benchmarks suggest it is measurably (if narrowly) faster than the
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* straightforward ternary expression.
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*/
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@VisibleForTesting
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static int lessThanBranchFree(int x, int y) {
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// The double negation is optimized away by normal Java, but is necessary for
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// GWT
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// to make sure bit twiddling works as expected.
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return ~~(x - y) >>> (Integer.SIZE - 1);
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}
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/**
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* Returns the base-2 logarithm of {@code x}, rounded according to the specified
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* rounding mode.
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*
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* @throws IllegalArgumentException if {@code x <= 0}
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* @throws ArithmeticException if {@code mode} is
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* {@link RoundingMode#UNNECESSARY} and
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* {@code x} is not a power of two
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*/
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@SuppressWarnings("fallthrough")
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// TODO(kevinb): remove after this warning is disabled globally
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public static int log2(int x, RoundingMode mode) {
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checkPositive("x", x);
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switch (mode) {
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case UNNECESSARY:
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checkRoundingUnnecessary(isPowerOfTwo(x));
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// fall through
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case DOWN:
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case FLOOR:
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return (Integer.SIZE - 1) - Integer.numberOfLeadingZeros(x);
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case UP:
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case CEILING:
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return Integer.SIZE - Integer.numberOfLeadingZeros(x - 1);
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case HALF_DOWN:
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case HALF_UP:
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case HALF_EVEN:
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// Since sqrt(2) is irrational, log2(x) - logFloor cannot be exactly 0.5
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int leadingZeros = Integer.numberOfLeadingZeros(x);
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int cmp = MAX_POWER_OF_SQRT2_UNSIGNED >>> leadingZeros;
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// floor(2^(logFloor + 0.5))
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int logFloor = (Integer.SIZE - 1) - leadingZeros;
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return logFloor + lessThanBranchFree(cmp, x);
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default:
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throw new AssertionError();
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}
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}
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/** The biggest half power of two that can fit in an unsigned int. */
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@VisibleForTesting
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static final int MAX_POWER_OF_SQRT2_UNSIGNED = 0xB504F333;
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/**
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* Returns the base-10 logarithm of {@code x}, rounded according to the
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* specified rounding mode.
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*
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* @throws IllegalArgumentException if {@code x <= 0}
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* @throws ArithmeticException if {@code mode} is
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* {@link RoundingMode#UNNECESSARY} and
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* {@code x} is not a power of ten
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*/
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@GwtIncompatible("need BigIntegerMath to adequately test")
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@SuppressWarnings("fallthrough")
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public static int log10(int x, RoundingMode mode) {
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checkPositive("x", x);
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int logFloor = log10Floor(x);
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int floorPow = powersOf10[logFloor];
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switch (mode) {
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case UNNECESSARY:
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checkRoundingUnnecessary(x == floorPow);
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// fall through
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case FLOOR:
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case DOWN:
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return logFloor;
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case CEILING:
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case UP:
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return logFloor + lessThanBranchFree(floorPow, x);
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case HALF_DOWN:
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case HALF_UP:
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case HALF_EVEN:
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// sqrt(10) is irrational, so log10(x) - logFloor is never exactly 0.5
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return logFloor + lessThanBranchFree(halfPowersOf10[logFloor], x);
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default:
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throw new AssertionError();
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}
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}
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private static int log10Floor(int x) {
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/*
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* Based on Hacker's Delight Fig. 11-5, the two-table-lookup, branch-free
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* implementation.
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*
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* The key idea is that based on the number of leading zeros (equivalently,
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* floor(log2(x))), we can narrow the possible floor(log10(x)) values to two.
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* For example, if floor(log2(x)) is 6, then 64 <= x < 128, so floor(log10(x))
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* is either 1 or 2.
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*/
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int y = maxLog10ForLeadingZeros[Integer.numberOfLeadingZeros(x)];
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/*
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* y is the higher of the two possible values of floor(log10(x)). If x < 10^y,
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* then we want the lower of the two possible values, or y - 1, otherwise, we
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* want y.
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*/
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return y - lessThanBranchFree(x, powersOf10[y]);
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}
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// maxLog10ForLeadingZeros[i] == floor(log10(2^(Long.SIZE - i)))
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@VisibleForTesting
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static final byte[] maxLog10ForLeadingZeros = { 9, 9, 9, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 5, 5, 5, 4, 4, 4, 3, 3, 3, 3,
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2, 2, 2, 1, 1, 1, 0, 0, 0, 0 };
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@VisibleForTesting
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static final int[] powersOf10 = { 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000 };
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// halfPowersOf10[i] = largest int less than 10^(i + 0.5)
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@VisibleForTesting
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static final int[] halfPowersOf10 = { 3, 31, 316, 3162, 31622, 316227, 3162277, 31622776, 316227766,
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Integer.MAX_VALUE };
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/**
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* Returns {@code b} to the {@code k}th power. Even if the result overflows, it
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* will be equal to {@code BigInteger.valueOf(b).pow(k).intValue()}. This
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* implementation runs in {@code O(log k)} time.
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*
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* <p>
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* Compare {@link #checkedPow}, which throws an {@link ArithmeticException} upon
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* overflow.
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*
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* @throws IllegalArgumentException if {@code k < 0}
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*/
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@GwtIncompatible("failing tests")
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public static int pow(int b, int k) {
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checkNonNegative("exponent", k);
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switch (b) {
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case 0:
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return (k == 0) ? 1 : 0;
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case 1:
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return 1;
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case (-1):
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return ((k & 1) == 0) ? 1 : -1;
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case 2:
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return (k < Integer.SIZE) ? (1 << k) : 0;
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case (-2):
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if (k < Integer.SIZE) {
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return ((k & 1) == 0) ? (1 << k) : -(1 << k);
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} else {
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return 0;
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}
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default:
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// continue below to handle the general case
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}
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for (int accum = 1;; k >>= 1) {
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switch (k) {
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case 0:
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return accum;
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case 1:
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return b * accum;
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default:
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accum *= ((k & 1) == 0) ? 1 : b;
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b *= b;
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}
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}
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}
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/**
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* Returns the square root of {@code x}, rounded with the specified rounding
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* mode.
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*
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* @throws IllegalArgumentException if {@code x < 0}
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* @throws ArithmeticException if {@code mode} is
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* {@link RoundingMode#UNNECESSARY} and
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* {@code sqrt(x)} is not an integer
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*/
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@GwtIncompatible("need BigIntegerMath to adequately test")
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@SuppressWarnings("fallthrough")
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public static int sqrt(int x, RoundingMode mode) {
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checkNonNegative("x", x);
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int sqrtFloor = sqrtFloor(x);
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switch (mode) {
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case UNNECESSARY:
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checkRoundingUnnecessary(sqrtFloor * sqrtFloor == x); // fall through
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case FLOOR:
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case DOWN:
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return sqrtFloor;
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case CEILING:
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case UP:
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return sqrtFloor + lessThanBranchFree(sqrtFloor * sqrtFloor, x);
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case HALF_DOWN:
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case HALF_UP:
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case HALF_EVEN:
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int halfSquare = sqrtFloor * sqrtFloor + sqrtFloor;
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/*
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* We wish to test whether or not x <= (sqrtFloor + 0.5)^2 = halfSquare + 0.25.
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* Since both x and halfSquare are integers, this is equivalent to testing
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* whether or not x <= halfSquare. (We have to deal with overflow, though.)
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*
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* If we treat halfSquare as an unsigned int, we know that sqrtFloor^2 <= x <
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* (sqrtFloor + 1)^2 halfSquare - sqrtFloor <= x < halfSquare + sqrtFloor + 1 so
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* |x - halfSquare| <= sqrtFloor. Therefore, it's safe to treat x - halfSquare
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* as a signed int, so lessThanBranchFree is safe for use.
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*/
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return sqrtFloor + lessThanBranchFree(halfSquare, x);
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default:
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throw new AssertionError();
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}
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}
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private static int sqrtFloor(int x) {
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// There is no loss of precision in converting an int to a double, according to
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// http://java.sun.com/docs/books/jls/third_edition/html/conversions.html#5.1.2
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return (int) Math.sqrt(x);
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}
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/**
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* Returns the result of dividing {@code p} by {@code q}, rounding using the
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* specified {@code RoundingMode}.
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*
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* @throws ArithmeticException if {@code q == 0}, or if
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* {@code mode == UNNECESSARY} and {@code a} is not
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* an integer multiple of {@code b}
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*/
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@SuppressWarnings("fallthrough")
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public static int divide(int p, int q, RoundingMode mode) {
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checkNotNull(mode);
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if (q == 0) {
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throw new ArithmeticException("/ by zero"); // for GWT
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}
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int div = p / q;
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int rem = p - q * div; // equal to p % q
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if (rem == 0) {
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return div;
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}
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/*
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* Normal Java division rounds towards 0, consistently with RoundingMode.DOWN.
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* We just have to deal with the cases where rounding towards 0 is wrong, which
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* typically depends on the sign of p / q.
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*
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* signum is 1 if p and q are both nonnegative or both negative, and -1
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* otherwise.
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*/
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int signum = 1 | ((p ^ q) >> (Integer.SIZE - 1));
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boolean increment;
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switch (mode) {
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case UNNECESSARY:
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checkRoundingUnnecessary(rem == 0);
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// fall through
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case DOWN:
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increment = false;
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break;
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case UP:
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increment = true;
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break;
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case CEILING:
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increment = signum > 0;
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break;
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case FLOOR:
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increment = signum < 0;
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break;
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case HALF_EVEN:
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case HALF_DOWN:
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case HALF_UP:
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int absRem = abs(rem);
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int cmpRemToHalfDivisor = absRem - (abs(q) - absRem);
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// subtracting two nonnegative ints can't overflow
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// cmpRemToHalfDivisor has the same sign as compare(abs(rem), abs(q) / 2).
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if (cmpRemToHalfDivisor == 0) { // exactly on the half mark
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increment = (mode == HALF_UP || (mode == HALF_EVEN & (div & 1) != 0));
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} else {
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increment = cmpRemToHalfDivisor > 0; // closer to the UP value
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}
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break;
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default:
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throw new AssertionError();
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}
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return increment ? div + signum : div;
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}
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/**
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* Returns {@code x mod m}, a non-negative value less than {@code m}. This
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* differs from {@code x % m}, which might be negative.
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*
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* <p>
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* For example:
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*
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* <pre>
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* {@code
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*
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* mod(7, 4) == 3
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* mod(-7, 4) == 1
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* mod(-1, 4) == 3
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* mod(-8, 4) == 0
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* mod(8, 4) == 0}
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* </pre>
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*
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* @throws ArithmeticException if {@code m <= 0}
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* @see <a href=
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* "http://docs.oracle.com/javase/specs/jls/se7/html/jls-15.html#jls-15.17.3">
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* Remainder Operator</a>
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*/
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public static int mod(int x, int m) {
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if (m <= 0) {
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throw new ArithmeticException("Modulus " + m + " must be > 0");
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}
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int result = x % m;
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return (result >= 0) ? result : result + m;
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}
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/**
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* Returns the greatest common divisor of {@code a, b}. Returns {@code 0} if
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* {@code a == 0 && b == 0}.
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*
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* @throws IllegalArgumentException if {@code a < 0} or {@code b < 0}
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*/
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public static int gcd(int a, int b) {
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/*
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* The reason we require both arguments to be >= 0 is because otherwise, what do
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* you return on gcd(0, Integer.MIN_VALUE)? BigInteger.gcd would return positive
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* 2^31, but positive 2^31 isn't an int.
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*/
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checkNonNegative("a", a);
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checkNonNegative("b", b);
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if (a == 0) {
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// 0 % b == 0, so b divides a, but the converse doesn't hold.
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// BigInteger.gcd is consistent with this decision.
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return b;
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} else if (b == 0) {
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return a; // similar logic
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}
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/*
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* Uses the binary GCD algorithm; see
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* http://en.wikipedia.org/wiki/Binary_GCD_algorithm. This is >40% faster than
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* the Euclidean algorithm in benchmarks.
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*/
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int aTwos = Integer.numberOfTrailingZeros(a);
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a >>= aTwos; // divide out all 2s
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int bTwos = Integer.numberOfTrailingZeros(b);
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b >>= bTwos; // divide out all 2s
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while (a != b) { // both a, b are odd
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// The key to the binary GCD algorithm is as follows:
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// Both a and b are odd. Assume a > b; then gcd(a - b, b) = gcd(a, b).
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// But in gcd(a - b, b), a - b is even and b is odd, so we can divide out powers
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// of two.
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// We bend over backwards to avoid branching, adapting a technique from
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// http://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMax
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int delta = a - b; // can't overflow, since a and b are nonnegative
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int minDeltaOrZero = delta & (delta >> (Integer.SIZE - 1));
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// equivalent to Math.min(delta, 0)
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a = delta - minDeltaOrZero - minDeltaOrZero; // sets a to Math.abs(a - b)
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// a is now nonnegative and even
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b += minDeltaOrZero; // sets b to min(old a, b)
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a >>= Integer.numberOfTrailingZeros(a); // divide out all 2s, since 2 doesn't divide b
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}
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return a << min(aTwos, bTwos);
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}
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/**
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* Returns the sum of {@code a} and {@code b}, provided it does not overflow.
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*
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* @throws ArithmeticException if {@code a + b} overflows in signed {@code int}
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* arithmetic
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*/
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public static int checkedAdd(int a, int b) {
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long result = (long) a + b;
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checkNoOverflow(result == (int) result);
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return (int) result;
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}
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/**
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* Returns the difference of {@code a} and {@code b}, provided it does not
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* overflow.
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*
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* @throws ArithmeticException if {@code a - b} overflows in signed {@code int}
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* arithmetic
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*/
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public static int checkedSubtract(int a, int b) {
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long result = (long) a - b;
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checkNoOverflow(result == (int) result);
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return (int) result;
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}
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/**
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* Returns the product of {@code a} and {@code b}, provided it does not
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* overflow.
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*
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* @throws ArithmeticException if {@code a * b} overflows in signed {@code int}
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* arithmetic
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*/
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public static int checkedMultiply(int a, int b) {
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long result = (long) a * b;
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checkNoOverflow(result == (int) result);
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return (int) result;
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}
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/**
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* Returns the {@code b} to the {@code k}th power, provided it does not
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* overflow.
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*
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* <p>
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* {@link #pow} may be faster, but does not check for overflow.
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*
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* @throws ArithmeticException if {@code b} to the {@code k}th power overflows
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* in signed {@code int} arithmetic
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*/
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public static int checkedPow(int b, int k) {
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checkNonNegative("exponent", k);
|
|
switch (b) {
|
|
case 0:
|
|
return (k == 0) ? 1 : 0;
|
|
case 1:
|
|
return 1;
|
|
case (-1):
|
|
return ((k & 1) == 0) ? 1 : -1;
|
|
case 2:
|
|
checkNoOverflow(k < Integer.SIZE - 1);
|
|
return 1 << k;
|
|
case (-2):
|
|
checkNoOverflow(k < Integer.SIZE);
|
|
return ((k & 1) == 0) ? 1 << k : -1 << k;
|
|
default:
|
|
// continue below to handle the general case
|
|
}
|
|
int accum = 1;
|
|
while (true) {
|
|
switch (k) {
|
|
case 0:
|
|
return accum;
|
|
case 1:
|
|
return checkedMultiply(accum, b);
|
|
default:
|
|
if ((k & 1) != 0) {
|
|
accum = checkedMultiply(accum, b);
|
|
}
|
|
k >>= 1;
|
|
if (k > 0) {
|
|
checkNoOverflow(-FLOOR_SQRT_MAX_INT <= b & b <= FLOOR_SQRT_MAX_INT);
|
|
b *= b;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
@VisibleForTesting
|
|
static final int FLOOR_SQRT_MAX_INT = 46340;
|
|
|
|
/**
|
|
* Returns {@code n!}, that is, the product of the first {@code n} positive
|
|
* integers, {@code 1} if {@code n == 0}, or {@link Integer#MAX_VALUE} if the
|
|
* result does not fit in a {@code int}.
|
|
*
|
|
* @throws IllegalArgumentException if {@code n < 0}
|
|
*/
|
|
public static int factorial(int n) {
|
|
checkNonNegative("n", n);
|
|
return (n < factorials.length) ? factorials[n] : Integer.MAX_VALUE;
|
|
}
|
|
|
|
private static final int[] factorials = { 1, 1, 1 * 2, 1 * 2 * 3, 1 * 2 * 3 * 4, 1 * 2 * 3 * 4 * 5,
|
|
1 * 2 * 3 * 4 * 5 * 6, 1 * 2 * 3 * 4 * 5 * 6 * 7, 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8,
|
|
1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9, 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10,
|
|
1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11, 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 };
|
|
|
|
/**
|
|
* Returns {@code n} choose {@code k}, also known as the binomial coefficient of
|
|
* {@code n} and {@code k}, or {@link Integer#MAX_VALUE} if the result does not
|
|
* fit in an {@code int}.
|
|
*
|
|
* @throws IllegalArgumentException if {@code n < 0}, {@code k < 0} or
|
|
* {@code k > n}
|
|
*/
|
|
@GwtIncompatible("need BigIntegerMath to adequately test")
|
|
public static int binomial(int n, int k) {
|
|
checkNonNegative("n", n);
|
|
checkNonNegative("k", k);
|
|
checkArgument(k <= n, "k (%s) > n (%s)", k, n);
|
|
if (k > (n >> 1)) {
|
|
k = n - k;
|
|
}
|
|
if (k >= biggestBinomials.length || n > biggestBinomials[k]) {
|
|
return Integer.MAX_VALUE;
|
|
}
|
|
switch (k) {
|
|
case 0:
|
|
return 1;
|
|
case 1:
|
|
return n;
|
|
default:
|
|
long result = 1;
|
|
for (int i = 0; i < k; i++) {
|
|
result *= n - i;
|
|
result /= i + 1;
|
|
}
|
|
return (int) result;
|
|
}
|
|
}
|
|
|
|
// binomial(biggestBinomials[k], k) fits in an int, but not
|
|
// binomial(biggestBinomials[k]+1,k).
|
|
@VisibleForTesting
|
|
static int[] biggestBinomials = { Integer.MAX_VALUE, Integer.MAX_VALUE, 65536, 2345, 477, 193, 110, 75, 58, 49, 43,
|
|
39, 37, 35, 34, 34, 33 };
|
|
|
|
/**
|
|
* Returns the arithmetic mean of {@code x} and {@code y}, rounded towards
|
|
* negative infinity. This method is overflow resilient.
|
|
*
|
|
* @since 14.0
|
|
*/
|
|
public static int mean(int x, int y) {
|
|
// Efficient method for computing the arithmetic mean.
|
|
// The alternative (x + y) / 2 fails for large values.
|
|
// The alternative (x + y) >>> 1 fails for negative values.
|
|
return (x & y) + ((x ^ y) >> 1);
|
|
}
|
|
|
|
private IntMath() {
|
|
}
|
|
}
|