https://github.com/google/guava/guava: Guava is a suite of core and expanded libraries that include utility classes, google's collections, io classes, and much much more. 
/*
 * 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.Beta;
import com.google.common.annotations.GwtCompatible;
import com.google.common.annotations.GwtIncompatible;
import com.google.common.annotations.VisibleForTesting;
import com.google.common.primitives.UnsignedLongs;
import java.math.BigInteger;
import java.math.RoundingMode;


A class for arithmetic on values of type long. Where possible, methods are defined and named analogously to their BigInteger counterparts. 
The implementations of many methods in this class are based on material from Henry S. Warren,
Jr.'s Hacker's Delight, (Addison Wesley, 2002).

Similar functionality for int and for BigInteger can be found in IntMath and BigIntegerMath respectively. For other common operations on long values, see Longs. 
Author:Louis Wasserman
Since:11.0
/**
 * 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 &&, ||

  @VisibleForTesting static final long MAX_SIGNED_POWER_OF_TWO = 1L << (Long.SIZE - 2);

  
Returns the smallest power of two greater than or equal to x. This is equivalent to checkedPow(2, log2(x, CEILING)). 
Throws:
Since:20.0
/**
   * Returns the smallest power of two greater than or equal to {@code x}. This is equivalent to
   * {@code checkedPow(2, log2(x, CEILING))}.
   *
   * @throws IllegalArgumentException if {@code x <= 0}
   * @throws ArithmeticException of the next-higher power of two is not representable as a {@code
   *     long}, i.e. when {@code x > 2^62}
   * @since 20.0
   */
  @Beta
  public static long ceilingPowerOfTwo(long x) {
    checkPositive("x", x);
    if (x > MAX_SIGNED_POWER_OF_TWO) {
      throw new ArithmeticException("ceilingPowerOfTwo(" + x + ") is not representable as a long");
    }
    return 1L << -Long.numberOfLeadingZeros(x - 1);
  }

  
Returns the largest power of two less than or equal to x. This is equivalent to 
checkedPow(2, log2(x, FLOOR)). 
Throws:
Since:20.0
/**
   * Returns the largest power of two less than or equal to {@code x}. This is equivalent to {@code
   * checkedPow(2, log2(x, FLOOR))}.
   *
   * @throws IllegalArgumentException if {@code x <= 0}
   * @since 20.0
   */
  @Beta
  public static long floorPowerOfTwo(long x) {
    checkPositive("x", x);

    // Long.highestOneBit was buggy on GWT.  We've fixed it, but I'm not certain when the fix will
    // be released.
    return 1L << ((Long.SIZE - 1) - Long.numberOfLeadingZeros(x));
  }

  
Returns true if x represents a power of two. 
This differs from Long.bitCount(x) == 1, because 
Long.bitCount(Long.MIN_VALUE) == 1, but Long.MIN_VALUE is not a power of two. 
/**
   * 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 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. 
/**
   * 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 x, rounded according to the specified rounding mode. 
Throws:
/**
   * 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 
/** 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 x, rounded according to the specified rounding mode. 
Throws:
/**
   * 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 b to the kth power. Even if the result overflows, it will be equal to BigInteger.valueOf(b).pow(k).longValue(). This implementation runs in O(log k) time. 
Throws:
/**
   * 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 x, rounded with the specified rounding mode. 
Throws:
/**
   * 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 p by q, rounding using the specified 
RoundingMode. 
Throws:
  • ArithmeticException – if q == 0, or if mode == UNNECESSARY and a is not an integer multiple of b
/**
   * 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 x mod m, a non-negative value less than m. This differs from x %
m, which might be negative. 
For example:


mod(7, 4) == 3
mod(-7, 4) == 1
mod(-1, 4) == 3
mod(-8, 4) == 0
mod(8, 4) == 0



Throws:
See Also:
/**
   * 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 x mod m, a non-negative value less than m. This differs from x %
m, which might be negative. 
For example:


mod(7, 4) == 3
mod(-7, 4) == 1
mod(-1, 4) == 3
mod(-8, 4) == 0
mod(8, 4) == 0



Throws:
See Also:
/**
   * 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 a, b. Returns 0 if a == 0 && b ==
0. 
Throws:
/**
   * 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 a and b, provided it does not overflow. 
Throws:
/**
   * 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, "checkedAdd", a, b);
    return result;
  }

  
Returns the difference of a and b, provided it does not overflow. 
Throws:
/**
   * 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, "checkedSubtract", a, b);
    return result;
  }

  
Returns the product of a and b, provided it does not overflow. 
Throws:
/**
   * 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
   */
  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, "checkedMultiply", a, b);
    checkNoOverflow(a >= 0 | b != Long.MIN_VALUE, "checkedMultiply", a, b);
    long result = a * b;
    checkNoOverflow(a == 0 || result / a == b, "checkedMultiply", a, b);
    return result;
  }

  
Returns the b to the kth power, provided it does not overflow. 
Throws:
/**
   * 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, "checkedPow", b, k);
          return 1L << k;
        case (-2):
          checkNoOverflow(k < Long.SIZE, "checkedPow", b, k);
          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(
                -FLOOR_SQRT_MAX_LONG <= b && b <= FLOOR_SQRT_MAX_LONG, "checkedPow", b, k);
            b *= b;
          }
      }
    }
  }

  
Returns the sum of a and b unless it would overflow or underflow in which case Long.MAX_VALUE or Long.MIN_VALUE is returned, respectively. 
Since:20.0
/**
   * Returns the sum of {@code a} and {@code b} unless it would overflow or underflow in which case
   * {@code Long.MAX_VALUE} or {@code Long.MIN_VALUE} is returned, respectively.
   *
   * @since 20.0
   */
  @Beta
  public static long saturatedAdd(long a, long b) {
    long naiveSum = a + b;
    if ((a ^ b) < 0 | (a ^ naiveSum) >= 0) {
      // If a and b have different signs or a has the same sign as the result then there was no
      // overflow, return.
      return naiveSum;
    }
    // we did over/under flow, if the sign is negative we should return MAX otherwise MIN
    return Long.MAX_VALUE + ((naiveSum >>> (Long.SIZE - 1)) ^ 1);
  }

  
Returns the difference of a and b unless it would overflow or underflow in which case Long.MAX_VALUE or Long.MIN_VALUE is returned, respectively. 
Since:20.0
/**
   * Returns the difference of {@code a} and {@code b} unless it would overflow or underflow in
   * which case {@code Long.MAX_VALUE} or {@code Long.MIN_VALUE} is returned, respectively.
   *
   * @since 20.0
   */
  @Beta
  public static long saturatedSubtract(long a, long b) {
    long naiveDifference = a - b;
    if ((a ^ b) >= 0 | (a ^ naiveDifference) >= 0) {
      // If a and b have the same signs or a has the same sign as the result then there was no
      // overflow, return.
      return naiveDifference;
    }
    // we did over/under flow
    return Long.MAX_VALUE + ((naiveDifference >>> (Long.SIZE - 1)) ^ 1);
  }

  
Returns the product of a and b unless it would overflow or underflow in which case Long.MAX_VALUE or Long.MIN_VALUE is returned, respectively. 
Since:20.0
/**
   * Returns the product of {@code a} and {@code b} unless it would overflow or underflow in which
   * case {@code Long.MAX_VALUE} or {@code Long.MIN_VALUE} is returned, respectively.
   *
   * @since 20.0
   */
  @Beta
  public static long saturatedMultiply(long a, long b) {
    // see checkedMultiply for explanation
    int leadingZeros =
        Long.numberOfLeadingZeros(a)
            + Long.numberOfLeadingZeros(~a)
            + Long.numberOfLeadingZeros(b)
            + Long.numberOfLeadingZeros(~b);
    if (leadingZeros > Long.SIZE + 1) {
      return a * b;
    }
    // the return value if we will overflow (which we calculate by overflowing a long :) )
    long limit = Long.MAX_VALUE + ((a ^ b) >>> (Long.SIZE - 1));
    if (leadingZeros < Long.SIZE | (a < 0 & b == Long.MIN_VALUE)) {
      // overflow
      return limit;
    }
    long result = a * b;
    if (a == 0 || result / a == b) {
      return result;
    }
    return limit;
  }

  
Returns the b to the kth power, unless it would overflow or underflow in which case Long.MAX_VALUE or Long.MIN_VALUE is returned, respectively. 
Since:20.0
/**
   * Returns the {@code b} to the {@code k}th power, unless it would overflow or underflow in which
   * case {@code Long.MAX_VALUE} or {@code Long.MIN_VALUE} is returned, respectively.
   *
   * @since 20.0
   */
  @Beta
  public static long saturatedPow(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:
          if (k >= Long.SIZE - 1) {
            return Long.MAX_VALUE;
          }
          return 1L << k;
        case (-2):
          if (k >= Long.SIZE) {
            return Long.MAX_VALUE + (k & 1);
          }
          return ((k & 1) == 0) ? (1L << k) : (-1L << k);
        default:
          throw new AssertionError();
      }
    }
    long accum = 1;
    // if b is negative and k is odd then the limit is MIN otherwise the limit is MAX
    long limit = Long.MAX_VALUE + ((b >>> Long.SIZE - 1) & (k & 1));
    while (true) {
      switch (k) {
        case 0:
          return accum;
        case 1:
          return saturatedMultiply(accum, b);
        default:
          if ((k & 1) != 0) {
            accum = saturatedMultiply(accum, b);
          }
          k >>= 1;
          if (k > 0) {
            if (-FLOOR_SQRT_MAX_LONG > b | b > FLOOR_SQRT_MAX_LONG) {
              return limit;
            }
            b *= b;
          }
      }
    }
  }

  @VisibleForTesting static final long FLOOR_SQRT_MAX_LONG = 3037000499L;

  
Returns n!, that is, the product of the first n positive integers, 1 if n == 0, or Long.MAX_VALUE if the result does not fit in a long. 
Throws:
/**
   * 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 n choose k, also known as the binomial coefficient of n and k, or Long.MAX_VALUE if the result does not fit in a long. 
Throws:
/**
   * 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. 
/** 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 x and y, rounded toward negative infinity. This method is resilient to overflow. 
Since:14.0
/**
   * 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);
  }

  /*
   * This bitmask is used as an optimization for cheaply testing for divisiblity by 2, 3, or 5.
   * Each bit is set to 1 for all remainders that indicate divisibility by 2, 3, or 5, so
   * 1, 7, 11, 13, 17, 19, 23, 29 are set to 0. 30 and up don't matter because they won't be hit.
   */
  private static final int SIEVE_30 =
      ~((1 << 1) | (1 << 7) | (1 << 11) | (1 << 13) | (1 << 17) | (1 << 19) | (1 << 23)
          | (1 << 29));

  
Returns true if n is a prime number: an integer greater
than one that cannot be factored into a product of smaller positive integers. Returns false if n is zero, one, or a composite number (one which can be
factored into smaller positive integers).

To test larger numbers, use BigInteger.isProbablePrime. 
Throws:
Since:20.0
/**
   * Returns {@code true} if {@code n} is a <a
   * href="http://mathworld.wolfram.com/PrimeNumber.html">prime number</a>: an integer <i>greater
   * than one</i> that cannot be factored into a product of <i>smaller</i> positive integers.
   * Returns {@code false} if {@code n} is zero, one, or a composite number (one which <i>can</i> be
   * factored into smaller positive integers).
   *
   * <p>To test larger numbers, use {@link BigInteger#isProbablePrime}.
   *
   * @throws IllegalArgumentException if {@code n} is negative
   * @since 20.0
   */
  @GwtIncompatible // TODO
  @Beta
  public static boolean isPrime(long n) {
    if (n < 2) {
      checkNonNegative("n", n);
      return false;
    }
    if (n == 2 || n == 3 || n == 5 || n == 7 || n == 11 || n == 13) {
      return true;
    }

    if ((SIEVE_30 & (1 << (n % 30))) != 0) {
      return false;
    }
    if (n % 7 == 0 || n % 11 == 0 || n % 13 == 0) {
      return false;
    }
    if (n < 17 * 17) {
      return true;
    }

    for (long[] baseSet : millerRabinBaseSets) {
      if (n <= baseSet[0]) {
        for (int i = 1; i < baseSet.length; i++) {
          if (!MillerRabinTester.test(baseSet[i], n)) {
            return false;
          }
        }
        return true;
      }
    }
    throw new AssertionError();
  }

  /*
   * If n <= millerRabinBases[i][0], then testing n against bases millerRabinBases[i][1..] suffices
   * to prove its primality. Values from miller-rabin.appspot.com.
   *
   * NOTE: We could get slightly better bases that would be treated as unsigned, but benchmarks
   * showed negligible performance improvements.
   */
  private static final long[][] millerRabinBaseSets = {
    {291830, 126401071349994536L},
    {885594168, 725270293939359937L, 3569819667048198375L},
    {273919523040L, 15, 7363882082L, 992620450144556L},
    {47636622961200L, 2, 2570940, 211991001, 3749873356L},
    {
      7999252175582850L,
      2,
      4130806001517L,
      149795463772692060L,
      186635894390467037L,
      3967304179347715805L
    },
    {
      585226005592931976L,
      2,
      123635709730000L,
      9233062284813009L,
      43835965440333360L,
      761179012939631437L,
      1263739024124850375L
    },
    {Long.MAX_VALUE, 2, 325, 9375, 28178, 450775, 9780504, 1795265022}
  };

  private enum MillerRabinTester {
    
Works for inputs ≤ FLOOR_SQRT_MAX_LONG. 
/** Works for inputs ≤ FLOOR_SQRT_MAX_LONG. */
    SMALL {
      @Override
      long mulMod(long a, long b, long m) {
        /*
         * NOTE(lowasser, 2015-Feb-12): Benchmarks suggest that changing this to
         * UnsignedLongs.remainder and increasing the threshold to 2^32 doesn't pay for itself, and
         * adding another enum constant hurts performance further -- I suspect because bimorphic
         * implementation is a sweet spot for the JVM.
         */
        return (a * b) % m;
      }

      @Override
      long squareMod(long a, long m) {
        return (a * a) % m;
      }
    },
    
Works for all nonnegative signed longs. 
/** Works for all nonnegative signed longs. */
    LARGE {
      
Returns (a + b) mod m. Precondition: 0 <= a, b < m < 2^63. 
/** Returns (a + b) mod m. Precondition: {@code 0 <= a}, {@code b < m < 2^63}. */
      private long plusMod(long a, long b, long m) {
        return (a >= m - b) ? (a + b - m) : (a + b);
      }

      
Returns (a * 2^32) mod m. a may be any unsigned long. 
/** Returns (a * 2^32) mod m. a may be any unsigned long. */
      private long times2ToThe32Mod(long a, long m) {
        int remainingPowersOf2 = 32;
        do {
          int shift = Math.min(remainingPowersOf2, Long.numberOfLeadingZeros(a));
          // shift is either the number of powers of 2 left to multiply a by, or the biggest shift
          // possible while keeping a in an unsigned long.
          a = UnsignedLongs.remainder(a << shift, m);
          remainingPowersOf2 -= shift;
        } while (remainingPowersOf2 > 0);
        return a;
      }

      @Override
      long mulMod(long a, long b, long m) {
        long aHi = a >>> 32; // < 2^31
        long bHi = b >>> 32; // < 2^31
        long aLo = a & 0xFFFFFFFFL; // < 2^32
        long bLo = b & 0xFFFFFFFFL; // < 2^32

        /*
         * a * b == aHi * bHi * 2^64 + (aHi * bLo + aLo * bHi) * 2^32 + aLo * bLo.
         *       == (aHi * bHi * 2^32 + aHi * bLo + aLo * bHi) * 2^32 + aLo * bLo
         *
         * We carry out this computation in modular arithmetic. Since times2ToThe32Mod accepts any
         * unsigned long, we don't have to do a mod on every operation, only when intermediate
         * results can exceed 2^63.
         */
        long result = times2ToThe32Mod(aHi * bHi /* < 2^62 */, m); // < m < 2^63
        result += aHi * bLo; // aHi * bLo < 2^63, result < 2^64
        if (result < 0) {
          result = UnsignedLongs.remainder(result, m);
        }
        // result < 2^63 again
        result += aLo * bHi; // aLo * bHi < 2^63, result < 2^64
        result = times2ToThe32Mod(result, m); // result < m < 2^63
        return plusMod(result, UnsignedLongs.remainder(aLo * bLo /* < 2^64 */, m), m);
      }

      @Override
      long squareMod(long a, long m) {
        long aHi = a >>> 32; // < 2^31
        long aLo = a & 0xFFFFFFFFL; // < 2^32

        /*
         * a^2 == aHi^2 * 2^64 + aHi * aLo * 2^33 + aLo^2
         *     == (aHi^2 * 2^32 + aHi * aLo * 2) * 2^32 + aLo^2
         * We carry out this computation in modular arithmetic.  Since times2ToThe32Mod accepts any
         * unsigned long, we don't have to do a mod on every operation, only when intermediate
         * results can exceed 2^63.
         */
        long result = times2ToThe32Mod(aHi * aHi /* < 2^62 */, m); // < m < 2^63
        long hiLo = aHi * aLo * 2;
        if (hiLo < 0) {
          hiLo = UnsignedLongs.remainder(hiLo, m);
        }
        // hiLo < 2^63
        result += hiLo; // result < 2^64
        result = times2ToThe32Mod(result, m); // result < m < 2^63
        return plusMod(result, UnsignedLongs.remainder(aLo * aLo /* < 2^64 */, m), m);
      }
    };

    static boolean test(long base, long n) {
      // Since base will be considered % n, it's okay if base > FLOOR_SQRT_MAX_LONG,
      // so long as n <= FLOOR_SQRT_MAX_LONG.
      return ((n <= FLOOR_SQRT_MAX_LONG) ? SMALL : LARGE).testWitness(base, n);
    }

    
Returns a * b mod m. 
/** Returns a * b mod m. */
    abstract long mulMod(long a, long b, long m);

    
Returns a^2 mod m. 
/** Returns a^2 mod m. */
    abstract long squareMod(long a, long m);

    
Returns a^p mod m. 
/** Returns a^p mod m. */
    private long powMod(long a, long p, long m) {
      long res = 1;
      for (; p != 0; p >>= 1) {
        if ((p & 1) != 0) {
          res = mulMod(res, a, m);
        }
        a = squareMod(a, m);
      }
      return res;
    }

    
Returns true if n is a strong probable prime relative to the specified base. 
/** Returns true if n is a strong probable prime relative to the specified base. */
    private boolean testWitness(long base, long n) {
      int r = Long.numberOfTrailingZeros(n - 1);
      long d = (n - 1) >> r;
      base %= n;
      if (base == 0) {
        return true;
      }
      // Calculate a := base^d mod n.
      long a = powMod(base, d, n);
      // n passes this test if
      //    base^d = 1 (mod n)
      // or base^(2^j * d) = -1 (mod n) for some 0 <= j < r.
      if (a == 1) {
        return true;
      }
      int j = 0;
      while (a != n - 1) {
        if (++j == r) {
          return false;
        }
        a = squareMod(a, n);
      }
      return true;
    }
  }

  private LongMath() {}
}