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/*
* Copyright 2009 Google Inc.
*
* 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.
*/
/*
* 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.
*
* INCLUDES MODIFICATIONS BY RICHARD ZSCHECH AS WELL AS GOOGLE.
*/
package java.math;
/**
* Static library that provides all multiplication of {@link BigInteger}
* methods.
*/
class Multiplication {
/**
* An array with the first powers of five in {@code BigInteger} version. (
* {@code 5^0,5^1,...,5^31})
*/
static final BigInteger bigFivePows[] = new BigInteger[32];
/**
* An array with the first powers of ten in {@code BigInteger} version. (
* {@code 10^0,10^1,...,10^31})
*/
static final BigInteger[] bigTenPows = new BigInteger[32];
/**
* An array with powers of five that fit in the type {@code int}. ({@code
* 5^0,5^1,...,5^13})
*/
static final int fivePows[] = {
1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625,
48828125, 244140625, 1220703125};
/**
* An array with powers of ten that fit in the type {@code int}. ({@code
* 10^0,10^1,...,10^9})
*/
static final int tenPows[] = {
1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000};
/**
* Break point in digits (number of {@code int} elements) between Karatsuba
* and Pencil and Paper multiply.
*/
static final int whenUseKaratsuba = 63; // an heuristic value
static {
int i;
long fivePow = 1L;
for (i = 0; i <= 18; i++) {
bigFivePows[i] = BigInteger.valueOf(fivePow);
bigTenPows[i] = BigInteger.valueOf(fivePow << i);
fivePow *= 5;
}
for (; i < bigTenPows.length; i++) {
bigFivePows[i] = bigFivePows[i - 1].multiply(bigFivePows[1]);
bigTenPows[i] = bigTenPows[i - 1].multiply(BigInteger.TEN);
}
}
/**
* Performs the multiplication with the Karatsuba's algorithm. Karatsuba's
* algorithm:
* u = u1 * B + u0
* v = v1 * B + v0
*
*
* u*v = (u1 * v1) * B2 + ((u1 - u0) * (v0 - v1) + u1 * v1 +
* u0 * v0 ) * B + u0 * v0
*
*
* @param op1 first factor of the product
* @param op2 second factor of the product
* @return {@code op1 * op2}
* @see #multiply(BigInteger, BigInteger)
*/
static BigInteger karatsuba(BigInteger op1, BigInteger op2) {
BigInteger temp;
if (op2.numberLength > op1.numberLength) {
temp = op1;
op1 = op2;
op2 = temp;
}
if (op2.numberLength < whenUseKaratsuba) {
return multiplyPAP(op1, op2);
}
/*
* Karatsuba: u = u1*B + u0 v = v1*B + v0 u*v = (u1*v1)*B^2 +
* ((u1-u0)*(v0-v1) + u1*v1 + u0*v0)*B + u0*v0
*/
// ndiv2 = (op1.numberLength / 2) * 32
int ndiv2 = (op1.numberLength & 0xFFFFFFFE) << 4;
BigInteger upperOp1 = op1.shiftRight(ndiv2);
BigInteger upperOp2 = op2.shiftRight(ndiv2);
BigInteger lowerOp1 = op1.subtract(upperOp1.shiftLeft(ndiv2));
BigInteger lowerOp2 = op2.subtract(upperOp2.shiftLeft(ndiv2));
BigInteger upper = karatsuba(upperOp1, upperOp2);
BigInteger lower = karatsuba(lowerOp1, lowerOp2);
BigInteger middle = karatsuba(upperOp1.subtract(lowerOp1),
lowerOp2.subtract(upperOp2));
middle = middle.add(upper).add(lower);
middle = middle.shiftLeft(ndiv2);
upper = upper.shiftLeft(ndiv2 << 1);
return upper.add(middle).add(lower);
}
static void multArraysPAP(int[] aDigits, int aLen, int[] bDigits, int bLen,
int[] resDigits) {
if (aLen == 0 || bLen == 0) {
return;
}
if (aLen == 1) {
resDigits[bLen] = multiplyByInt(resDigits, bDigits, bLen, aDigits[0]);
} else if (bLen == 1) {
resDigits[aLen] = multiplyByInt(resDigits, aDigits, aLen, bDigits[0]);
} else {
multPAP(aDigits, bDigits, resDigits, aLen, bLen);
}
}
/**
* Performs a multiplication of two BigInteger and hides the algorithm used.
*
* @see BigInteger#multiply(BigInteger)
*/
static BigInteger multiply(BigInteger x, BigInteger y) {
return karatsuba(x, y);
}
/**
* Multiplies a number by a power of five. This method is used in {@code
* BigDecimal} class.
*
* @param val the number to be multiplied
* @param exp a positive {@code int} exponent
* @return {@code val * 5exp}
*/
static BigInteger multiplyByFivePow(BigInteger val, int exp) {
// PRE: exp >= 0
if (exp < fivePows.length) {
return multiplyByPositiveInt(val, fivePows[exp]);
} else if (exp < bigFivePows.length) {
return val.multiply(bigFivePows[exp]);
} else {
// Large powers of five
return val.multiply(bigFivePows[1].pow(exp));
}
}
/**
* Multiplies an array of integers by an integer value.
*
* @param a the array of integers
* @param aSize the number of elements of intArray to be multiplied
* @param factor the multiplier
* @return the top digit of production
*/
static int multiplyByInt(int a[], final int aSize, final int factor) {
return multiplyByInt(a, a, aSize, factor);
}
/**
* Multiplies a number by a positive integer.
*
* @param val an arbitrary {@code BigInteger}
* @param factor a positive {@code int} number
* @return {@code val * factor}
*/
static BigInteger multiplyByPositiveInt(BigInteger val, int factor) {
int resSign = val.sign;
if (resSign == 0) {
return BigInteger.ZERO;
}
int aNumberLength = val.numberLength;
int[] aDigits = val.digits;
if (aNumberLength == 1) {
long res = unsignedMultAddAdd(aDigits[0], factor, 0, 0);
int resLo = (int) res;
int resHi = (int) (res >>> 32);
return ((resHi == 0) ? new BigInteger(resSign, resLo) : new BigInteger(
resSign, 2, new int[] {resLo, resHi}));
}
// Common case
int resLength = aNumberLength + 1;
int resDigits[] = new int[resLength];
resDigits[aNumberLength] = multiplyByInt(resDigits, aDigits, aNumberLength,
factor);
BigInteger result = new BigInteger(resSign, resLength, resDigits);
result.cutOffLeadingZeroes();
return result;
}
/**
* Multiplies a number by a power of ten. This method is used in {@code
* BigDecimal} class.
*
* @param val the number to be multiplied
* @param exp a positive {@code long} exponent
* @return {@code val * 10exp}
*/
static BigInteger multiplyByTenPow(BigInteger val, int exp) {
// PRE: exp >= 0
return ((exp < tenPows.length) ? multiplyByPositiveInt(val,
tenPows[(int) exp]) : val.multiply(powerOf10(exp)));
}
/**
* Multiplies two BigIntegers. Implements traditional scholar algorithm
* described by Knuth.
*
*
*
*
*
*
*
*
*
A=
*
a3
*
a2
*
a1
*
a0
*
*
*
*
*
*
B=
*
*
b2
*
b1
*
b1
*
*
*
*
*
*
*
*
*
b0*a3
*
b0*a2
*
b0*a1
*
b0*a0
*
*
*
*
*
*
b1*a3
*
b1*a2
*
b1*a1
*
b1*a0
*
*
*
*
+
*
b2*a3
*
b2*a2
*
b2*a1
*
b2*a0
*
*
*
*
*
______
*
______
*
______
*
______
*
______
*
______
*
*
*
*
*
A*B=R=
*
r5
*
r4
*
r3
*
r2
*
r1
*
r0
*
*
*
*
*
*
*
*
* @param op1 first factor of the multiplication {@code op1 >= 0}
* @param op2 second factor of the multiplication {@code op2 >= 0}
* @return a {@code BigInteger} of value {@code op1 * op2}
*/
static BigInteger multiplyPAP(BigInteger a, BigInteger b) {
// PRE: a >= b
int aLen = a.numberLength;
int bLen = b.numberLength;
int resLength = aLen + bLen;
int resSign = (a.sign != b.sign) ? -1 : 1;
// A special case when both numbers don't exceed int
if (resLength == 2) {
long val = unsignedMultAddAdd(a.digits[0], b.digits[0], 0, 0);
int valueLo = (int) val;
int valueHi = (int) (val >>> 32);
return ((valueHi == 0) ? new BigInteger(resSign, valueLo)
: new BigInteger(resSign, 2, new int[] {valueLo, valueHi}));
}
int[] aDigits = a.digits;
int[] bDigits = b.digits;
int resDigits[] = new int[resLength];
// Common case
multArraysPAP(aDigits, aLen, bDigits, bLen, resDigits);
BigInteger result = new BigInteger(resSign, resLength, resDigits);
result.cutOffLeadingZeroes();
return result;
}
static void multPAP(int a[], int b[], int t[], int aLen, int bLen) {
if (a == b && aLen == bLen) {
square(a, aLen, t);
return;
}
for (int i = 0; i < aLen; i++) {
long carry = 0;
int aI = a[i];
for (int j = 0; j < bLen; j++) {
carry = unsignedMultAddAdd(aI, b[j], t[i + j], (int) carry);
t[i + j] = (int) carry;
carry >>>= 32;
}
t[i + bLen] = (int) carry;
}
}
static BigInteger pow(BigInteger base, int exponent) {
// PRE: exp > 0
BigInteger res = BigInteger.ONE;
BigInteger acc = base;
for (; exponent > 1; exponent >>= 1) {
if ((exponent & 1) != 0) {
// if odd, multiply one more time by acc
res = res.multiply(acc);
}
// acc = base^(2^i)
// a limit where karatsuba performs a faster square than the square
// algorithm
if (acc.numberLength == 1) {
acc = acc.multiply(acc); // square
} else {
acc = new BigInteger(1, square(acc.digits, acc.numberLength,
new int[acc.numberLength << 1]));
}
}
// exponent == 1, multiply one more time
res = res.multiply(acc);
return res;
}
/**
* It calculates a power of ten, which exponent could be out of 32-bit range.
* Note that internally this method will be used in the worst case with an
* exponent equals to: {@code Integer.MAX_VALUE - Integer.MIN_VALUE}.
*
* @param exp the exponent of power of ten, it must be positive.
* @return a {@code BigInteger} with value {@code 10exp}.
*/
static BigInteger powerOf10(double exp) {
// PRE: exp >= 0
int intExp = (int) exp;
// "SMALL POWERS"
if (exp < bigTenPows.length) {
// The largest power that fit in 'long' type
return bigTenPows[intExp];
} else if (exp <= 50) {
// To calculate: 10^exp
return BigInteger.TEN.pow(intExp);
} else if (exp <= 1000) {
// To calculate: 5^exp * 2^exp
return bigFivePows[1].pow(intExp).shiftLeft(intExp);
}
// "LARGE POWERS"
/*
* To check if there is free memory to allocate a BigInteger of the
* estimated size, measured in bytes: 1 + [exp / log10(2)]
*/
if (exp > 1000000) {
throw new ArithmeticException("power of ten too big"); //$NON-NLS-1$
}
if (exp <= Integer.MAX_VALUE) {
// To calculate: 5^exp * 2^exp
return bigFivePows[1].pow(intExp).shiftLeft(intExp);
}
/*
* "HUGE POWERS"
*
* This branch probably won't be executed since the power of ten is too big.
*/
// To calculate: 5^exp
BigInteger powerOfFive = bigFivePows[1].pow(Integer.MAX_VALUE);
BigInteger res = powerOfFive;
long longExp = (long) (exp - Integer.MAX_VALUE);
intExp = (int) (exp % Integer.MAX_VALUE);
while (longExp > Integer.MAX_VALUE) {
res = res.multiply(powerOfFive);
longExp -= Integer.MAX_VALUE;
}
res = res.multiply(bigFivePows[1].pow(intExp));
// To calculate: 5^exp << exp
res = res.shiftLeft(Integer.MAX_VALUE);
longExp = (long) (exp - Integer.MAX_VALUE);
while (longExp > Integer.MAX_VALUE) {
res = res.shiftLeft(Integer.MAX_VALUE);
longExp -= Integer.MAX_VALUE;
}
res = res.shiftLeft(intExp);
return res;
}
/**
* Performs a2.
*
* @param a The number to square.
* @param aLen The length of the number to square.
*/
static int[] square(int[] a, int aLen, int[] res) {
long carry;
for (int i = 0; i < aLen; i++) {
carry = 0;
for (int j = i + 1; j < aLen; j++) {
carry = unsignedMultAddAdd(a[i], a[j], res[i + j], (int) carry);
res[i + j] = (int) carry;
carry >>>= 32;
}
res[i + aLen] = (int) carry;
}
BitLevel.shiftLeftOneBit(res, res, aLen << 1);
carry = 0;
for (int i = 0, index = 0; i < aLen; i++, index++) {
carry = unsignedMultAddAdd(a[i], a[i], res[index], (int) carry);
res[index] = (int) carry;
carry >>>= 32;
index++;
carry += res[index] & 0xFFFFFFFFL;
res[index] = (int) carry;
carry >>>= 32;
}
return res;
}
/**
* Computes the value unsigned ((uint)a*(uint)b + (uint)c + (uint)d). This
* method could improve the readability and performance of the code.
*
* @param a parameter 1
* @param b parameter 2
* @param c parameter 3
* @param d parameter 4
* @return value of expression
*/
static long unsignedMultAddAdd(int a, int b, int c, int d) {
return (a & 0xFFFFFFFFL) * (b & 0xFFFFFFFFL) + (c & 0xFFFFFFFFL)
+ (d & 0xFFFFFFFFL);
}
/**
* Multiplies an array of integers by an integer value and saves the result in
* {@code res}.
*
* @param a the array of integers
* @param aSize the number of elements of intArray to be multiplied
* @param factor the multiplier
* @return the top digit of production
*/
private static int multiplyByInt(int res[], int a[], final int aSize,
final int factor) {
long carry = 0;
for (int i = 0; i < aSize; i++) {
carry = unsignedMultAddAdd(a[i], factor, (int) carry, 0);
res[i] = (int) carry;
carry >>>= 32;
}
return (int) carry;
}
/**
* Just to denote that this class can't be instantiated.
*/
private Multiplication() {
}
}
