mirror of
https://github.com/moparisthebest/mailiverse
synced 2024-11-22 16:52:24 -05:00
535 lines
16 KiB
Java
535 lines
16 KiB
Java
/*
|
|
* 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. <b>Karatsuba's
|
|
* algorithm:</b> <tt>
|
|
* u = u<sub>1</sub> * B + u<sub>0</sub><br>
|
|
* v = v<sub>1</sub> * B + v<sub>0</sub><br>
|
|
*
|
|
*
|
|
* u*v = (u<sub>1</sub> * v<sub>1</sub>) * B<sub>2</sub> + ((u<sub>1</sub> - u<sub>0</sub>) * (v<sub>0</sub> - v<sub>1</sub>) + u<sub>1</sub> * v<sub>1</sub> +
|
|
* u<sub>0</sub> * v<sub>0</sub> ) * B + u<sub>0</sub> * v<sub>0</sub><br>
|
|
*</tt>
|
|
*
|
|
* @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 * 5<sup>exp</sup>}
|
|
*/
|
|
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 * 10<sup>exp</sup>}
|
|
*/
|
|
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.
|
|
*
|
|
* <br>
|
|
* <tt>
|
|
* <table border="0">
|
|
* <tbody>
|
|
*
|
|
*
|
|
* <tr>
|
|
* <td align="center">A=</td>
|
|
* <td>a<sub>3</sub></td>
|
|
* <td>a<sub>2</sub></td>
|
|
* <td>a<sub>1</sub></td>
|
|
* <td>a<sub>0</sub></td>
|
|
* <td></td>
|
|
* <td></td>
|
|
* </tr>
|
|
*
|
|
* <tr>
|
|
* <td align="center">B=</td>
|
|
* <td></td>
|
|
* <td>b<sub>2</sub></td>
|
|
* <td>b<sub>1</sub></td>
|
|
* <td>b<sub>1</sub></td>
|
|
* <td></td>
|
|
* <td></td>
|
|
* </tr>
|
|
*
|
|
* <tr>
|
|
* <td></td>
|
|
* <td></td>
|
|
* <td></td>
|
|
* <td>b<sub>0</sub>*a<sub>3</sub></td>
|
|
* <td>b<sub>0</sub>*a<sub>2</sub></td>
|
|
* <td>b<sub>0</sub>*a<sub>1</sub></td>
|
|
* <td>b<sub>0</sub>*a<sub>0</sub></td>
|
|
* </tr>
|
|
*
|
|
* <tr>
|
|
* <td></td>
|
|
* <td></td>
|
|
* <td>b<sub>1</sub>*a<sub>3</sub></td>
|
|
* <td>b<sub>1</sub>*a<sub>2</sub></td>
|
|
* <td>b<sub>1</sub>*a1</td>
|
|
* <td>b<sub>1</sub>*a0</td>
|
|
* </tr>
|
|
*
|
|
* <tr>
|
|
* <td>+</td>
|
|
* <td>b<sub>2</sub>*a<sub>3</sub></td>
|
|
* <td>b<sub>2</sub>*a<sub>2</sub></td>
|
|
* <td>b<sub>2</sub>*a<sub>1</sub></td>
|
|
* <td>b<sub>2</sub>*a<sub>0</sub></td>
|
|
* </tr>
|
|
*
|
|
* <tr>
|
|
* <td></td>
|
|
* <td>______</td>
|
|
* <td>______</td>
|
|
* <td>______</td>
|
|
* <td>______</td>
|
|
* <td>______</td>
|
|
* <td>______</td>
|
|
* </tr>
|
|
*
|
|
* <tr>
|
|
*
|
|
* <td align="center">A*B=R=</td>
|
|
* <td align="center">r<sub>5</sub></td>
|
|
* <td align="center">r<sub>4</sub></td>
|
|
* <td align="center">r<sub>3</sub></td>
|
|
* <td align="center">r<sub>2</sub></td>
|
|
* <td align="center">r<sub>1</sub></td>
|
|
* <td align="center">r<sub>0</sub></td>
|
|
* <td></td>
|
|
* </tr>
|
|
*
|
|
* </tbody>
|
|
* </table>
|
|
*
|
|
*</tt>
|
|
*
|
|
* @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 10<sup>exp</sup>}.
|
|
*/
|
|
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 a<sup>2</sup>.
|
|
*
|
|
* @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() {
|
|
}
|
|
|
|
}
|