| /* |
| * 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.base.Preconditions.checkNotNull; |
| import static com.google.common.math.MathPreconditions.checkNoOverflow; |
| import static com.google.common.math.MathPreconditions.checkNonNegative; |
| import static com.google.common.math.MathPreconditions.checkPositive; |
| import static com.google.common.math.MathPreconditions.checkRoundingUnnecessary; |
| import static java.lang.Math.abs; |
| import static java.lang.Math.min; |
| import static java.math.RoundingMode.HALF_EVEN; |
| import static java.math.RoundingMode.HALF_UP; |
| |
| import com.google.common.annotations.GwtCompatible; |
| import com.google.common.annotations.GwtIncompatible; |
| import com.google.common.annotations.VisibleForTesting; |
| |
| import java.math.BigInteger; |
| import java.math.RoundingMode; |
| |
| /** |
| * A class for arithmetic on values of type {@code long}. Where possible, methods are defined and |
| * named analogously to their {@code BigInteger} counterparts. |
| * |
| * <p>The implementations of many methods in this class are based on material from Henry S. Warren, |
| * Jr.'s <i>Hacker's Delight</i>, (Addison Wesley, 2002). |
| * |
| * <p>Similar functionality for {@code int} and for {@link BigInteger} can be found in |
| * {@link IntMath} and {@link BigIntegerMath} respectively. For other common operations on |
| * {@code long} values, see {@link com.google.common.primitives.Longs}. |
| * |
| * @author Louis Wasserman |
| * @since 11.0 |
| */ |
| @GwtCompatible(emulated = true) |
| public final class LongMath { |
| // NOTE: Whenever both tests are cheap and functional, it's faster to use &, | instead of &&, || |
| |
| /** |
| * Returns {@code true} if {@code x} represents a power of two. |
| * |
| * <p>This differs from {@code Long.bitCount(x) == 1}, because |
| * {@code Long.bitCount(Long.MIN_VALUE) == 1}, but {@link Long#MIN_VALUE} is not a power of two. |
| */ |
| public static boolean isPowerOfTwo(long x) { |
| return x > 0 & (x & (x - 1)) == 0; |
| } |
| |
| /** |
| * Returns 1 if {@code x < y} as unsigned longs, and 0 otherwise. Assumes that x - y fits into a |
| * signed long. The implementation is branch-free, and benchmarks suggest it is measurably |
| * faster than the straightforward ternary expression. |
| */ |
| @VisibleForTesting |
| static int lessThanBranchFree(long x, long y) { |
| // Returns the sign bit of x - y. |
| return (int) (~~(x - y) >>> (Long.SIZE - 1)); |
| } |
| |
| /** |
| * Returns the base-2 logarithm of {@code x}, rounded according to the specified rounding mode. |
| * |
| * @throws IllegalArgumentException if {@code x <= 0} |
| * @throws ArithmeticException if {@code mode} is {@link RoundingMode#UNNECESSARY} and {@code x} |
| * is not a power of two |
| */ |
| @SuppressWarnings("fallthrough") |
| // TODO(kevinb): remove after this warning is disabled globally |
| public static int log2(long x, RoundingMode mode) { |
| checkPositive("x", x); |
| switch (mode) { |
| case UNNECESSARY: |
| checkRoundingUnnecessary(isPowerOfTwo(x)); |
| // fall through |
| case DOWN: |
| case FLOOR: |
| return (Long.SIZE - 1) - Long.numberOfLeadingZeros(x); |
| |
| case UP: |
| case CEILING: |
| return Long.SIZE - Long.numberOfLeadingZeros(x - 1); |
| |
| case HALF_DOWN: |
| case HALF_UP: |
| case HALF_EVEN: |
| // Since sqrt(2) is irrational, log2(x) - logFloor cannot be exactly 0.5 |
| int leadingZeros = Long.numberOfLeadingZeros(x); |
| long cmp = MAX_POWER_OF_SQRT2_UNSIGNED >>> leadingZeros; |
| // floor(2^(logFloor + 0.5)) |
| int logFloor = (Long.SIZE - 1) - leadingZeros; |
| return logFloor + lessThanBranchFree(cmp, x); |
| |
| default: |
| throw new AssertionError("impossible"); |
| } |
| } |
| |
| /** The biggest half power of two that fits into an unsigned long */ |
| @VisibleForTesting static final long MAX_POWER_OF_SQRT2_UNSIGNED = 0xB504F333F9DE6484L; |
| |
| /** |
| * Returns the base-10 logarithm of {@code x}, rounded according to the specified rounding mode. |
| * |
| * @throws IllegalArgumentException if {@code x <= 0} |
| * @throws ArithmeticException if {@code mode} is {@link RoundingMode#UNNECESSARY} and {@code x} |
| * is not a power of ten |
| */ |
| @GwtIncompatible("TODO") |
| @SuppressWarnings("fallthrough") |
| // TODO(kevinb): remove after this warning is disabled globally |
| public static int log10(long x, RoundingMode mode) { |
| checkPositive("x", x); |
| int logFloor = log10Floor(x); |
| long floorPow = powersOf10[logFloor]; |
| switch (mode) { |
| case UNNECESSARY: |
| checkRoundingUnnecessary(x == floorPow); |
| // fall through |
| case FLOOR: |
| case DOWN: |
| return logFloor; |
| case CEILING: |
| case UP: |
| return logFloor + lessThanBranchFree(floorPow, x); |
| case HALF_DOWN: |
| case HALF_UP: |
| case HALF_EVEN: |
| // sqrt(10) is irrational, so log10(x)-logFloor is never exactly 0.5 |
| return logFloor + lessThanBranchFree(halfPowersOf10[logFloor], x); |
| default: |
| throw new AssertionError(); |
| } |
| } |
| |
| @GwtIncompatible("TODO") |
| static int log10Floor(long x) { |
| /* |
| * Based on Hacker's Delight Fig. 11-5, the two-table-lookup, branch-free implementation. |
| * |
| * The key idea is that based on the number of leading zeros (equivalently, floor(log2(x))), |
| * we can narrow the possible floor(log10(x)) values to two. For example, if floor(log2(x)) |
| * is 6, then 64 <= x < 128, so floor(log10(x)) is either 1 or 2. |
| */ |
| int y = maxLog10ForLeadingZeros[Long.numberOfLeadingZeros(x)]; |
| /* |
| * y is the higher of the two possible values of floor(log10(x)). If x < 10^y, then we want the |
| * lower of the two possible values, or y - 1, otherwise, we want y. |
| */ |
| return y - lessThanBranchFree(x, powersOf10[y]); |
| } |
| |
| // maxLog10ForLeadingZeros[i] == floor(log10(2^(Long.SIZE - i))) |
| @VisibleForTesting static final byte[] maxLog10ForLeadingZeros = { |
| 19, 18, 18, 18, 18, 17, 17, 17, 16, 16, 16, 15, 15, 15, 15, 14, 14, 14, 13, 13, 13, 12, 12, |
| 12, 12, 11, 11, 11, 10, 10, 10, 9, 9, 9, 9, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 5, 5, 5, 4, 4, 4, |
| 3, 3, 3, 3, 2, 2, 2, 1, 1, 1, 0, 0, 0 }; |
| |
| @GwtIncompatible("TODO") |
| @VisibleForTesting |
| static final long[] powersOf10 = { |
| 1L, |
| 10L, |
| 100L, |
| 1000L, |
| 10000L, |
| 100000L, |
| 1000000L, |
| 10000000L, |
| 100000000L, |
| 1000000000L, |
| 10000000000L, |
| 100000000000L, |
| 1000000000000L, |
| 10000000000000L, |
| 100000000000000L, |
| 1000000000000000L, |
| 10000000000000000L, |
| 100000000000000000L, |
| 1000000000000000000L |
| }; |
| |
| // halfPowersOf10[i] = largest long less than 10^(i + 0.5) |
| @GwtIncompatible("TODO") |
| @VisibleForTesting |
| static final long[] halfPowersOf10 = { |
| 3L, |
| 31L, |
| 316L, |
| 3162L, |
| 31622L, |
| 316227L, |
| 3162277L, |
| 31622776L, |
| 316227766L, |
| 3162277660L, |
| 31622776601L, |
| 316227766016L, |
| 3162277660168L, |
| 31622776601683L, |
| 316227766016837L, |
| 3162277660168379L, |
| 31622776601683793L, |
| 316227766016837933L, |
| 3162277660168379331L |
| }; |
| |
| /** |
| * Returns {@code b} to the {@code k}th power. Even if the result overflows, it will be equal to |
| * {@code BigInteger.valueOf(b).pow(k).longValue()}. This implementation runs in {@code O(log k)} |
| * time. |
| * |
| * @throws IllegalArgumentException if {@code k < 0} |
| */ |
| @GwtIncompatible("TODO") |
| public static long pow(long b, int k) { |
| checkNonNegative("exponent", k); |
| if (-2 <= b && b <= 2) { |
| switch ((int) b) { |
| case 0: |
| return (k == 0) ? 1 : 0; |
| case 1: |
| return 1; |
| case (-1): |
| return ((k & 1) == 0) ? 1 : -1; |
| case 2: |
| return (k < Long.SIZE) ? 1L << k : 0; |
| case (-2): |
| if (k < Long.SIZE) { |
| return ((k & 1) == 0) ? 1L << k : -(1L << k); |
| } else { |
| return 0; |
| } |
| default: |
| throw new AssertionError(); |
| } |
| } |
| for (long accum = 1;; k >>= 1) { |
| switch (k) { |
| case 0: |
| return accum; |
| case 1: |
| return accum * b; |
| default: |
| accum *= ((k & 1) == 0) ? 1 : b; |
| b *= b; |
| } |
| } |
| } |
| |
| /** |
| * Returns the square root of {@code x}, rounded with the specified rounding mode. |
| * |
| * @throws IllegalArgumentException if {@code x < 0} |
| * @throws ArithmeticException if {@code mode} is {@link RoundingMode#UNNECESSARY} and |
| * {@code sqrt(x)} is not an integer |
| */ |
| @GwtIncompatible("TODO") |
| @SuppressWarnings("fallthrough") |
| public static long sqrt(long x, RoundingMode mode) { |
| checkNonNegative("x", x); |
| if (fitsInInt(x)) { |
| return IntMath.sqrt((int) x, mode); |
| } |
| /* |
| * Let k be the true value of floor(sqrt(x)), so that |
| * |
| * k * k <= x < (k + 1) * (k + 1) |
| * (double) (k * k) <= (double) x <= (double) ((k + 1) * (k + 1)) |
| * since casting to double is nondecreasing. |
| * Note that the right-hand inequality is no longer strict. |
| * Math.sqrt(k * k) <= Math.sqrt(x) <= Math.sqrt((k + 1) * (k + 1)) |
| * since Math.sqrt is monotonic. |
| * (long) Math.sqrt(k * k) <= (long) Math.sqrt(x) <= (long) Math.sqrt((k + 1) * (k + 1)) |
| * since casting to long is monotonic |
| * k <= (long) Math.sqrt(x) <= k + 1 |
| * since (long) Math.sqrt(k * k) == k, as checked exhaustively in |
| * {@link LongMathTest#testSqrtOfPerfectSquareAsDoubleIsPerfect} |
| */ |
| long guess = (long) Math.sqrt(x); |
| // Note: guess is always <= FLOOR_SQRT_MAX_LONG. |
| long guessSquared = guess * guess; |
| // Note (2013-2-26): benchmarks indicate that, inscrutably enough, using if statements is |
| // faster here than using lessThanBranchFree. |
| switch (mode) { |
| case UNNECESSARY: |
| checkRoundingUnnecessary(guessSquared == x); |
| return guess; |
| case FLOOR: |
| case DOWN: |
| if (x < guessSquared) { |
| return guess - 1; |
| } |
| return guess; |
| case CEILING: |
| case UP: |
| if (x > guessSquared) { |
| return guess + 1; |
| } |
| return guess; |
| case HALF_DOWN: |
| case HALF_UP: |
| case HALF_EVEN: |
| long sqrtFloor = guess - ((x < guessSquared) ? 1 : 0); |
| long halfSquare = sqrtFloor * sqrtFloor + sqrtFloor; |
| /* |
| * We wish to test whether or not x <= (sqrtFloor + 0.5)^2 = halfSquare + 0.25. Since both |
| * x and halfSquare are integers, this is equivalent to testing whether or not x <= |
| * halfSquare. (We have to deal with overflow, though.) |
| * |
| * If we treat halfSquare as an unsigned long, we know that |
| * sqrtFloor^2 <= x < (sqrtFloor + 1)^2 |
| * halfSquare - sqrtFloor <= x < halfSquare + sqrtFloor + 1 |
| * so |x - halfSquare| <= sqrtFloor. Therefore, it's safe to treat x - halfSquare as a |
| * signed long, so lessThanBranchFree is safe for use. |
| */ |
| return sqrtFloor + lessThanBranchFree(halfSquare, x); |
| default: |
| throw new AssertionError(); |
| } |
| } |
| |
| /** |
| * Returns the result of dividing {@code p} by {@code q}, rounding using the specified |
| * {@code RoundingMode}. |
| * |
| * @throws ArithmeticException if {@code q == 0}, or if {@code mode == UNNECESSARY} and {@code a} |
| * is not an integer multiple of {@code b} |
| */ |
| @GwtIncompatible("TODO") |
| @SuppressWarnings("fallthrough") |
| public static long divide(long p, long q, RoundingMode mode) { |
| checkNotNull(mode); |
| long div = p / q; // throws if q == 0 |
| long rem = p - q * div; // equals p % q |
| |
| if (rem == 0) { |
| return div; |
| } |
| |
| /* |
| * Normal Java division rounds towards 0, consistently with RoundingMode.DOWN. We just have to |
| * deal with the cases where rounding towards 0 is wrong, which typically depends on the sign of |
| * p / q. |
| * |
| * signum is 1 if p and q are both nonnegative or both negative, and -1 otherwise. |
| */ |
| int signum = 1 | (int) ((p ^ q) >> (Long.SIZE - 1)); |
| boolean increment; |
| switch (mode) { |
| case UNNECESSARY: |
| checkRoundingUnnecessary(rem == 0); |
| // fall through |
| case DOWN: |
| increment = false; |
| break; |
| case UP: |
| increment = true; |
| break; |
| case CEILING: |
| increment = signum > 0; |
| break; |
| case FLOOR: |
| increment = signum < 0; |
| break; |
| case HALF_EVEN: |
| case HALF_DOWN: |
| case HALF_UP: |
| long absRem = abs(rem); |
| long cmpRemToHalfDivisor = absRem - (abs(q) - absRem); |
| // subtracting two nonnegative longs can't overflow |
| // cmpRemToHalfDivisor has the same sign as compare(abs(rem), abs(q) / 2). |
| if (cmpRemToHalfDivisor == 0) { // exactly on the half mark |
| increment = (mode == HALF_UP | (mode == HALF_EVEN & (div & 1) != 0)); |
| } else { |
| increment = cmpRemToHalfDivisor > 0; // closer to the UP value |
| } |
| break; |
| default: |
| throw new AssertionError(); |
| } |
| return increment ? div + signum : div; |
| } |
| |
| /** |
| * Returns {@code x mod m}, a non-negative value less than {@code m}. |
| * This differs from {@code x % m}, which might be negative. |
| * |
| * <p>For example: |
| * |
| * <pre> {@code |
| * |
| * mod(7, 4) == 3 |
| * mod(-7, 4) == 1 |
| * mod(-1, 4) == 3 |
| * mod(-8, 4) == 0 |
| * mod(8, 4) == 0}</pre> |
| * |
| * @throws ArithmeticException if {@code m <= 0} |
| * @see <a href="http://docs.oracle.com/javase/specs/jls/se7/html/jls-15.html#jls-15.17.3"> |
| * Remainder Operator</a> |
| */ |
| @GwtIncompatible("TODO") |
| public static int mod(long x, int m) { |
| // Cast is safe because the result is guaranteed in the range [0, m) |
| return (int) mod(x, (long) m); |
| } |
| |
| /** |
| * Returns {@code x mod m}, a non-negative value less than {@code m}. |
| * This differs from {@code x % m}, which might be negative. |
| * |
| * <p>For example: |
| * |
| * <pre> {@code |
| * |
| * mod(7, 4) == 3 |
| * mod(-7, 4) == 1 |
| * mod(-1, 4) == 3 |
| * mod(-8, 4) == 0 |
| * mod(8, 4) == 0}</pre> |
| * |
| * @throws ArithmeticException if {@code m <= 0} |
| * @see <a href="http://docs.oracle.com/javase/specs/jls/se7/html/jls-15.html#jls-15.17.3"> |
| * Remainder Operator</a> |
| */ |
| @GwtIncompatible("TODO") |
| public static long mod(long x, long m) { |
| if (m <= 0) { |
| throw new ArithmeticException("Modulus must be positive"); |
| } |
| long result = x % m; |
| return (result >= 0) ? result : result + m; |
| } |
| |
| /** |
| * Returns the greatest common divisor of {@code a, b}. Returns {@code 0} if |
| * {@code a == 0 && b == 0}. |
| * |
| * @throws IllegalArgumentException if {@code a < 0} or {@code b < 0} |
| */ |
| public static long gcd(long a, long b) { |
| /* |
| * The reason we require both arguments to be >= 0 is because otherwise, what do you return on |
| * gcd(0, Long.MIN_VALUE)? BigInteger.gcd would return positive 2^63, but positive 2^63 isn't |
| * an int. |
| */ |
| checkNonNegative("a", a); |
| checkNonNegative("b", b); |
| if (a == 0) { |
| // 0 % b == 0, so b divides a, but the converse doesn't hold. |
| // BigInteger.gcd is consistent with this decision. |
| return b; |
| } else if (b == 0) { |
| return a; // similar logic |
| } |
| /* |
| * Uses the binary GCD algorithm; see http://en.wikipedia.org/wiki/Binary_GCD_algorithm. |
| * This is >60% faster than the Euclidean algorithm in benchmarks. |
| */ |
| int aTwos = Long.numberOfTrailingZeros(a); |
| a >>= aTwos; // divide out all 2s |
| int bTwos = Long.numberOfTrailingZeros(b); |
| b >>= bTwos; // divide out all 2s |
| while (a != b) { // both a, b are odd |
| // The key to the binary GCD algorithm is as follows: |
| // Both a and b are odd. Assume a > b; then gcd(a - b, b) = gcd(a, b). |
| // But in gcd(a - b, b), a - b is even and b is odd, so we can divide out powers of two. |
| |
| // We bend over backwards to avoid branching, adapting a technique from |
| // http://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMax |
| |
| long delta = a - b; // can't overflow, since a and b are nonnegative |
| |
| long minDeltaOrZero = delta & (delta >> (Long.SIZE - 1)); |
| // equivalent to Math.min(delta, 0) |
| |
| a = delta - minDeltaOrZero - minDeltaOrZero; // sets a to Math.abs(a - b) |
| // a is now nonnegative and even |
| |
| b += minDeltaOrZero; // sets b to min(old a, b) |
| a >>= Long.numberOfTrailingZeros(a); // divide out all 2s, since 2 doesn't divide b |
| } |
| return a << min(aTwos, bTwos); |
| } |
| |
| /** |
| * Returns the sum of {@code a} and {@code b}, provided it does not overflow. |
| * |
| * @throws ArithmeticException if {@code a + b} overflows in signed {@code long} arithmetic |
| */ |
| @GwtIncompatible("TODO") |
| public static long checkedAdd(long a, long b) { |
| long result = a + b; |
| checkNoOverflow((a ^ b) < 0 | (a ^ result) >= 0); |
| return result; |
| } |
| |
| /** |
| * Returns the difference of {@code a} and {@code b}, provided it does not overflow. |
| * |
| * @throws ArithmeticException if {@code a - b} overflows in signed {@code long} arithmetic |
| */ |
| @GwtIncompatible("TODO") |
| public static long checkedSubtract(long a, long b) { |
| long result = a - b; |
| checkNoOverflow((a ^ b) >= 0 | (a ^ result) >= 0); |
| return result; |
| } |
| |
| /** |
| * Returns the product of {@code a} and {@code b}, provided it does not overflow. |
| * |
| * @throws ArithmeticException if {@code a * b} overflows in signed {@code long} arithmetic |
| */ |
| @GwtIncompatible("TODO") |
| public static long checkedMultiply(long a, long b) { |
| // Hacker's Delight, Section 2-12 |
| int leadingZeros = Long.numberOfLeadingZeros(a) + Long.numberOfLeadingZeros(~a) |
| + Long.numberOfLeadingZeros(b) + Long.numberOfLeadingZeros(~b); |
| /* |
| * If leadingZeros > Long.SIZE + 1 it's definitely fine, if it's < Long.SIZE it's definitely |
| * bad. We do the leadingZeros check to avoid the division below if at all possible. |
| * |
| * Otherwise, if b == Long.MIN_VALUE, then the only allowed values of a are 0 and 1. We take |
| * care of all a < 0 with their own check, because in particular, the case a == -1 will |
| * incorrectly pass the division check below. |
| * |
| * In all other cases, we check that either a is 0 or the result is consistent with division. |
| */ |
| if (leadingZeros > Long.SIZE + 1) { |
| return a * b; |
| } |
| checkNoOverflow(leadingZeros >= Long.SIZE); |
| checkNoOverflow(a >= 0 | b != Long.MIN_VALUE); |
| long result = a * b; |
| checkNoOverflow(a == 0 || result / a == b); |
| return result; |
| } |
| |
| /** |
| * Returns the {@code b} to the {@code k}th power, provided it does not overflow. |
| * |
| * @throws ArithmeticException if {@code b} to the {@code k}th power overflows in signed |
| * {@code long} arithmetic |
| */ |
| @GwtIncompatible("TODO") |
| public static long checkedPow(long b, int k) { |
| checkNonNegative("exponent", k); |
| if (b >= -2 & b <= 2) { |
| switch ((int) b) { |
| case 0: |
| return (k == 0) ? 1 : 0; |
| case 1: |
| return 1; |
| case (-1): |
| return ((k & 1) == 0) ? 1 : -1; |
| case 2: |
| checkNoOverflow(k < Long.SIZE - 1); |
| return 1L << k; |
| case (-2): |
| checkNoOverflow(k < Long.SIZE); |
| return ((k & 1) == 0) ? (1L << k) : (-1L << k); |
| default: |
| throw new AssertionError(); |
| } |
| } |
| long 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(b <= FLOOR_SQRT_MAX_LONG); |
| b *= b; |
| } |
| } |
| } |
| } |
| |
| @VisibleForTesting static final long FLOOR_SQRT_MAX_LONG = 3037000499L; |
| |
| /** |
| * Returns {@code n!}, that is, the product of the first {@code n} positive |
| * integers, {@code 1} if {@code n == 0}, or {@link Long#MAX_VALUE} if the |
| * result does not fit in a {@code long}. |
| * |
| * @throws IllegalArgumentException if {@code n < 0} |
| */ |
| @GwtIncompatible("TODO") |
| public static long factorial(int n) { |
| checkNonNegative("n", n); |
| return (n < factorials.length) ? factorials[n] : Long.MAX_VALUE; |
| } |
| |
| static final long[] factorials = { |
| 1L, |
| 1L, |
| 1L * 2, |
| 1L * 2 * 3, |
| 1L * 2 * 3 * 4, |
| 1L * 2 * 3 * 4 * 5, |
| 1L * 2 * 3 * 4 * 5 * 6, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19, |
| 1L * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10 * 11 * 12 * 13 * 14 * 15 * 16 * 17 * 18 * 19 * 20 |
| }; |
| |
| /** |
| * Returns {@code n} choose {@code k}, also known as the binomial coefficient of {@code n} and |
| * {@code k}, or {@link Long#MAX_VALUE} if the result does not fit in a {@code long}. |
| * |
| * @throws IllegalArgumentException if {@code n < 0}, {@code k < 0}, or {@code k > n} |
| */ |
| public static long 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; |
| } |
| switch (k) { |
| case 0: |
| return 1; |
| case 1: |
| return n; |
| default: |
| if (n < factorials.length) { |
| return factorials[n] / (factorials[k] * factorials[n - k]); |
| } else if (k >= biggestBinomials.length || n > biggestBinomials[k]) { |
| return Long.MAX_VALUE; |
| } else if (k < biggestSimpleBinomials.length && n <= biggestSimpleBinomials[k]) { |
| // guaranteed not to overflow |
| long result = n--; |
| for (int i = 2; i <= k; n--, i++) { |
| result *= n; |
| result /= i; |
| } |
| return result; |
| } else { |
| int nBits = LongMath.log2(n, RoundingMode.CEILING); |
| |
| long result = 1; |
| long numerator = n--; |
| long denominator = 1; |
| |
| int numeratorBits = nBits; |
| // This is an upper bound on log2(numerator, ceiling). |
| |
| /* |
| * We want to do this in long math for speed, but want to avoid overflow. We adapt the |
| * technique previously used by BigIntegerMath: maintain separate numerator and |
| * denominator accumulators, multiplying the fraction into result when near overflow. |
| */ |
| for (int i = 2; i <= k; i++, n--) { |
| if (numeratorBits + nBits < Long.SIZE - 1) { |
| // It's definitely safe to multiply into numerator and denominator. |
| numerator *= n; |
| denominator *= i; |
| numeratorBits += nBits; |
| } else { |
| // It might not be safe to multiply into numerator and denominator, |
| // so multiply (numerator / denominator) into result. |
| result = multiplyFraction(result, numerator, denominator); |
| numerator = n; |
| denominator = i; |
| numeratorBits = nBits; |
| } |
| } |
| return multiplyFraction(result, numerator, denominator); |
| } |
| } |
| } |
| |
| /** |
| * Returns (x * numerator / denominator), which is assumed to come out to an integral value. |
| */ |
| static long multiplyFraction(long x, long numerator, long denominator) { |
| if (x == 1) { |
| return numerator / denominator; |
| } |
| long commonDivisor = gcd(x, denominator); |
| x /= commonDivisor; |
| denominator /= commonDivisor; |
| // We know gcd(x, denominator) = 1, and x * numerator / denominator is exact, |
| // so denominator must be a divisor of numerator. |
| return x * (numerator / denominator); |
| } |
| |
| /* |
| * binomial(biggestBinomials[k], k) fits in a long, but not |
| * binomial(biggestBinomials[k] + 1, k). |
| */ |
| static final int[] biggestBinomials = |
| {Integer.MAX_VALUE, Integer.MAX_VALUE, Integer.MAX_VALUE, 3810779, 121977, 16175, 4337, 1733, |
| 887, 534, 361, 265, 206, 169, 143, 125, 111, 101, 94, 88, 83, 79, 76, 74, 72, 70, 69, 68, |
| 67, 67, 66, 66, 66, 66}; |
| |
| /* |
| * binomial(biggestSimpleBinomials[k], k) doesn't need to use the slower GCD-based impl, |
| * but binomial(biggestSimpleBinomials[k] + 1, k) does. |
| */ |
| @VisibleForTesting static final int[] biggestSimpleBinomials = |
| {Integer.MAX_VALUE, Integer.MAX_VALUE, Integer.MAX_VALUE, 2642246, 86251, 11724, 3218, 1313, |
| 684, 419, 287, 214, 169, 139, 119, 105, 95, 87, 81, 76, 73, 70, 68, 66, 64, 63, 62, 62, |
| 61, 61, 61}; |
| // These values were generated by using checkedMultiply to see when the simple multiply/divide |
| // algorithm would lead to an overflow. |
| |
| static boolean fitsInInt(long x) { |
| return (int) x == x; |
| } |
| |
| /** |
| * Returns the arithmetic mean of {@code x} and {@code y}, rounded toward |
| * negative infinity. This method is resilient to overflow. |
| * |
| * @since 14.0 |
| */ |
| public static long mean(long x, long 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 LongMath() {} |
| } |
