/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You 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 org.apache.commons.math3.util;

import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.Localizable;
import org.apache.commons.math3.exception.util.LocalizedFormats;

import java.math.BigInteger;

/** Some useful, arithmetics related, additions to the built-in functions in {@link Math}. */
public final class ArithmeticUtils {

    /** Private constructor. */
    private ArithmeticUtils() {
        super();
    }

    /**
     * Add two integers, checking for overflow.
     *
     * @param x an addend
     * @param y an addend
     * @return the sum {@code x+y}
     * @throws MathArithmeticException if the result can not be represented as an {@code int}.
     * @since 1.1
     */
    public static int addAndCheck(int x, int y) throws MathArithmeticException {
        long s = (long) x + (long) y;
        if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
            throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_ADDITION, x, y);
        }
        return (int) s;
    }

    /**
     * Add two long integers, checking for overflow.
     *
     * @param a an addend
     * @param b an addend
     * @return the sum {@code a+b}
     * @throws MathArithmeticException if the result can not be represented as an long
     * @since 1.2
     */
    public static long addAndCheck(long a, long b) throws MathArithmeticException {
        return addAndCheck(a, b, LocalizedFormats.OVERFLOW_IN_ADDITION);
    }

    /**
     * Returns an exact representation of the <a
     * href="http://mathworld.wolfram.com/BinomialCoefficient.html">Binomial Coefficient</a>,
     * "{@code n choose k}", the number of {@code k}-element subsets that can be selected from an
     * {@code n}-element set.
     *
     * <p><Strong>Preconditions</strong>:
     *
     * <ul>
     *   <li>{@code 0 <= k <= n } (otherwise {@code IllegalArgumentException} is thrown)
     *   <li>The result is small enough to fit into a {@code long}. The largest value of {@code n}
     *       for which all coefficients are {@code < Long.MAX_VALUE} is 66. If the computed value
     *       exceeds {@code Long.MAX_VALUE} an {@code ArithMeticException} is thrown.
     * </ul>
     *
     * @param n the size of the set
     * @param k the size of the subsets to be counted
     * @return {@code n choose k}
     * @throws NotPositiveException if {@code n < 0}.
     * @throws NumberIsTooLargeException if {@code k > n}.
     * @throws MathArithmeticException if the result is too large to be represented by a long
     *     integer.
     * @deprecated use {@link CombinatoricsUtils#binomialCoefficient(int, int)}
     */
    @Deprecated
    public static long binomialCoefficient(final int n, final int k)
            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
        return CombinatoricsUtils.binomialCoefficient(n, k);
    }

    /**
     * Returns a {@code double} representation of the <a
     * href="http://mathworld.wolfram.com/BinomialCoefficient.html">Binomial Coefficient</a>,
     * "{@code n choose k}", the number of {@code k}-element subsets that can be selected from an
     * {@code n}-element set.
     *
     * <p><Strong>Preconditions</strong>:
     *
     * <ul>
     *   <li>{@code 0 <= k <= n } (otherwise {@code IllegalArgumentException} is thrown)
     *   <li>The result is small enough to fit into a {@code double}. The largest value of {@code n}
     *       for which all coefficients are < Double.MAX_VALUE is 1029. If the computed value
     *       exceeds Double.MAX_VALUE, Double.POSITIVE_INFINITY is returned
     * </ul>
     *
     * @param n the size of the set
     * @param k the size of the subsets to be counted
     * @return {@code n choose k}
     * @throws NotPositiveException if {@code n < 0}.
     * @throws NumberIsTooLargeException if {@code k > n}.
     * @throws MathArithmeticException if the result is too large to be represented by a long
     *     integer.
     * @deprecated use {@link CombinatoricsUtils#binomialCoefficientDouble(int, int)}
     */
    @Deprecated
    public static double binomialCoefficientDouble(final int n, final int k)
            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
        return CombinatoricsUtils.binomialCoefficientDouble(n, k);
    }

    /**
     * Returns the natural {@code log} of the <a
     * href="http://mathworld.wolfram.com/BinomialCoefficient.html">Binomial Coefficient</a>,
     * "{@code n choose k}", the number of {@code k}-element subsets that can be selected from an
     * {@code n}-element set.
     *
     * <p><Strong>Preconditions</strong>:
     *
     * <ul>
     *   <li>{@code 0 <= k <= n } (otherwise {@code IllegalArgumentException} is thrown)
     * </ul>
     *
     * @param n the size of the set
     * @param k the size of the subsets to be counted
     * @return {@code n choose k}
     * @throws NotPositiveException if {@code n < 0}.
     * @throws NumberIsTooLargeException if {@code k > n}.
     * @throws MathArithmeticException if the result is too large to be represented by a long
     *     integer.
     * @deprecated use {@link CombinatoricsUtils#binomialCoefficientLog(int, int)}
     */
    @Deprecated
    public static double binomialCoefficientLog(final int n, final int k)
            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
        return CombinatoricsUtils.binomialCoefficientLog(n, k);
    }

    /**
     * Returns n!. Shorthand for {@code n} <a href="http://mathworld.wolfram.com/Factorial.html">
     * Factorial</a>, the product of the numbers {@code 1,...,n}.
     *
     * <p><Strong>Preconditions</strong>:
     *
     * <ul>
     *   <li>{@code n >= 0} (otherwise {@code IllegalArgumentException} is thrown)
     *   <li>The result is small enough to fit into a {@code long}. The largest value of {@code n}
     *       for which {@code n!} < Long.MAX_VALUE} is 20. If the computed value exceeds {@code
     *       Long.MAX_VALUE} an {@code ArithMeticException } is thrown.
     * </ul>
     *
     * @param n argument
     * @return {@code n!}
     * @throws MathArithmeticException if the result is too large to be represented by a {@code
     *     long}.
     * @throws NotPositiveException if {@code n < 0}.
     * @throws MathArithmeticException if {@code n > 20}: The factorial value is too large to fit in
     *     a {@code long}.
     * @deprecated use {@link CombinatoricsUtils#factorial(int)}
     */
    @Deprecated
    public static long factorial(final int n) throws NotPositiveException, MathArithmeticException {
        return CombinatoricsUtils.factorial(n);
    }

    /**
     * Compute n!, the<a href="http://mathworld.wolfram.com/Factorial.html">factorial</a> of {@code
     * n} (the product of the numbers 1 to n), as a {@code double}. The result should be small
     * enough to fit into a {@code double}: The largest {@code n} for which {@code n! <
     * Double.MAX_VALUE} is 170. If the computed value exceeds {@code Double.MAX_VALUE}, {@code
     * Double.POSITIVE_INFINITY} is returned.
     *
     * @param n Argument.
     * @return {@code n!}
     * @throws NotPositiveException if {@code n < 0}.
     * @deprecated use {@link CombinatoricsUtils#factorialDouble(int)}
     */
    @Deprecated
    public static double factorialDouble(final int n) throws NotPositiveException {
        return CombinatoricsUtils.factorialDouble(n);
    }

    /**
     * Compute the natural logarithm of the factorial of {@code n}.
     *
     * @param n Argument.
     * @return {@code n!}
     * @throws NotPositiveException if {@code n < 0}.
     * @deprecated use {@link CombinatoricsUtils#factorialLog(int)}
     */
    @Deprecated
    public static double factorialLog(final int n) throws NotPositiveException {
        return CombinatoricsUtils.factorialLog(n);
    }

    /**
     * Computes the greatest common divisor of the absolute value of two numbers, using a modified
     * version of the "binary gcd" method. See Knuth 4.5.2 algorithm B. The algorithm is due to
     * Josef Stein (1961). <br>
     * Special cases:
     *
     * <ul>
     *   <li>The invocations {@code gcd(Integer.MIN_VALUE, Integer.MIN_VALUE)}, {@code
     *       gcd(Integer.MIN_VALUE, 0)} and {@code gcd(0, Integer.MIN_VALUE)} throw an {@code
     *       ArithmeticException}, because the result would be 2^31, which is too large for an int
     *       value.
     *   <li>The result of {@code gcd(x, x)}, {@code gcd(0, x)} and {@code gcd(x, 0)} is the
     *       absolute value of {@code x}, except for the special cases above.
     *   <li>The invocation {@code gcd(0, 0)} is the only one which returns {@code 0}.
     * </ul>
     *
     * @param p Number.
     * @param q Number.
     * @return the greatest common divisor (never negative).
     * @throws MathArithmeticException if the result cannot be represented as a non-negative {@code
     *     int} value.
     * @since 1.1
     */
    public static int gcd(int p, int q) throws MathArithmeticException {
        int a = p;
        int b = q;
        if (a == 0 || b == 0) {
            if (a == Integer.MIN_VALUE || b == Integer.MIN_VALUE) {
                throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS, p, q);
            }
            return FastMath.abs(a + b);
        }

        long al = a;
        long bl = b;
        boolean useLong = false;
        if (a < 0) {
            if (Integer.MIN_VALUE == a) {
                useLong = true;
            } else {
                a = -a;
            }
            al = -al;
        }
        if (b < 0) {
            if (Integer.MIN_VALUE == b) {
                useLong = true;
            } else {
                b = -b;
            }
            bl = -bl;
        }
        if (useLong) {
            if (al == bl) {
                throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS, p, q);
            }
            long blbu = bl;
            bl = al;
            al = blbu % al;
            if (al == 0) {
                if (bl > Integer.MAX_VALUE) {
                    throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS, p, q);
                }
                return (int) bl;
            }
            blbu = bl;

            // Now "al" and "bl" fit in an "int".
            b = (int) al;
            a = (int) (blbu % al);
        }

        return gcdPositive(a, b);
    }

    /**
     * Computes the greatest common divisor of two <em>positive</em> numbers (this precondition is
     * <em>not</em> checked and the result is undefined if not fulfilled) using the "binary gcd"
     * method which avoids division and modulo operations. See Knuth 4.5.2 algorithm B. The
     * algorithm is due to Josef Stein (1961). <br>
     * Special cases:
     *
     * <ul>
     *   <li>The result of {@code gcd(x, x)}, {@code gcd(0, x)} and {@code gcd(x, 0)} is the value
     *       of {@code x}.
     *   <li>The invocation {@code gcd(0, 0)} is the only one which returns {@code 0}.
     * </ul>
     *
     * @param a Positive number.
     * @param b Positive number.
     * @return the greatest common divisor.
     */
    private static int gcdPositive(int a, int b) {
        if (a == 0) {
            return b;
        } else if (b == 0) {
            return a;
        }

        // Make "a" and "b" odd, keeping track of common power of 2.
        final int aTwos = Integer.numberOfTrailingZeros(a);
        a >>= aTwos;
        final int bTwos = Integer.numberOfTrailingZeros(b);
        b >>= bTwos;
        final int shift = FastMath.min(aTwos, bTwos);

        // "a" and "b" are positive.
        // If a > b then "gdc(a, b)" is equal to "gcd(a - b, b)".
        // If a < b then "gcd(a, b)" is equal to "gcd(b - a, a)".
        // Hence, in the successive iterations:
        //  "a" becomes the absolute difference of the current values,
        //  "b" becomes the minimum of the current values.
        while (a != b) {
            final int delta = a - b;
            b = Math.min(a, b);
            a = Math.abs(delta);

            // Remove any power of 2 in "a" ("b" is guaranteed to be odd).
            a >>= Integer.numberOfTrailingZeros(a);
        }

        // Recover the common power of 2.
        return a << shift;
    }

    /**
     * Gets the greatest common divisor of the absolute value of two numbers, using the "binary gcd"
     * method which avoids division and modulo operations. See Knuth 4.5.2 algorithm B. This
     * algorithm is due to Josef Stein (1961). Special cases:
     *
     * <ul>
     *   <li>The invocations {@code gcd(Long.MIN_VALUE, Long.MIN_VALUE)}, {@code gcd(Long.MIN_VALUE,
     *       0L)} and {@code gcd(0L, Long.MIN_VALUE)} throw an {@code ArithmeticException}, because
     *       the result would be 2^63, which is too large for a long value.
     *   <li>The result of {@code gcd(x, x)}, {@code gcd(0L, x)} and {@code gcd(x, 0L)} is the
     *       absolute value of {@code x}, except for the special cases above.
     *   <li>The invocation {@code gcd(0L, 0L)} is the only one which returns {@code 0L}.
     * </ul>
     *
     * @param p Number.
     * @param q Number.
     * @return the greatest common divisor, never negative.
     * @throws MathArithmeticException if the result cannot be represented as a non-negative {@code
     *     long} value.
     * @since 2.1
     */
    public static long gcd(final long p, final long q) throws MathArithmeticException {
        long u = p;
        long v = q;
        if ((u == 0) || (v == 0)) {
            if ((u == Long.MIN_VALUE) || (v == Long.MIN_VALUE)) {
                throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_64_BITS, p, q);
            }
            return FastMath.abs(u) + FastMath.abs(v);
        }
        // keep u and v negative, as negative integers range down to
        // -2^63, while positive numbers can only be as large as 2^63-1
        // (i.e. we can't necessarily negate a negative number without
        // overflow)
        /* assert u!=0 && v!=0; */
        if (u > 0) {
            u = -u;
        } // make u negative
        if (v > 0) {
            v = -v;
        } // make v negative
        // B1. [Find power of 2]
        int k = 0;
        while ((u & 1) == 0 && (v & 1) == 0 && k < 63) { // while u and v are
            // both even...
            u /= 2;
            v /= 2;
            k++; // cast out twos.
        }
        if (k == 63) {
            throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_64_BITS, p, q);
        }
        // B2. Initialize: u and v have been divided by 2^k and at least
        // one is odd.
        long t = ((u & 1) == 1) ? v : -(u / 2) /* B3 */;
        // t negative: u was odd, v may be even (t replaces v)
        // t positive: u was even, v is odd (t replaces u)
        do {
            /* assert u<0 && v<0; */
            // B4/B3: cast out twos from t.
            while ((t & 1) == 0) { // while t is even..
                t /= 2; // cast out twos
            }
            // B5 [reset max(u,v)]
            if (t > 0) {
                u = -t;
            } else {
                v = t;
            }
            // B6/B3. at this point both u and v should be odd.
            t = (v - u) / 2;
            // |u| larger: t positive (replace u)
            // |v| larger: t negative (replace v)
        } while (t != 0);
        return -u * (1L << k); // gcd is u*2^k
    }

    /**
     * Returns the least common multiple of the absolute value of two numbers, using the formula
     * {@code lcm(a,b) = (a / gcd(a,b)) * b}. Special cases:
     *
     * <ul>
     *   <li>The invocations {@code lcm(Integer.MIN_VALUE, n)} and {@code lcm(n,
     *       Integer.MIN_VALUE)}, where {@code abs(n)} is a power of 2, throw an {@code
     *       ArithmeticException}, because the result would be 2^31, which is too large for an int
     *       value.
     *   <li>The result of {@code lcm(0, x)} and {@code lcm(x, 0)} is {@code 0} for any {@code x}.
     * </ul>
     *
     * @param a Number.
     * @param b Number.
     * @return the least common multiple, never negative.
     * @throws MathArithmeticException if the result cannot be represented as a non-negative {@code
     *     int} value.
     * @since 1.1
     */
    public static int lcm(int a, int b) throws MathArithmeticException {
        if (a == 0 || b == 0) {
            return 0;
        }
        int lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));
        if (lcm == Integer.MIN_VALUE) {
            throw new MathArithmeticException(LocalizedFormats.LCM_OVERFLOW_32_BITS, a, b);
        }
        return lcm;
    }

    /**
     * Returns the least common multiple of the absolute value of two numbers, using the formula
     * {@code lcm(a,b) = (a / gcd(a,b)) * b}. Special cases:
     *
     * <ul>
     *   <li>The invocations {@code lcm(Long.MIN_VALUE, n)} and {@code lcm(n, Long.MIN_VALUE)},
     *       where {@code abs(n)} is a power of 2, throw an {@code ArithmeticException}, because the
     *       result would be 2^63, which is too large for an int value.
     *   <li>The result of {@code lcm(0L, x)} and {@code lcm(x, 0L)} is {@code 0L} for any {@code
     *       x}.
     * </ul>
     *
     * @param a Number.
     * @param b Number.
     * @return the least common multiple, never negative.
     * @throws MathArithmeticException if the result cannot be represented as a non-negative {@code
     *     long} value.
     * @since 2.1
     */
    public static long lcm(long a, long b) throws MathArithmeticException {
        if (a == 0 || b == 0) {
            return 0;
        }
        long lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));
        if (lcm == Long.MIN_VALUE) {
            throw new MathArithmeticException(LocalizedFormats.LCM_OVERFLOW_64_BITS, a, b);
        }
        return lcm;
    }

    /**
     * Multiply two integers, checking for overflow.
     *
     * @param x Factor.
     * @param y Factor.
     * @return the product {@code x * y}.
     * @throws MathArithmeticException if the result can not be represented as an {@code int}.
     * @since 1.1
     */
    public static int mulAndCheck(int x, int y) throws MathArithmeticException {
        long m = ((long) x) * ((long) y);
        if (m < Integer.MIN_VALUE || m > Integer.MAX_VALUE) {
            throw new MathArithmeticException();
        }
        return (int) m;
    }

    /**
     * Multiply two long integers, checking for overflow.
     *
     * @param a Factor.
     * @param b Factor.
     * @return the product {@code a * b}.
     * @throws MathArithmeticException if the result can not be represented as a {@code long}.
     * @since 1.2
     */
    public static long mulAndCheck(long a, long b) throws MathArithmeticException {
        long ret;
        if (a > b) {
            // use symmetry to reduce boundary cases
            ret = mulAndCheck(b, a);
        } else {
            if (a < 0) {
                if (b < 0) {
                    // check for positive overflow with negative a, negative b
                    if (a >= Long.MAX_VALUE / b) {
                        ret = a * b;
                    } else {
                        throw new MathArithmeticException();
                    }
                } else if (b > 0) {
                    // check for negative overflow with negative a, positive b
                    if (Long.MIN_VALUE / b <= a) {
                        ret = a * b;
                    } else {
                        throw new MathArithmeticException();
                    }
                } else {
                    // assert b == 0
                    ret = 0;
                }
            } else if (a > 0) {
                // assert a > 0
                // assert b > 0

                // check for positive overflow with positive a, positive b
                if (a <= Long.MAX_VALUE / b) {
                    ret = a * b;
                } else {
                    throw new MathArithmeticException();
                }
            } else {
                // assert a == 0
                ret = 0;
            }
        }
        return ret;
    }

    /**
     * Subtract two integers, checking for overflow.
     *
     * @param x Minuend.
     * @param y Subtrahend.
     * @return the difference {@code x - y}.
     * @throws MathArithmeticException if the result can not be represented as an {@code int}.
     * @since 1.1
     */
    public static int subAndCheck(int x, int y) throws MathArithmeticException {
        long s = (long) x - (long) y;
        if (s < Integer.MIN_VALUE || s > Integer.MAX_VALUE) {
            throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_SUBTRACTION, x, y);
        }
        return (int) s;
    }

    /**
     * Subtract two long integers, checking for overflow.
     *
     * @param a Value.
     * @param b Value.
     * @return the difference {@code a - b}.
     * @throws MathArithmeticException if the result can not be represented as a {@code long}.
     * @since 1.2
     */
    public static long subAndCheck(long a, long b) throws MathArithmeticException {
        long ret;
        if (b == Long.MIN_VALUE) {
            if (a < 0) {
                ret = a - b;
            } else {
                throw new MathArithmeticException(LocalizedFormats.OVERFLOW_IN_ADDITION, a, -b);
            }
        } else {
            // use additive inverse
            ret = addAndCheck(a, -b, LocalizedFormats.OVERFLOW_IN_ADDITION);
        }
        return ret;
    }

    /**
     * Raise an int to an int power.
     *
     * @param k Number to raise.
     * @param e Exponent (must be positive or zero).
     * @return \( k^e \)
     * @throws NotPositiveException if {@code e < 0}.
     * @throws MathArithmeticException if the result would overflow.
     */
    public static int pow(final int k, final int e)
            throws NotPositiveException, MathArithmeticException {
        if (e < 0) {
            throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
        }

        try {
            int exp = e;
            int result = 1;
            int k2p = k;
            while (true) {
                if ((exp & 0x1) != 0) {
                    result = mulAndCheck(result, k2p);
                }

                exp >>= 1;
                if (exp == 0) {
                    break;
                }

                k2p = mulAndCheck(k2p, k2p);
            }

            return result;
        } catch (MathArithmeticException mae) {
            // Add context information.
            mae.getContext().addMessage(LocalizedFormats.OVERFLOW);
            mae.getContext().addMessage(LocalizedFormats.BASE, k);
            mae.getContext().addMessage(LocalizedFormats.EXPONENT, e);

            // Rethrow.
            throw mae;
        }
    }

    /**
     * Raise an int to a long power.
     *
     * @param k Number to raise.
     * @param e Exponent (must be positive or zero).
     * @return k<sup>e</sup>
     * @throws NotPositiveException if {@code e < 0}.
     * @deprecated As of 3.3. Please use {@link #pow(int,int)} instead.
     */
    @Deprecated
    public static int pow(final int k, long e) throws NotPositiveException {
        if (e < 0) {
            throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
        }

        int result = 1;
        int k2p = k;
        while (e != 0) {
            if ((e & 0x1) != 0) {
                result *= k2p;
            }
            k2p *= k2p;
            e >>= 1;
        }

        return result;
    }

    /**
     * Raise a long to an int power.
     *
     * @param k Number to raise.
     * @param e Exponent (must be positive or zero).
     * @return \( k^e \)
     * @throws NotPositiveException if {@code e < 0}.
     * @throws MathArithmeticException if the result would overflow.
     */
    public static long pow(final long k, final int e)
            throws NotPositiveException, MathArithmeticException {
        if (e < 0) {
            throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
        }

        try {
            int exp = e;
            long result = 1;
            long k2p = k;
            while (true) {
                if ((exp & 0x1) != 0) {
                    result = mulAndCheck(result, k2p);
                }

                exp >>= 1;
                if (exp == 0) {
                    break;
                }

                k2p = mulAndCheck(k2p, k2p);
            }

            return result;
        } catch (MathArithmeticException mae) {
            // Add context information.
            mae.getContext().addMessage(LocalizedFormats.OVERFLOW);
            mae.getContext().addMessage(LocalizedFormats.BASE, k);
            mae.getContext().addMessage(LocalizedFormats.EXPONENT, e);

            // Rethrow.
            throw mae;
        }
    }

    /**
     * Raise a long to a long power.
     *
     * @param k Number to raise.
     * @param e Exponent (must be positive or zero).
     * @return k<sup>e</sup>
     * @throws NotPositiveException if {@code e < 0}.
     * @deprecated As of 3.3. Please use {@link #pow(long,int)} instead.
     */
    @Deprecated
    public static long pow(final long k, long e) throws NotPositiveException {
        if (e < 0) {
            throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
        }

        long result = 1l;
        long k2p = k;
        while (e != 0) {
            if ((e & 0x1) != 0) {
                result *= k2p;
            }
            k2p *= k2p;
            e >>= 1;
        }

        return result;
    }

    /**
     * Raise a BigInteger to an int power.
     *
     * @param k Number to raise.
     * @param e Exponent (must be positive or zero).
     * @return k<sup>e</sup>
     * @throws NotPositiveException if {@code e < 0}.
     */
    public static BigInteger pow(final BigInteger k, int e) throws NotPositiveException {
        if (e < 0) {
            throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
        }

        return k.pow(e);
    }

    /**
     * Raise a BigInteger to a long power.
     *
     * @param k Number to raise.
     * @param e Exponent (must be positive or zero).
     * @return k<sup>e</sup>
     * @throws NotPositiveException if {@code e < 0}.
     */
    public static BigInteger pow(final BigInteger k, long e) throws NotPositiveException {
        if (e < 0) {
            throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
        }

        BigInteger result = BigInteger.ONE;
        BigInteger k2p = k;
        while (e != 0) {
            if ((e & 0x1) != 0) {
                result = result.multiply(k2p);
            }
            k2p = k2p.multiply(k2p);
            e >>= 1;
        }

        return result;
    }

    /**
     * Raise a BigInteger to a BigInteger power.
     *
     * @param k Number to raise.
     * @param e Exponent (must be positive or zero).
     * @return k<sup>e</sup>
     * @throws NotPositiveException if {@code e < 0}.
     */
    public static BigInteger pow(final BigInteger k, BigInteger e) throws NotPositiveException {
        if (e.compareTo(BigInteger.ZERO) < 0) {
            throw new NotPositiveException(LocalizedFormats.EXPONENT, e);
        }

        BigInteger result = BigInteger.ONE;
        BigInteger k2p = k;
        while (!BigInteger.ZERO.equals(e)) {
            if (e.testBit(0)) {
                result = result.multiply(k2p);
            }
            k2p = k2p.multiply(k2p);
            e = e.shiftRight(1);
        }

        return result;
    }

    /**
     * Returns the <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">
     * Stirling number of the second kind</a>, "{@code S(n,k)}", the number of ways of partitioning
     * an {@code n}-element set into {@code k} non-empty subsets.
     *
     * <p>The preconditions are {@code 0 <= k <= n } (otherwise {@code NotPositiveException} is
     * thrown)
     *
     * @param n the size of the set
     * @param k the number of non-empty subsets
     * @return {@code S(n,k)}
     * @throws NotPositiveException if {@code k < 0}.
     * @throws NumberIsTooLargeException if {@code k > n}.
     * @throws MathArithmeticException if some overflow happens, typically for n exceeding 25 and k
     *     between 20 and n-2 (S(n,n-1) is handled specifically and does not overflow)
     * @since 3.1
     * @deprecated use {@link CombinatoricsUtils#stirlingS2(int, int)}
     */
    @Deprecated
    public static long stirlingS2(final int n, final int k)
            throws NotPositiveException, NumberIsTooLargeException, MathArithmeticException {
        return CombinatoricsUtils.stirlingS2(n, k);
    }

    /**
     * Add two long integers, checking for overflow.
     *
     * @param a Addend.
     * @param b Addend.
     * @param pattern Pattern to use for any thrown exception.
     * @return the sum {@code a + b}.
     * @throws MathArithmeticException if the result cannot be represented as a {@code long}.
     * @since 1.2
     */
    private static long addAndCheck(long a, long b, Localizable pattern)
            throws MathArithmeticException {
        final long result = a + b;
        if (!((a ^ b) < 0 | (a ^ result) >= 0)) {
            throw new MathArithmeticException(pattern, a, b);
        }
        return result;
    }

    /**
     * Returns true if the argument is a power of two.
     *
     * @param n the number to test
     * @return true if the argument is a power of two
     */
    public static boolean isPowerOfTwo(long n) {
        return (n > 0) && ((n & (n - 1)) == 0);
    }
}