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/* * 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.checkNonNegative; import static com.google.common.math.MathPreconditions.checkPositive; import static com.google.common.math.MathPreconditions.checkRoundingUnnecessary; import static java.math.RoundingMode.CEILING; import static java.math.RoundingMode.FLOOR; import static java.math.RoundingMode.HALF_EVEN; import com.google.common.annotations.GwtCompatible; import com.google.common.annotations.GwtIncompatible; import com.google.common.annotations.VisibleForTesting; import java.math.BigDecimal; import java.math.BigInteger; import java.math.RoundingMode; import java.util.ArrayList; import java.util.List; /** * A class for arithmetic on values of type {@code BigInteger}. * * <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 {@code long} can be found in * {@link IntMath} and {@link LongMath} respectively. * * @author Louis Wasserman * @since 11.0 */ @GwtCompatible(emulated = true) public final class BigIntegerMath { /** * Returns {@code true} if {@code x} represents a power of two. */ public static boolean isPowerOfTwo(BigInteger x) { checkNotNull(x); return x.signum() > 0 && x.getLowestSetBit() == x.bitLength() - 1; } /** * 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(BigInteger x, RoundingMode mode) { checkPositive("x", checkNotNull(x)); int logFloor = x.bitLength() - 1; switch (mode) { case UNNECESSARY: checkRoundingUnnecessary(isPowerOfTwo(x)); // fall through case DOWN: case FLOOR: return logFloor; case UP: case CEILING: return isPowerOfTwo(x) ? logFloor : logFloor + 1; case HALF_DOWN: case HALF_UP: case HALF_EVEN: if (logFloor < SQRT2_PRECOMPUTE_THRESHOLD) { BigInteger halfPower = SQRT2_PRECOMPUTED_BITS.shiftRight(SQRT2_PRECOMPUTE_THRESHOLD - logFloor); if (x.compareTo(halfPower) <= 0) { return logFloor; } else { return logFloor + 1; } } /* * Since sqrt(2) is irrational, log2(x) - logFloor cannot be exactly 0.5 * * To determine which side of logFloor.5 the logarithm is, we compare x^2 to 2^(2 * * logFloor + 1). */ BigInteger x2 = x.pow(2); int logX2Floor = x2.bitLength() - 1; return (logX2Floor < 2 * logFloor + 1) ? logFloor : logFloor + 1; default: throw new AssertionError(); } } /* * The maximum number of bits in a square root for which we'll precompute an explicit half power * of two. This can be any value, but higher values incur more class load time and linearly * increasing memory consumption. */ @VisibleForTesting static final int SQRT2_PRECOMPUTE_THRESHOLD = 256; @VisibleForTesting static final BigInteger SQRT2_PRECOMPUTED_BITS = new BigInteger( "16a09e667f3bcc908b2fb1366ea957d3e3adec17512775099da2f590b0667322a", 16); /** * 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") public static int log10(BigInteger x, RoundingMode mode) { checkPositive("x", x); if (fitsInLong(x)) { return LongMath.log10(x.longValue(), mode); } int approxLog10 = (int) (log2(x, FLOOR) * LN_2 / LN_10); BigInteger approxPow = BigInteger.TEN.pow(approxLog10); int approxCmp = approxPow.compareTo(x); /* * We adjust approxLog10 and approxPow until they're equal to floor(log10(x)) and * 10^floor(log10(x)). */ if (approxCmp > 0) { /* * The code is written so that even completely incorrect approximations will still yield the * correct answer eventually, but in practice this branch should almost never be entered, * and even then the loop should not run more than once. */ do { approxLog10--; approxPow = approxPow.divide(BigInteger.TEN); approxCmp = approxPow.compareTo(x); } while (approxCmp > 0); } else { BigInteger nextPow = BigInteger.TEN.multiply(approxPow); int nextCmp = nextPow.compareTo(x); while (nextCmp <= 0) { approxLog10++; approxPow = nextPow; approxCmp = nextCmp; nextPow = BigInteger.TEN.multiply(approxPow); nextCmp = nextPow.compareTo(x); } } int floorLog = approxLog10; BigInteger floorPow = approxPow; int floorCmp = approxCmp; switch (mode) { case UNNECESSARY: checkRoundingUnnecessary(floorCmp == 0); // fall through case FLOOR: case DOWN: return floorLog; case CEILING: case UP: return floorPow.equals(x) ? floorLog : floorLog + 1; case HALF_DOWN: case HALF_UP: case HALF_EVEN: // Since sqrt(10) is irrational, log10(x) - floorLog can never be exactly 0.5 BigInteger x2 = x.pow(2); BigInteger halfPowerSquared = floorPow.pow(2).multiply(BigInteger.TEN); return (x2.compareTo(halfPowerSquared) <= 0) ? floorLog : floorLog + 1; default: throw new AssertionError(); } } private static final double LN_10 = Math.log(10); private static final double LN_2 = Math.log(2); /** * 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 BigInteger sqrt(BigInteger x, RoundingMode mode) { checkNonNegative("x", x); if (fitsInLong(x)) { return BigInteger.valueOf(LongMath.sqrt(x.longValue(), mode)); } BigInteger sqrtFloor = sqrtFloor(x); switch (mode) { case UNNECESSARY: checkRoundingUnnecessary(sqrtFloor.pow(2).equals(x)); // fall through case FLOOR: case DOWN: return sqrtFloor; case CEILING: case UP: int sqrtFloorInt = sqrtFloor.intValue(); boolean sqrtFloorIsExact = (sqrtFloorInt * sqrtFloorInt == x.intValue()) // fast check mod 2^32 && sqrtFloor.pow(2).equals(x); // slow exact check return sqrtFloorIsExact ? sqrtFloor : sqrtFloor.add(BigInteger.ONE); case HALF_DOWN: case HALF_UP: case HALF_EVEN: BigInteger halfSquare = sqrtFloor.pow(2).add(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. */ return (halfSquare.compareTo(x) >= 0) ? sqrtFloor : sqrtFloor.add(BigInteger.ONE); default: throw new AssertionError(); } } @GwtIncompatible("TODO") private static BigInteger sqrtFloor(BigInteger x) { /* * Adapted from Hacker's Delight, Figure 11-1. * * Using DoubleUtils.bigToDouble, getting a double approximation of x is extremely fast, and * then we can get a double approximation of the square root. Then, we iteratively improve this * guess with an application of Newton's method, which sets guess := (guess + (x / guess)) / 2. * This iteration has the following two properties: * * a) every iteration (except potentially the first) has guess >= floor(sqrt(x)). This is * because guess' is the arithmetic mean of guess and x / guess, sqrt(x) is the geometric mean, * and the arithmetic mean is always higher than the geometric mean. * * b) this iteration converges to floor(sqrt(x)). In fact, the number of correct digits doubles * with each iteration, so this algorithm takes O(log(digits)) iterations. * * We start out with a double-precision approximation, which may be higher or lower than the * true value. Therefore, we perform at least one Newton iteration to get a guess that's * definitely >= floor(sqrt(x)), and then continue the iteration until we reach a fixed point. */ BigInteger sqrt0; int log2 = log2(x, FLOOR); if (log2 < Double.MAX_EXPONENT) { sqrt0 = sqrtApproxWithDoubles(x); } else { int shift = (log2 - DoubleUtils.SIGNIFICAND_BITS) & ~1; // even! /* * We have that x / 2^shift < 2^54. Our initial approximation to sqrtFloor(x) will be * 2^(shift/2) * sqrtApproxWithDoubles(x / 2^shift). */ sqrt0 = sqrtApproxWithDoubles(x.shiftRight(shift)).shiftLeft(shift >> 1); } BigInteger sqrt1 = sqrt0.add(x.divide(sqrt0)).shiftRight(1); if (sqrt0.equals(sqrt1)) { return sqrt0; } do { sqrt0 = sqrt1; sqrt1 = sqrt0.add(x.divide(sqrt0)).shiftRight(1); } while (sqrt1.compareTo(sqrt0) < 0); return sqrt0; } @GwtIncompatible("TODO") private static BigInteger sqrtApproxWithDoubles(BigInteger x) { return DoubleMath.roundToBigInteger(Math.sqrt(DoubleUtils.bigToDouble(x)), HALF_EVEN); } /** * 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") public static BigInteger divide(BigInteger p, BigInteger q, RoundingMode mode) { BigDecimal pDec = new BigDecimal(p); BigDecimal qDec = new BigDecimal(q); return pDec.divide(qDec, 0, mode).toBigIntegerExact(); } /** * Returns {@code n!}, that is, the product of the first {@code n} positive * integers, or {@code 1} if {@code n == 0}. * * <p><b>Warning:</b> the result takes <i>O(n log n)</i> space, so use cautiously. * * <p>This uses an efficient binary recursive algorithm to compute the factorial * with balanced multiplies. It also removes all the 2s from the intermediate * products (shifting them back in at the end). * * @throws IllegalArgumentException if {@code n < 0} */ public static BigInteger factorial(int n) { checkNonNegative("n", n); // If the factorial is small enough, just use LongMath to do it. if (n < LongMath.factorials.length) { return BigInteger.valueOf(LongMath.factorials[n]); } // Pre-allocate space for our list of intermediate BigIntegers. int approxSize = IntMath.divide(n * IntMath.log2(n, CEILING), Long.SIZE, CEILING); ArrayList<BigInteger> bignums = new ArrayList<BigInteger>(approxSize); // Start from the pre-computed maximum long factorial. int startingNumber = LongMath.factorials.length; long product = LongMath.factorials[startingNumber - 1]; // Strip off 2s from this value. int shift = Long.numberOfTrailingZeros(product); product >>= shift; // Use floor(log2(num)) + 1 to prevent overflow of multiplication. int productBits = LongMath.log2(product, FLOOR) + 1; int bits = LongMath.log2(startingNumber, FLOOR) + 1; // Check for the next power of two boundary, to save us a CLZ operation. int nextPowerOfTwo = 1 << (bits - 1); // Iteratively multiply the longs as big as they can go. for (long num = startingNumber; num <= n; num++) { // Check to see if the floor(log2(num)) + 1 has changed. if ((num & nextPowerOfTwo) != 0) { nextPowerOfTwo <<= 1; bits++; } // Get rid of the 2s in num. int tz = Long.numberOfTrailingZeros(num); long normalizedNum = num >> tz; shift += tz; // Adjust floor(log2(num)) + 1. int normalizedBits = bits - tz; // If it won't fit in a long, then we store off the intermediate product. if (normalizedBits + productBits >= Long.SIZE) { bignums.add(BigInteger.valueOf(product)); product = 1; productBits = 0; } product *= normalizedNum; productBits = LongMath.log2(product, FLOOR) + 1; } // Check for leftovers. if (product > 1) { bignums.add(BigInteger.valueOf(product)); } // Efficiently multiply all the intermediate products together. return listProduct(bignums).shiftLeft(shift); } static BigInteger listProduct(List<BigInteger> nums) { return listProduct(nums, 0, nums.size()); } static BigInteger listProduct(List<BigInteger> nums, int start, int end) { switch (end - start) { case 0: return BigInteger.ONE; case 1: return nums.get(start); case 2: return nums.get(start).multiply(nums.get(start + 1)); case 3: return nums.get(start).multiply(nums.get(start + 1)).multiply(nums.get(start + 2)); default: // Otherwise, split the list in half and recursively do this. int m = (end + start) >>> 1; return listProduct(nums, start, m).multiply(listProduct(nums, m, end)); } } /** * Returns {@code n} choose {@code k}, also known as the binomial coefficient of {@code n} and * {@code k}, that is, {@code n! / (k! (n - k)!)}. * * <p><b>Warning:</b> the result can take as much as <i>O(k log n)</i> space. * * @throws IllegalArgumentException if {@code n < 0}, {@code k < 0}, or {@code k > n} */ public static BigInteger 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; } if (k < LongMath.biggestBinomials.length && n <= LongMath.biggestBinomials[k]) { return BigInteger.valueOf(LongMath.binomial(n, k)); } BigInteger accum = BigInteger.ONE; long numeratorAccum = n; long denominatorAccum = 1; int bits = LongMath.log2(n, RoundingMode.CEILING); int numeratorBits = bits; for (int i = 1; i < k; i++) { int p = n - i; int q = i + 1; // log2(p) >= bits - 1, because p >= n/2 if (numeratorBits + bits >= Long.SIZE - 1) { // The numerator is as big as it can get without risking overflow. // Multiply numeratorAccum / denominatorAccum into accum. accum = accum.multiply(BigInteger.valueOf(numeratorAccum)) .divide(BigInteger.valueOf(denominatorAccum)); numeratorAccum = p; denominatorAccum = q; numeratorBits = bits; } else { // We can definitely multiply into the long accumulators without overflowing them. numeratorAccum *= p; denominatorAccum *= q; numeratorBits += bits; } } return accum.multiply(BigInteger.valueOf(numeratorAccum)).divide(BigInteger.valueOf(denominatorAccum)); } // Returns true if BigInteger.valueOf(x.longValue()).equals(x). @GwtIncompatible("TODO") static boolean fitsInLong(BigInteger x) { return x.bitLength() <= Long.SIZE - 1; } private BigIntegerMath() { } }