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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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 java.math.BigInteger;
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;
/**
* Some useful, arithmetics related, additions to the built-in functions in
* {@link Math}.
*
* @version $Id: ArithmeticUtils.java 1244107 2012-02-14 16:17:55Z erans $
*/
public final class ArithmeticUtils {
/** All long-representable factorials */
static final long[] FACTORIALS = new long[] {
1l, 1l, 2l,
6l, 24l, 120l,
720l, 5040l, 40320l,
362880l, 3628800l, 39916800l,
479001600l, 6227020800l, 87178291200l,
1307674368000l, 20922789888000l, 355687428096000l,
6402373705728000l, 121645100408832000l, 2432902008176640000l };
/** 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) {
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) {
return ArithmeticUtils.addAndCheck(a, b, LocalizedFormats.OVERFLOW_IN_ADDITION);
}
/**
* Returns an exact representation of the Binomial
* Coefficient, "{@code n choose k}", the number of
* {@code k}-element subsets that can be selected from an
* {@code n}-element set.
*
* Preconditions:
*
* - {@code 0 <= k <= n } (otherwise
* {@code IllegalArgumentException} is thrown)
* - 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.
*
*
* @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.
*/
public static long binomialCoefficient(final int n, final int k) {
ArithmeticUtils.checkBinomial(n, k);
if ((n == k) || (k == 0)) {
return 1;
}
if ((k == 1) || (k == n - 1)) {
return n;
}
// Use symmetry for large k
if (k > n / 2) {
return binomialCoefficient(n, n - k);
}
// We use the formula
// (n choose k) = n! / (n-k)! / k!
// (n choose k) == ((n-k+1)*...*n) / (1*...*k)
// which could be written
// (n choose k) == (n-1 choose k-1) * n / k
long result = 1;
if (n <= 61) {
// For n <= 61, the naive implementation cannot overflow.
int i = n - k + 1;
for (int j = 1; j <= k; j++) {
result = result * i / j;
i++;
}
} else if (n <= 66) {
// For n > 61 but n <= 66, the result cannot overflow,
// but we must take care not to overflow intermediate values.
int i = n - k + 1;
for (int j = 1; j <= k; j++) {
// We know that (result * i) is divisible by j,
// but (result * i) may overflow, so we split j:
// Filter out the gcd, d, so j/d and i/d are integer.
// result is divisible by (j/d) because (j/d)
// is relative prime to (i/d) and is a divisor of
// result * (i/d).
final long d = gcd(i, j);
result = (result / (j / d)) * (i / d);
i++;
}
} else {
// For n > 66, a result overflow might occur, so we check
// the multiplication, taking care to not overflow
// unnecessary.
int i = n - k + 1;
for (int j = 1; j <= k; j++) {
final long d = gcd(i, j);
result = mulAndCheck(result / (j / d), i / d);
i++;
}
}
return result;
}
/**
* Returns a {@code double} representation of the Binomial
* Coefficient, "{@code n choose k}", the number of
* {@code k}-element subsets that can be selected from an
* {@code n}-element set.
*
* Preconditions:
*
* - {@code 0 <= k <= n } (otherwise
* {@code IllegalArgumentException} is thrown)
* - 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
*
*
* @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}.
*/
public static double binomialCoefficientDouble(final int n, final int k) {
ArithmeticUtils.checkBinomial(n, k);
if ((n == k) || (k == 0)) {
return 1d;
}
if ((k == 1) || (k == n - 1)) {
return n;
}
if (k > n/2) {
return binomialCoefficientDouble(n, n - k);
}
if (n < 67) {
return binomialCoefficient(n,k);
}
double result = 1d;
for (int i = 1; i <= k; i++) {
result *= (double)(n - k + i) / (double)i;
}
return FastMath.floor(result + 0.5);
}
/**
* Returns the natural {@code log} of the Binomial
* Coefficient, "{@code n choose k}", the number of
* {@code k}-element subsets that can be selected from an
* {@code n}-element set.
*
* Preconditions:
*
* - {@code 0 <= k <= n } (otherwise
* {@code IllegalArgumentException} is thrown)
*
*
* @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}.
*/
public static double binomialCoefficientLog(final int n, final int k) {
ArithmeticUtils.checkBinomial(n, k);
if ((n == k) || (k == 0)) {
return 0;
}
if ((k == 1) || (k == n - 1)) {
return FastMath.log(n);
}
/*
* For values small enough to do exact integer computation,
* return the log of the exact value
*/
if (n < 67) {
return FastMath.log(binomialCoefficient(n,k));
}
/*
* Return the log of binomialCoefficientDouble for values that will not
* overflow binomialCoefficientDouble
*/
if (n < 1030) {
return FastMath.log(binomialCoefficientDouble(n, k));
}
if (k > n / 2) {
return binomialCoefficientLog(n, n - k);
}
/*
* Sum logs for values that could overflow
*/
double logSum = 0;
// n!/(n-k)!
for (int i = n - k + 1; i <= n; i++) {
logSum += FastMath.log(i);
}
// divide by k!
for (int i = 2; i <= k; i++) {
logSum -= FastMath.log(i);
}
return logSum;
}
/**
* Returns n!. Shorthand for {@code n} Factorial, the
* product of the numbers {@code 1,...,n}.
*
* Preconditions:
*
* - {@code n >= 0} (otherwise
* {@code IllegalArgumentException} is thrown)
* - 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.
*
*
*
* @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}.
*/
public static long factorial(final int n) {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n > 20) {
throw new MathArithmeticException();
}
return FACTORIALS[n];
}
/**
* Compute n!, the
* factorial 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}.
*/
public static double factorialDouble(final int n) {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n < 21) {
return factorial(n);
}
return FastMath.floor(FastMath.exp(ArithmeticUtils.factorialLog(n)) + 0.5);
}
/**
* Compute the natural logarithm of the factorial of {@code n}.
*
* @param n Argument.
* @return {@code n!}
* @throws NotPositiveException if {@code n < 0}.
*/
public static double factorialLog(final int n) {
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.FACTORIAL_NEGATIVE_PARAMETER,
n);
}
if (n < 21) {
return FastMath.log(factorial(n));
}
double logSum = 0;
for (int i = 2; i <= n; i++) {
logSum += FastMath.log(i);
}
return logSum;
}
/**
*
* 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:
*
* - 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.
* - 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.
*
- The invocation {@code gcd(0, 0)} is the only one which returns
* {@code 0}.
*
*
* @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(final int p, final int q) {
int u = p;
int v = q;
if ((u == 0) || (v == 0)) {
if ((u == Integer.MIN_VALUE) || (v == Integer.MIN_VALUE)) {
throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,
p, q);
}
return FastMath.abs(u) + FastMath.abs(v);
}
// keep u and v negative, as negative integers range down to
// -2^31, while positive numbers can only be as large as 2^31-1
// (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 < 31) { // while u and v are
// both even...
u /= 2;
v /= 2;
k++; // cast out twos.
}
if (k == 31) {
throw new MathArithmeticException(LocalizedFormats.GCD_OVERFLOW_32_BITS,
p, q);
}
// B2. Initialize: u and v have been divided by 2^k and at least
// one is odd.
int 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 * (1 << k); // gcd is u*2^k
}
/**
*
* 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:
*
* - 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.
* - 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.
*
- The invocation {@code gcd(0L, 0L)} is the only one which returns
* {@code 0L}.
*
*
* @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) {
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:
*
* - 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.
* - The result of {@code lcm(0, x)} and {@code lcm(x, 0)} is
* {@code 0} for any {@code x}.
*
*
* @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) {
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:
*
* - 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.
* - The result of {@code lcm(0L, x)} and {@code lcm(x, 0L)} is
* {@code 0L} for any {@code x}.
*
*
* @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) {
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) {
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) {
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) {
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) {
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 ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static int pow(final int k, int e) {
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 = e >> 1;
}
return result;
}
/**
* Raise an int to a long power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static int pow(final int k, long e) {
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 = 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 ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static long pow(final long k, int e) {
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 = e >> 1;
}
return result;
}
/**
* Raise a long to a long power.
*
* @param k Number to raise.
* @param e Exponent (must be positive or zero).
* @return ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static long pow(final long k, long e) {
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 = 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 ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static BigInteger pow(final BigInteger k, int e) {
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 ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static BigInteger pow(final BigInteger k, long e) {
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 = 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 ke
* @throws NotPositiveException if {@code e < 0}.
*/
public static BigInteger pow(final BigInteger k, BigInteger e) {
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;
}
/**
* 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) {
long ret;
if (a > b) {
// use symmetry to reduce boundary cases
ret = addAndCheck(b, a, pattern);
} else {
// assert a <= b
if (a < 0) {
if (b < 0) {
// check for negative overflow
if (Long.MIN_VALUE - b <= a) {
ret = a + b;
} else {
throw new MathArithmeticException(pattern, a, b);
}
} else {
// opposite sign addition is always safe
ret = a + b;
}
} else {
// assert a >= 0
// assert b >= 0
// check for positive overflow
if (a <= Long.MAX_VALUE - b) {
ret = a + b;
} else {
throw new MathArithmeticException(pattern, a, b);
}
}
}
return ret;
}
/**
* Check binomial preconditions.
*
* @param n Size of the set.
* @param k Size of the subsets to be counted.
* @throws NotPositiveException if {@code n < 0}.
* @throws NumberIsTooLargeException if {@code k > n}.
*/
private static void checkBinomial(final int n, final int k) {
if (n < k) {
throw new NumberIsTooLargeException(LocalizedFormats.BINOMIAL_INVALID_PARAMETERS_ORDER,
k, n, true);
}
if (n < 0) {
throw new NotPositiveException(LocalizedFormats.BINOMIAL_NEGATIVE_PARAMETER, n);
}
}
/**
* 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);
}
}