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Kotlin Multiplatform BigNum library
/*
* Copyright 2019 Ugljesa Jovanovic
*
* 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.ionspin.kotlin.bignum.integer.base63
import com.ionspin.kotlin.bignum.Endianness
import com.ionspin.kotlin.bignum.integer.BigInteger
import com.ionspin.kotlin.bignum.integer.BigIntegerArithmetic
import com.ionspin.kotlin.bignum.integer.Quadruple
import com.ionspin.kotlin.bignum.integer.Sextuple
import com.ionspin.kotlin.bignum.integer.Sign
import com.ionspin.kotlin.bignum.integer.base32.BigInteger32Arithmetic
import com.ionspin.kotlin.bignum.integer.util.toDigit
import com.ionspin.kotlin.bignum.modular.ModularBigInteger
import kotlin.math.absoluteValue
import kotlin.math.ceil
import kotlin.math.floor
/**
* Created by Ugljesa Jovanovic
* [email protected]
* on 10-Mar-2019
*
* Word order is big endian
*/
@ExperimentalUnsignedTypes
internal object BigInteger63InPlaceArithmetic : BigIntegerArithmetic {
override val ZERO: ULongArray = ulongArrayOf(0u)
override val ONE: ULongArray = ulongArrayOf(1u)
override val TWO: ULongArray = ulongArrayOf(2u)
override val TEN: ULongArray = ulongArrayOf(10UL)
val reciprocalOf3In2ToThePowerOf63 = ulongArrayOf(3074457345618258603U)
override val basePowerOfTwo: Int = 63
val wordSizeInBits = 63
val baseMask: ULong = 0x7FFFFFFFFFFFFFFFUL
val baseMaskArray: ULongArray = ulongArrayOf(0x7FFFFFFFFFFFFFFFUL)
val lowMask = 0x00000000FFFFFFFFUL
val highMask = 0x7FFFFFFF00000000UL
val overflowMask = 0x8000000000000000UL
const val karatsubaThreshold = 120
const val toomCookThreshold = 15_000
const val debugOperandSize = true
override fun numberOfLeadingZerosInAWord(value: ULong): Int {
var x = value
var y: ULong
var n = 63
y = x shr 32
if (y != 0UL) {
n = n - 32
x = y
}
y = x shr 16
if (y != 0UL) {
n = n - 16
x = y
}
y = x shr 8
if (y != 0UL) {
n = n - 8
x = y
}
y = x shr 4
if (y != 0UL) {
n = n - 4
x = y
}
y = x shr 2
if (y != 0UL) {
n = n - 2
x = y
}
y = x shr 1
if (y != 0UL) {
return n - 2
}
return n - x.toInt()
}
fun numberOfTrailingZerosInAWord(value: ULong): Int {
var x = value
var y: ULong
var n = 63
y = (x shl 32) and baseMask
if (y != 0UL) {
n -= 32
x = y
}
y = (x shl 16) and baseMask
if (y != 0UL) {
n -= 16
x = y
}
y = (x shl 8) and baseMask
if (y != 0UL) {
n = n - 8
x = y
}
y = (x shl 4) and baseMask
if (y != 0UL) {
n = n - 4
x = y
}
y = (x shl 2) and baseMask
if (y != 0UL) {
n = n - 2
x = y
}
y = (x shl 1) and baseMask
if (y != 0UL) {
return n - 2
}
return n - x.toInt()
}
override fun bitLength(value: ULongArray): Int {
if (value.contentEquals(ZERO)) {
return 0
}
val start = value.size - countLeadingZeroWords(value) - 1
val mostSignificant = value[start]
return bitLength(mostSignificant) + (start) * 63
}
fun bitLength(value: ULong): Int {
return 63 - numberOfLeadingZerosInAWord(value)
}
fun trailingZeroBits(value: ULong): Int {
return numberOfTrailingZerosInAWord(value)
}
override fun trailingZeroBits(value: ULongArray): Int {
if (value.contentEquals(ZERO)) {
return 0
}
val zeroWordsCount = value.takeWhile { it == 0UL }.count()
if (zeroWordsCount == value.size) {
return 0
}
return trailingZeroBits(value[zeroWordsCount]) + (zeroWordsCount * 63)
}
fun removeLeadingZeros(bigInteger: ULongArray): ULongArray {
val firstEmpty = bigInteger.indexOfLast { it != 0UL } + 1
if (firstEmpty == -1 || firstEmpty == 0) {
// Array is equal to zero, so we return array with zero elements
return ZERO
}
if (bigInteger.size == firstEmpty) {
return bigInteger
}
// println("RLZ original array : ${bigInteger.size} contains: ${bigInteger.size - firstEmpty} zeros")
return bigInteger.copyOfRange(0, firstEmpty)
}
fun countLeadingZeroWords(bigInteger: ULongArray): Int {
// Presume there are no leading zeros
var lastNonEmptyIndex = bigInteger.size - 1
// Check if it's an empty array
if (lastNonEmptyIndex <= 0) {
return 0
}
// Get the last element (Word order is high endian so leading zeros are only on highest indexes
var element = bigInteger[lastNonEmptyIndex]
while (element == 0UL && lastNonEmptyIndex > 0) {
lastNonEmptyIndex -= 1
element = bigInteger[lastNonEmptyIndex]
}
if (bigInteger[lastNonEmptyIndex] == 0UL) {
lastNonEmptyIndex -= 1
}
return bigInteger.size - lastNonEmptyIndex - 1
}
override fun shiftLeft(operand: ULongArray, places: Int): ULongArray {
if (operand.contentEquals(ZERO)) {
return operand
}
if (places == 0) {
return operand
}
if (operand.isEmpty()) {
return ZERO
}
val leadingZeroWords = countLeadingZeroWords(operand)
if (operand.size == leadingZeroWords) {
return ZERO
}
val originalSize = operand.size - leadingZeroWords
val leadingZeros =
numberOfLeadingZerosInAWord(operand[originalSize - 1])
val shiftWords = places / basePowerOfTwo
val shiftBits = places % basePowerOfTwo
val wordsNeeded = if (shiftBits > leadingZeros) {
shiftWords + 1
} else {
shiftWords
}
if (shiftBits == 0) {
return ULongArray(originalSize + wordsNeeded) {
when (it) {
in 0 until shiftWords -> 0U
else -> operand[it - shiftWords]
}
}
}
return ULongArray(originalSize + wordsNeeded) {
when (it) {
in 0 until shiftWords -> 0U
shiftWords -> {
(operand[it - shiftWords] shl shiftBits) and baseMask
}
in (shiftWords + 1) until (originalSize + shiftWords) -> {
((operand[it - shiftWords] shl shiftBits) and baseMask) or (operand[it - shiftWords - 1] shr (basePowerOfTwo - shiftBits))
}
originalSize + wordsNeeded - 1 -> {
(operand[it - wordsNeeded] shr (basePowerOfTwo - shiftBits))
}
else -> {
throw RuntimeException("Invalid case $it")
}
}
}
}
override fun shiftRight(operand: ULongArray, places: Int): ULongArray {
if (operand.isEmpty() || places == 0) {
return operand
}
val leadingZeroWords = countLeadingZeroWords(operand)
val realOperandSize = operand.size - leadingZeroWords
val shiftBits = (places % basePowerOfTwo)
val wordsToDiscard = places / basePowerOfTwo
if (wordsToDiscard >= realOperandSize) {
return ZERO
}
if (shiftBits == 0) {
operand.copyOfRange(realOperandSize - wordsToDiscard, realOperandSize)
}
if (realOperandSize > 1 && realOperandSize - wordsToDiscard == 1) {
return ulongArrayOf((operand[realOperandSize - 1] shr shiftBits))
}
val newLength = realOperandSize - wordsToDiscard
if (newLength == 0) {
return ZERO
}
val result = ULongArray(realOperandSize - wordsToDiscard) {
when (it) {
in 0 until (realOperandSize - 1 - wordsToDiscard) -> {
((operand[it + wordsToDiscard] shr shiftBits)) or
((operand[it + wordsToDiscard + 1] shl (basePowerOfTwo - shiftBits) and baseMask))
}
realOperandSize - 1 - wordsToDiscard -> {
(operand[it + wordsToDiscard] shr shiftBits)
}
else -> {
throw RuntimeException("Invalid case $it")
}
}
}
return result
}
fun compareWithStartIndexes(first: ULongArray, second: ULongArray, firstStart: Int, secondStart: Int): Int {
// debugOperandsCheck(first, second)
if (firstStart > secondStart) {
return 1
}
if (secondStart > firstStart) {
return -1
}
var counter = firstStart - 1
var firstIsLarger = false
var bothAreEqual = true
while (counter >= 0) {
if (first[counter] > second[counter]) {
firstIsLarger = true
bothAreEqual = false
break
}
if (first[counter] < second[counter]) {
firstIsLarger = false
bothAreEqual = false
break
}
counter--
}
if (bothAreEqual) {
return 0
}
if (firstIsLarger) {
return 1
} else {
return -1
}
}
override fun compare(first: ULongArray, second: ULongArray): Int {
val firstStart = first.size - countLeadingZeroWords(first)
val secondStart = second.size - countLeadingZeroWords(second)
return compareWithStartIndexes(first, second, firstStart, secondStart)
}
override fun numberOfDecimalDigits(operand: ULongArray): Long {
val bitLenght = bitLength(operand)
val minDigit = ceil((bitLenght - 1) * BigInteger.LOG_10_OF_2)
// val maxDigit = floor(bitLenght * LOG_10_OF_2) + 1
// val correct = this / 10.toBigInteger().pow(maxDigit.toInt())
// return when {
// correct == ZERO -> maxDigit.toInt() - 1
// correct > 0 && correct < 10 -> maxDigit.toInt()
// else -> -1
// }
var tmp = operand / pow(TEN, minDigit.toLong())
var counter = 0L
while (compare(tmp, ZERO) != 0) {
tmp /= TEN
counter++
}
return counter + minDigit.toInt()
}
// private fun addWithInPlaceBuffer TODO start here
override fun add(first: ULongArray, second: ULongArray): ULongArray {
// debugOperandsCheck(first, second)
if (first.size == 1 && first[0] == 0UL) return second
if (second.size == 1 && second[0] == 0UL) return first
val firstStart = first.size - countLeadingZeroWords(first)
val secondStart = second.size - countLeadingZeroWords(second)
val (largerLength, smallerLength, largerData, smallerData, largerStart, smallerStart) = if (firstStart > secondStart) {
Sextuple(first.size, second.size, first, second, firstStart, secondStart)
} else {
Sextuple(second.size, first.size, second, first, secondStart, firstStart)
}
val result = ULongArray(largerStart + 1) { 0u }
var i = 0
var sum: ULong = 0u
while (i < smallerStart) {
sum = sum + largerData[i] + smallerData[i]
result[i] = sum and baseMask
sum = sum shr 63
i++
}
while (true) {
if (sum == 0UL) {
while (i < largerStart) {
result[i] = largerData[i]
i++
}
val final = if (result[result.size - 1] == 0UL) {
if ((result.size - 1) == 0) {
return ZERO
}
result.copyOfRange(0, result.size - 1)
} else {
result
}
return final
}
if (i == largerLength) {
result[largerLength] = sum
return result
}
sum = sum + largerData[i]
result[i] = (sum and baseMask)
sum = sum shr 63
i++
}
}
fun subtractWithStartIndexes(
first: ULongArray,
second: ULongArray,
firstStart: Int,
secondStart: Int
): ULongArray {
val comparison = compareWithStartIndexes(first, second, firstStart, secondStart)
val firstSize = firstStart + 1
val secondSize = secondStart + 1
val firstIsLarger = comparison == 1
if (comparison == 0) return ZERO
if (secondSize == 1 && second[0] == 0UL) {
return first
}
// Lets throw this just to catch when we didn't prepare the operands correctly
if (!firstIsLarger) {
throw RuntimeException("subtract result less than zero")
}
val (largerData, smallerData, largerStart, smallerStart) = if (firstIsLarger) {
Quadruple(first, second, firstStart, secondStart)
} else {
Quadruple(second, first, secondStart, firstStart)
}
val result = ULongArray(largerStart) { 0U }
var i = 0
var diff: ULong = 0u
while (i < smallerStart) {
diff = largerData[i] - smallerData[i] - diff
result[i] = (diff and baseMask)
diff = diff shr 63
i++
}
while (diff != 0UL) {
diff = largerData[i] - diff
result[i] = (diff and baseMask)
diff = diff shr 63
i++
}
while (i < largerStart) {
result[i] = largerData[i]
i++
}
if (countLeadingZeroWords(result) == (result.size - 1) && result[0] == 0UL) {
return ZERO
}
return result
}
override fun subtract(first: ULongArray, second: ULongArray): ULongArray {
// debugOperandsCheck(first, second)
val firstStart = first.size - countLeadingZeroWords(first)
val secondStart = second.size - countLeadingZeroWords(second)
return subtractWithStartIndexes(first, second, firstStart, secondStart)
}
override fun multiply(first: ULongArray, second: ULongArray): ULongArray {
// debugOperandsCheck(first, second)
val firstRealSize = first.size - countLeadingZeroWords(first)
val seondRealSize = second.size - countLeadingZeroWords(second)
if (first.contentEquals(ZERO) || second.contentEquals(ZERO)) {
return ZERO
}
if ((firstRealSize >= karatsubaThreshold || seondRealSize >= karatsubaThreshold) &&
(firstRealSize <= toomCookThreshold || seondRealSize < toomCookThreshold)
) {
return karatsubaMultiply(first, second)
}
if (firstRealSize >= toomCookThreshold && seondRealSize >= toomCookThreshold) {
return toomCook3Multiply(first, second)
}
return basecaseMultiply(first, second)
}
fun basecaseMultiply(first: ULongArray, second: ULongArray): ULongArray {
val secondStart = second.size - countLeadingZeroWords(second)
var resultArray = ZERO
second.forEachIndexed { index: Int, element: ULong ->
if (index > secondStart) {
resultArray
} else {
resultArray = resultArray + (baseMultiply(first, element) shl (index * basePowerOfTwo))
}
}
return resultArray
}
fun combaMultiply(first: ULongArray, second: ULongArray) {
// TODO
}
fun karatsubaMultiply(firstUnsigned: ULongArray, secondUnsigned: ULongArray): ULongArray {
val first = SignedULongArray(firstUnsigned, true)
val second = SignedULongArray(secondUnsigned, true)
val firstRealSize = first.unsignedValue.size - countLeadingZeroWords(first.unsignedValue)
val secondRealSize = second.unsignedValue.size - countLeadingZeroWords(second.unsignedValue)
val halfLength = (kotlin.math.max(firstRealSize, secondRealSize) + 1) / 2
val mask = (ONE shl (halfLength * wordSizeInBits)) - 1UL
val firstLower = first and mask
val firstHigher = first shr halfLength * wordSizeInBits
val secondLower = second and mask
val secondHigher = second shr halfLength * wordSizeInBits
val higherProduct = firstHigher * secondHigher
val lowerProduct = firstLower * secondLower
val middleProduct = (firstHigher + firstLower) * (secondHigher + secondLower)
val result =
(higherProduct shl (2 * wordSizeInBits * halfLength)) + ((middleProduct - higherProduct - lowerProduct) shl (wordSizeInBits * halfLength)) + lowerProduct
return result.unsignedValue
}
fun prependULongArray(original: ULongArray, numberOfWords: Int, value: ULong): ULongArray {
return ULongArray(original.size + numberOfWords) {
when {
it < numberOfWords -> value
else -> original[it - numberOfWords]
}
}
}
fun extendULongArray(original: ULongArray, numberOfWords: Int, value: ULong): ULongArray {
return ULongArray(original.size + numberOfWords) {
when {
it < original.size -> original[it]
else -> value
}
}
}
@Suppress("DuplicatedCode")
fun toomCook3Multiply(firstUnchecked: ULongArray, secondUnchecked: ULongArray): ULongArray {
val first = if (firstUnchecked.size % 3 != 0) {
firstUnchecked.plus(ULongArray((((firstUnchecked.size + 2) / 3) * 3) - firstUnchecked.size) { 0U }.asIterable())
} else {
firstUnchecked
}.toULongArray()
val second = if (secondUnchecked.size % 3 != 0) {
secondUnchecked.plus(ULongArray((((secondUnchecked.size + 2) / 3) * 3) - secondUnchecked.size) { 0U }.asIterable())
} else {
secondUnchecked
}.toULongArray()
val firstLength = first.size
val secondLength = second.size
val (firstPrepared, secondPrepared) = when {
firstLength > secondLength -> {
val prepared = extendULongArray(second, firstLength - secondLength, 0U)
Pair(first, prepared)
}
firstLength < secondLength -> {
val prepared = extendULongArray(first, secondLength - firstLength, 0U)
Pair(prepared, second)
}
else -> Pair(first, second)
}
val longestLength = kotlin.math.max(first.size, second.size)
val extendedDigit = (longestLength + 2) / 3
val m0 = SignedULongArray(firstPrepared.slice(0 until extendedDigit).toULongArray(), true)
val m1 = SignedULongArray(firstPrepared.slice(extendedDigit until extendedDigit * 2).toULongArray(), true)
val m2 = SignedULongArray(firstPrepared.slice(extendedDigit * 2 until extendedDigit * 3).toULongArray(), true)
val n0 = SignedULongArray(secondPrepared.slice(0 until extendedDigit).toULongArray(), true)
val n1 = SignedULongArray(secondPrepared.slice(extendedDigit until extendedDigit * 2).toULongArray(), true)
val n2 = SignedULongArray(secondPrepared.slice(extendedDigit * 2 until extendedDigit * 3).toULongArray(), true)
val p0 = m0 + m2
// p(0)
val pe0 = m0
// p(1)
val pe1 = p0 + m1
// p(-1)
val pem1 = p0 - m1
// p(-2)
val doublePemM2 = (pem1 + m2) * SIGNED_POSITIVE_TWO
val pem2 = doublePemM2 - m0
// p(inf)
val pinf = m2
val q0 = n0 + n2
// q(0)
val qe0 = n0
// q(1)
val qe1 = q0 + n1
// q(-1)
val qem1 = q0 - n1
// q(-2)
val doubleQemN2 = (qem1 + n2) * SIGNED_POSITIVE_TWO
val qem2 = doubleQemN2 - n0
// q(inf)
val qinf = n2
val re0 = pe0 * qe0
val re1 = pe1 * qe1
val rem1 = pem1 * qem1
val rem2 = pem2 * qem2
val rinf = pinf * qinf
var r0 = re0
var r4 = rinf
val rem2re1diff = (rem2 - re1)
// var r3 = SignedULongArray(exactDivideBy3(rem2re1diff.unsignedValue), rem2re1diff.sign)
var r3 = rem2re1diff / SignedULongArray(ulongArrayOf(3U), true)
// println("R3 ${r3.sign} ${r3.unsignedValue}")
var r1 = (re1 - rem1) shr 1
var r2 = rem1 - r0
r3 = ((r2 - r3) shr 1) + SIGNED_POSITIVE_TWO * rinf
r2 = r2 + r1 - r4
r1 = r1 - r3
val bShiftAmount = extendedDigit * 63
val rb0 = r0
val rb1 = (r1 shl (bShiftAmount))
val rb2 = (r2 shl (bShiftAmount * 2))
val rb3 = (r3 shl (bShiftAmount * 3))
val rb4 = (r4 shl (bShiftAmount * 4))
val rb = rb0 +
rb1 +
rb2 +
rb3 +
rb4
return rb.unsignedValue
}
// Signed operations TODO evaluate if we really want to do this to support Toom-Cook or just move it out of arithmetic
data class SignedULongArray(val unsignedValue: ULongArray, val sign: Boolean)
private fun signedAdd(first: SignedULongArray, second: SignedULongArray) = if (first.sign xor second.sign) {
if (first.unsignedValue > second.unsignedValue) {
SignedULongArray(first.unsignedValue - second.unsignedValue, first.sign)
} else {
SignedULongArray(second.unsignedValue - first.unsignedValue, second.sign)
}
} else {
// Same sign
SignedULongArray(first.unsignedValue + second.unsignedValue, first.sign)
}
val SIGNED_POSITIVE_TWO = SignedULongArray(TWO, true)
private fun signedSubtract(first: SignedULongArray, second: SignedULongArray) =
signedAdd(first, second.copy(sign = !second.sign))
private fun signedMultiply(first: SignedULongArray, second: SignedULongArray) =
SignedULongArray(first.unsignedValue * second.unsignedValue, !(first.sign xor second.sign))
private fun signedDivide(first: SignedULongArray, second: SignedULongArray) =
SignedULongArray(first.unsignedValue / second.unsignedValue, !(first.sign xor second.sign))
private fun signedRemainder(first: SignedULongArray, second: SignedULongArray) =
SignedULongArray(first.unsignedValue % second.unsignedValue, !(first.sign xor second.sign))
internal operator fun SignedULongArray.plus(other: SignedULongArray): SignedULongArray {
return signedAdd(this, other)
}
internal operator fun SignedULongArray.minus(other: SignedULongArray): SignedULongArray {
return signedSubtract(this, other)
}
internal operator fun SignedULongArray.times(other: SignedULongArray): SignedULongArray {
return signedMultiply(this, other)
}
internal operator fun SignedULongArray.div(other: SignedULongArray): SignedULongArray {
return signedDivide(this, other)
}
internal operator fun SignedULongArray.rem(other: SignedULongArray): SignedULongArray {
return signedRemainder(this, other)
}
internal infix fun SignedULongArray.shr(places: Int) = SignedULongArray(unsignedValue shr places, sign)
internal infix fun SignedULongArray.shl(places: Int) = SignedULongArray(unsignedValue shl places, sign)
internal infix fun SignedULongArray.and(operand: ULongArray) = SignedULongArray(and(unsignedValue, operand), sign)
// End of signed operations
fun fftMultiply(first: ULongArray, second: ULongArray): ULongArray {
return first
}
fun baseMultiply(first: ULongArray, second: ULong): ULongArray {
val firstStart = first.size - countLeadingZeroWords(first)
val secondLow = second and lowMask
val secondHigh = second shr 32
val result = ULongArray(firstStart + 1)
var carryIntoNextRound = 0UL
var i = 0
var j = 0
while (i < firstStart) {
val firstLow = first[i] and lowMask
val firstHigh = first[i] shr 32
i++
// Calculate low part product
val lowerProduct = (firstLow * secondLow)
var lowerCarry = lowerProduct shr 63
var lowResult = carryIntoNextRound + (lowerProduct and baseMask)
lowerCarry += lowResult shr 63
lowResult = lowResult and baseMask
val middleProduct = firstLow * secondHigh + secondLow * firstHigh
var middleCarry = lowerCarry
middleCarry += (middleProduct shr 31)
lowResult += (middleProduct shl 32) and baseMask
middleCarry += (lowResult shr 63)
result[j] = lowResult and baseMask
var highResult = middleCarry
val higherProduct = (firstHigh * secondHigh) shl 1
highResult = highResult + higherProduct
carryIntoNextRound = highResult
j++
}
if (carryIntoNextRound != 0UL) {
result[j] = carryIntoNextRound
}
return result
}
/*
Useful when we want to do a ULong * ULong -> ULongArray
*/
fun multiply(first: ULong, second: ULong): ULongArray {
if (first == 0UL || second == 0UL) {
return ulongArrayOf(0UL)
}
// Split the operands
val firstLow = first and lowMask
val firstHigh = first shr 32
val secondLow = second and lowMask
val secondHigh = second shr 32
// Calculate low part product
val lowerProduct = firstLow * secondLow
val lowCarry = lowerProduct shr 63
var lowResult = lowerProduct and baseMask
val middleProduct = firstLow * secondHigh + secondLow * firstHigh
var middleCarry = lowCarry
middleCarry += (middleProduct shr 31)
lowResult += (middleProduct shl 32) and baseMask
middleCarry += (lowResult shr 63)
var highResult = middleCarry
val higherProduct = (firstHigh * secondHigh) shl 1
highResult = highResult + higherProduct
return removeLeadingZeros(ulongArrayOf(lowResult and baseMask, highResult))
}
override fun pow(base: ULongArray, exponent: Long): ULongArray {
if (exponent == 0L) {
return ONE
}
if (exponent == 1L) {
return base
}
if (base.size == 1 && base[0] == 10UL && exponent < powersOf10.size) {
return powersOf10[exponent.toInt()]
}
return (0 until exponent).fold(ONE) { acc, _ ->
acc * base
}
}
fun normalize(dividend: ULongArray, divisor: ULongArray): Triple {
val divisorSize = divisor.size
val normalizationShift = numberOfLeadingZerosInAWord(divisor[divisorSize - 1])
val divisorNormalized = divisor.shl(normalizationShift)
val dividendNormalized = dividend.shl(normalizationShift)
return Triple(dividendNormalized, divisorNormalized, normalizationShift)
}
fun normalize(operand: ULongArray): Pair {
val normalizationShift = numberOfLeadingZerosInAWord(operand[operand.size - 1])
return Pair(operand.shl(normalizationShift), normalizationShift)
}
fun denormalize(
remainderNormalized: ULongArray,
normalizationShift: Int
): ULongArray {
val remainder = remainderNormalized shr normalizationShift
return remainder
}
/**
* Based on Basecase DivRem algorithm from
* Modern Computer Arithmetic, Richard Brent and Paul Zimmermann, Cambridge University Press, 2010.
* Version 0.5.9
* https://members.loria.fr/PZimmermann/mca/pub226.html
*/
fun baseDivide(
unnormalizedDividend: ULongArray,
unnormalizedDivisor: ULongArray
): Pair {
if (unnormalizedDivisor > unnormalizedDividend) {
return Pair(ZERO, unnormalizedDividend)
}
if (unnormalizedDivisor.size == 1 && unnormalizedDividend.size == 1) {
return Pair(
removeLeadingZeros(
ulongArrayOf(
unnormalizedDividend[0] / unnormalizedDivisor[0]
)
),
removeLeadingZeros(
ulongArrayOf(
unnormalizedDividend[0] % unnormalizedDivisor[0]
)
)
)
}
val bitPrecision = bitLength(unnormalizedDividend) - bitLength(
unnormalizedDivisor
)
if (bitPrecision == 0) {
return Pair(ONE, unnormalizedDividend - unnormalizedDivisor)
}
var (dividend, divisor, normalizationShift) = normalize(
unnormalizedDividend,
unnormalizedDivisor
)
val dividendSize = dividend.size
val divisorSize = divisor.size
var wordPrecision = dividendSize - divisorSize
var qjhat: ULongArray
var reconstructedQuotient: ULongArray
var quotient = ULongArray(wordPrecision)
val divisorTimesBaseToPowerOfM = (divisor shl (wordPrecision * basePowerOfTwo))
if (dividend >= divisorTimesBaseToPowerOfM) {
quotient = ULongArray(wordPrecision + 1)
quotient[wordPrecision] = 1U
dividend = dividend - divisorTimesBaseToPowerOfM
}
for (j in (wordPrecision - 1) downTo 0) {
val twoDigit = if (divisorSize + j < dividend.size) {
((ulongArrayOf(dividend[divisorSize + j]) shl basePowerOfTwo) + dividend[divisorSize + j - 1])
} else {
if (divisorSize + j == dividend.size) {
ulongArrayOf(dividend[divisorSize + j - 1])
} else {
ZERO
}
}
val convertedResult =
BigInteger32Arithmetic.divide(twoDigit.to32Bit(), ulongArrayOf(divisor[divisorSize - 1]).to32Bit())
qjhat = convertedResult.first.from32Bit()
quotient[j] = if (qjhat < (baseMask - 1UL)) {
qjhat[0]
} else {
baseMask
}
// We don't have signed integers here so we need to check if reconstructed quotient is larger than the dividend
// instead of just doing (dividend = dividend − qj * β^j * divisor) and then looping. Final effect is the same.
reconstructedQuotient = ((divisor * quotient[j]) shl (j * basePowerOfTwo))
while (reconstructedQuotient > dividend) {
quotient[j] = quotient[j] - 1U
reconstructedQuotient = ((divisor * quotient[j]) shl (j * basePowerOfTwo))
}
dividend = dividend - reconstructedQuotient
}
while (dividend >= divisor) {
quotient += 1UL
dividend -= divisor
}
val denormRemainder =
denormalize(dividend, normalizationShift)
return Pair(removeLeadingZeros(quotient), denormRemainder)
}
fun basicDivide2(
unnormalizedDividend: ULongArray,
unnormalizedDivisor: ULongArray
): Pair {
var (a, b, shift) = normalize(unnormalizedDividend, unnormalizedDivisor)
val m = a.size - b.size
val bmb = b shl (m * wordSizeInBits)
var q = ULongArray(m + 1) { 0U }
if (a > bmb) {
q[m] = 1U
a = a - bmb
}
var qjhat = ZERO
var qjhatULong = ZERO
var bjb = ZERO
var delta = ZERO
for (j in m - 1 downTo 0) {
qjhatULong = BigInteger32Arithmetic.divide(
(a.copyOfRange(b.size - 1, b.size + 1)).to32Bit(),
ulongArrayOf(b[b.size - 1]).to32Bit()
).first.from32Bit()
q[j] = min(qjhatULong, baseMaskArray)[0]
bjb = b shl (j * BigInteger32Arithmetic.wordSizeInBits)
val qjBjb = (b * q[j]) shl (j * wordSizeInBits)
if (qjBjb > a) {
delta = qjBjb - a
while (delta > qjBjb) {
q[j] = q[j] - 1U
delta = delta - bjb
}
// quotient is now such that q[j] * b*B^j won't be larger than divisor
a = a - (b * q[j]) shl (j * BigInteger32Arithmetic.wordSizeInBits)
} else {
a = a - qjBjb
}
}
val denormRemainder =
denormalize(a, shift)
return Pair(removeLeadingZeros(q), denormRemainder)
}
/**
* When division is known to be exact ( no remainder, we can use this, especially in Toom-Cook)
* TODO Need to move modInverse from BigInteger to arithmetic, and then replace here
*/
fun exactDivideBy3(operand: ULongArray): ULongArray {
val base = BigInteger.ONE.shl(operand.size * 63)
val creator = ModularBigInteger.creatorForModulo(base)
val reciprocalOf3 = creator.fromInt(3).inverse()
val multipliedByInverse = multiply(operand, reciprocalOf3.toBigInteger().magnitude.toULongArray())
return multipliedByInverse.slice(operand.indices).toULongArray()
}
fun exactDivideBy3Better(operand: ULongArray): ULongArray {
// TODO
return operand
}
override fun reciprocal(operand: ULongArray): Pair {
return d1ReciprocalRecursiveWordVersion(operand)
}
fun d1ReciprocalRecursive(a: ULongArray): Pair {
val fullBitLenght = bitLength(a)
val n = if (fullBitLenght > 63) {
fullBitLenght - 63
} else {
fullBitLenght
}
if (n <= 30) {
val rhoPowered = 1UL shl (n * 2)
val longA = a[0]
val x = rhoPowered / longA
val r = rhoPowered - x * longA
return Pair(ulongArrayOf(x), ulongArrayOf(r))
}
val l = floor((n - 1).toDouble() / 2).toInt()
val h = n - l
val mask = (ONE shl l) - ONE
val ah = a shr l
val al = and(a, mask)
var (xh, rh) = d1ReciprocalRecursive(ah)
val s = al * xh
// val rhoL = (ONE shl l)
val rhRhoL = rh shl l
val t = if (rhRhoL >= s) {
rhRhoL - s
} else {
xh = xh - ONE
(rhRhoL + a) - s
}
val tm = t shr h
val d = (xh * tm) shr h
var x = (xh shl l) + d
var r = (t shl l) - a * d
if (r >= a) {
x = x + ONE
r = r - a
if (r >= a) {
x = x + ONE
r = r - a
}
}
return Pair(x, r)
}
fun d1ReciprocalRecursiveWordVersion(a: ULongArray): Pair {
val n = a.size - 1
if (n <= 2) {
val corrected = if (n == 0) {
1
} else {
n
}
val rhoPowered = ONE shl (corrected * 2 * wordSizeInBits)
val x = rhoPowered / a
val r = rhoPowered - (x * a)
return Pair(x, r)
}
val l = floor((n - 1).toDouble() / 2).toInt()
val h = n - l
val ah = a.copyOfRange(a.size - h - 1, a.size)
val al = a.copyOfRange(0, l)
var (xh, rh) = d1ReciprocalRecursiveWordVersion(ah)
val s = al * xh
// val rhoL = (ONE shl l)
val rhRhoL = rh shl (l * wordSizeInBits)
val t = if (rhRhoL >= s) {
rhRhoL - s
} else {
xh = xh - ONE
(rhRhoL + a) - s
}
val tm = t shr (h * wordSizeInBits)
val d = (xh * tm) shr (h * wordSizeInBits)
var x = (xh shl (l * wordSizeInBits)) + d
var r = (t shl (l * wordSizeInBits)) - a * d
if (r >= a) {
x = x + ONE
r = r - a
if (r >= a) {
x = x + ONE
r = r - a
}
}
return Pair(x, r)
}
private fun unbalancedReciprocal(a: ULongArray, diff: Int): Pair {
val n = a.size - 1 - diff
val a0 = a.copyOfRange(n + 1, a.size)
val a1 = a.copyOfRange(0, n)
var (x, r) = d1ReciprocalRecursiveWordVersion(a0)
if (x == ONE shl (n * 63)) {
if (a1.compareTo(ZERO) == 0) {
r = ZERO
} else {
x = x - ONE
r = a - (a1 shl (n * 63))
}
} else {
val rRhoD = r shl diff
val a1x = a1 * x
if (rRhoD > a1x) {
r = rRhoD - a1x
} else {
x = x - ONE
r = rRhoD - (a1 * x)
}
}
return Pair(x, r)
}
internal fun convertTo64BitRepresentation(operand: ULongArray): ULongArray {
if (operand == ZERO) return ZERO
val length = bitLength(operand)
val requiredLength = if (length % 64 == 0) {
length / 64
} else {
(length / 64) + 1
}
var wordStep: Int
var shiftAmount: Int
val result = ULongArray(requiredLength)
for (i in 0 until requiredLength) {
wordStep = i / 63
shiftAmount = i % 63
if (i + wordStep + 1 < operand.size) {
result[i] =
(operand[i + wordStep] shr shiftAmount) or ((operand[i + wordStep + 1] shl (63 - shiftAmount)))
} else {
result[i] = (operand[i + wordStep] shr shiftAmount)
}
}
return result
}
internal fun convertTo32BitRepresentation(operand: ULongArray): UIntArray {
val power64Representation = convertTo64BitRepresentation(operand)
val result = UIntArray(power64Representation.size * 2)
for (i in 0 until power64Representation.size) {
result[2 * i] = (power64Representation[i] and BigInteger32Arithmetic.base.toULong()).toUInt()
result[2 * i + 1] = (power64Representation[i] shr 32).toUInt()
}
return BigInteger32Arithmetic.removeLeadingZeros(result)
}
internal fun convertFrom32BitRepresentation(operand: UIntArray): ULongArray {
if (operand.size == 0) {
return ZERO
}
if (operand.size == 1) {
return ulongArrayOf(operand[0].toULong())
}
val length = BigInteger32Arithmetic.bitLength(operand)
val requiredLength = if (length % 63 == 0) {
length / 63
} else {
(length / 63) + 1
}
val result = ULongArray(requiredLength)
var skipWordCount: Int
for (i in 0 until requiredLength) {
skipWordCount = i / 32
val shiftAmount = i % 32
val position = (i * 2) - skipWordCount
if (requiredLength == 2) {
result[0] = operand[0].toULong() or ((operand[1].toULong() shl 32) and highMask)
result[i] =
(operand[1].toULong() shr 31) or (operand[2].toULong() shl 1) or (operand[3].toULong() shl 33)
} else {
when (i) {
0 -> {
result[i] = operand[0].toULong() or ((operand[1].toULong() shl 32) and highMask)
}
in 1 until requiredLength - 1 -> {
result[i] =
(operand[position - 1].toULong() shr (32 - shiftAmount)) or
(operand[position].toULong() shl shiftAmount) or
((operand[position + 1].toULong() shl (32 + shiftAmount)) and highMask)
}
requiredLength - 1 -> {
if (position < operand.size) {
result[i] =
(operand[position - 1].toULong() shr (32 - shiftAmount)) or
(operand[position].toULong() shl shiftAmount)
} else {
result[i] =
(operand[position - 1].toULong() shr (32 - shiftAmount))
}
}
}
}
}
return result
}
override fun divide(first: ULongArray, second: ULongArray): Pair {
// debugOperandsCheck(first, second)
return baseDivide(first, second)
}
internal fun reciprocalDivision(first: ULongArray, second: ULongArray): Pair {
if (first.size < second.size) {
throw RuntimeException("Invalid division: ${first.size} words / ${second.size} words")
}
val shift = if (second.size == 1) {
1
} else {
second.size - 1
}
val precisionExtension = (first.size - second.size + 1)
val secondHigherPrecision = ULongArray(second.size + precisionExtension) {
when {
it >= precisionExtension -> second[it - precisionExtension]
else -> 0UL
}
}
val secondReciprocalWithRemainder = d1ReciprocalRecursiveWordVersion(secondHigherPrecision)
val secondReciprocal = secondReciprocalWithRemainder.first
var product = first * secondReciprocal
// TODO Proper rounding
if (product.compareTo(0UL) == 0) {
return Pair(ZERO, first)
}
if (product.size == 1) {
if (product >= baseMask - 1UL) {
product = product + ONE
}
} else {
val importantWord = product[product.size - second.size]
if (importantWord >= baseMask) {
product = ULongArray(product.size) {
when (it) {
product.size - 1 -> product[product.size - 1] + 1UL
else -> 0UL
}
}
}
}
val result = product.copyOfRange(2 * shift + precisionExtension, product.size)
val remainder = first - (result * second)
return Pair(result, remainder)
}
override fun sqrt(operand: ULongArray): Pair {
return reqursiveSqrt(operand)
}
private fun reqursiveSqrt(operand: ULongArray): Pair {
val n = operand.size
val l = floor((n - 1).toDouble() / 4).toInt()
if (l == 0) {
return basecaseSqrt(operand)
}
val step = n / 4
val stepRemainder = n % 4
val baseLPowerShift = 63 * l
val a1 = operand.copyOfRange(n - ((3 * step) + stepRemainder), n - ((2 * step) + stepRemainder))
val a0 = operand.copyOfRange(0, n - ((3 * step) + stepRemainder))
val a3a2 = operand.copyOfRange(n - ((2 * step) + stepRemainder), n)
val (sPrim, rPrim) = reqursiveSqrt(a3a2)
val (q, u) = ((rPrim shl baseLPowerShift) + a1) divrem (sPrim shl 1)
var s = (sPrim shl baseLPowerShift) + q
var r = (u shl baseLPowerShift) + a0 - (q * q)
return Pair(s, r)
}
internal fun basecaseSqrt(operand: ULongArray): Pair {
val sqrt = sqrtInt(operand)
val remainder = operand - (sqrt * sqrt)
return Pair(sqrt, remainder)
}
internal fun sqrtInt(operand: ULongArray): ULongArray {
var u = operand
var s = ZERO
var tmp = ZERO
do {
s = u
tmp = s + (operand / s)
u = tmp shr 1
} while (u < s)
return s
}
override fun gcd(first: ULongArray, second: ULongArray): ULongArray {
return if (first.size > 150 || second.size > 150) {
euclideanGcd(first, second)
} else {
binaryGcd(first, second)
}
}
private fun euclideanGcd(first: ULongArray, second: ULongArray): ULongArray {
var u = first
var v = second
while (v != ZERO) {
val tmpU = u
u = v
v = tmpU % v
}
return u
}
private tailrec fun binaryGcd(first: ULongArray, second: ULongArray): ULongArray {
// debugOperandsCheck(first, second)
if (first.contentEquals(second)) {
return first
}
if (first.contentEquals(ZERO)) {
return second
}
if (second.contentEquals(ZERO)) {
return first
}
if (and(first, ONE).contentEquals(ZERO)) { // first is even
if (and(second, ONE).contentEquals(ZERO)) { // second is even
return binaryGcd(first shr 1, second shr 1) shl 1
} else { // second is odd
return binaryGcd(first shr 1, second)
}
}
if (and(second, ONE).contentEquals(ZERO)) {
return binaryGcd(first, second shr 1)
}
return if (compare(first, second) == 1) {
binaryGcd(subtract(first, second) shr 1, second)
} else {
binaryGcd(subtract(second, first) shr 1, first)
}
}
fun min(first: ULongArray, second: ULongArray): ULongArray {
return if (first < second) {
first
} else {
second
}
}
fun max(first: ULongArray, second: ULongArray): ULongArray {
return if (first > second) {
first
} else {
second
}
}
override fun parseForBase(number: String, base: Int): ULongArray {
var parsed = ZERO
number.toLowerCase().forEach { char ->
parsed = (parsed * base.toULong()) + (char.toDigit()).toULong()
}
return removeLeadingZeros(parsed)
}
override fun toString(operand: ULongArray, base: Int): String {
var copy = operand.copyOf()
val baseArray = ulongArrayOf(base.toULong())
val stringBuilder = StringBuilder()
while (copy != ZERO) {
val divremResult = (copy divrem baseArray)
if (divremResult.second.isEmpty()) {
stringBuilder.append(0)
} else {
stringBuilder.append(divremResult.second[0].toString(base))
}
copy = divremResult.first
}
return stringBuilder.toString().reversed()
}
override fun and(operand: ULongArray, mask: ULongArray): ULongArray {
return ULongArray(operand.size) {
if (it < mask.size) {
operand[it] and mask[it]
} else {
0UL
}
}
}
override fun or(operand: ULongArray, mask: ULongArray): ULongArray {
return removeLeadingZeros(
ULongArray(operand.size) {
if (it < mask.size) {
operand[it] or mask[it]
} else {
operand[it]
}
}
)
}
override fun xor(operand: ULongArray, mask: ULongArray): ULongArray {
return removeLeadingZeros(
ULongArray(operand.size) {
if (it < mask.size) {
operand[it] xor mask[it]
} else {
operand[it] xor 0UL
}
}
)
}
override fun not(operand: ULongArray): ULongArray {
val leadingZeros = numberOfLeadingZerosInAWord(operand[operand.size - 1])
val cleanupMask = (((1UL shl leadingZeros + 1) - 1U) shl (basePowerOfTwo - leadingZeros)).inv()
val inverted = ULongArray(operand.size) {
if (it < operand.size - 2) {
operand[it].inv() and baseMask
} else {
operand[it].inv() and cleanupMask
}
}
return inverted
}
// -------------- Bitwise ---------------- //
internal infix fun ULongArray.shl(places: Int): ULongArray {
return shiftLeft(this, places)
}
internal infix fun ULongArray.shr(places: Int): ULongArray {
return shiftRight(this, places)
}
override fun bitAt(operand: ULongArray, position: Long): Boolean {
if (position / 63 > Int.MAX_VALUE) {
throw RuntimeException("Invalid bit index, too large, cannot access word (Word position > Int.MAX_VALUE")
}
val wordPosition = position / 63
if (wordPosition >= operand.size) {
return false
}
val bitPosition = position % 63
val word = operand[wordPosition.toInt()]
return (word and (1UL shl bitPosition.toInt()) == 1UL)
}
override fun setBitAt(operand: ULongArray, position: Long, bit: Boolean): ULongArray {
if (position / 63 > Int.MAX_VALUE) {
throw RuntimeException("Invalid bit index, too large, cannot access word (Word position > Int.MAX_VALUE")
}
val wordPosition = position / 63
if (wordPosition >= operand.size) {
throw IndexOutOfBoundsException("Invalid position, addressed word $wordPosition larger than number of words ${operand.size}")
}
val bitPosition = position % 63
val setMask = 1UL shl bitPosition.toInt()
return ULongArray(operand.size) {
if (it == wordPosition.toInt()) {
if (bit) {
operand[it] or setMask
} else {
operand[it] xor setMask
}
} else {
operand[it]
}
}
}
// -------------- Operations ---------------- //
internal operator fun ULongArray.plus(other: ULongArray): ULongArray {
return add(this, other)
}
internal operator fun ULongArray.minus(other: ULongArray): ULongArray {
return subtract(this, other)
}
internal operator fun ULongArray.times(other: ULongArray): ULongArray {
return multiply(this, other)
}
internal operator fun ULongArray.plus(other: ULong): ULongArray {
return add(this, ulongArrayOf(other))
}
internal operator fun ULongArray.minus(other: ULong): ULongArray {
return subtract(this, ulongArrayOf(other))
}
internal operator fun ULongArray.times(other: ULong): ULongArray {
return multiply(this, ulongArrayOf(other))
}
internal operator fun ULongArray.div(other: ULong): ULongArray {
return divide(this, ulongArrayOf(other)).first
}
internal operator fun ULongArray.rem(other: ULong): ULongArray {
return divide(this, ulongArrayOf(other)).second
}
internal operator fun ULongArray.div(other: ULongArray): ULongArray {
return divide(this, other).first
}
internal operator fun ULongArray.rem(other: ULongArray): ULongArray {
return divide(this, other).second
}
internal infix fun ULongArray.divrem(other: ULongArray): Pair {
return divide(this, other)
}
internal operator fun ULongArray.compareTo(other: ULongArray): Int {
return compare(this, other)
}
internal operator fun ULongArray.compareTo(other: ULong): Int {
return compare(this, ulongArrayOf(other))
}
internal fun ULongArray.to32Bit(): UIntArray {
return convertTo32BitRepresentation(this)
}
internal fun UIntArray.from32Bit(): ULongArray {
return convertFrom32BitRepresentation(this)
}
override fun fromULong(uLong: ULong): ULongArray {
return fromLong(uLong.toLong())
}
override fun fromUInt(uInt: UInt): ULongArray = ulongArrayOf(uInt.toULong())
override fun fromUShort(uShort: UShort): ULongArray = ulongArrayOf(uShort.toULong())
override fun fromUByte(uByte: UByte): ULongArray = ulongArrayOf(uByte.toULong())
override fun fromLong(long: Long): ULongArray {
if ((long.absoluteValue.toULong() and overflowMask shr 63) == 1UL) {
return ulongArrayOf(baseMask) + 1U
}
return ulongArrayOf((long.absoluteValue.toULong() and baseMask))
}
override fun fromInt(int: Int): ULongArray = ulongArrayOf(int.absoluteValue.toULong())
override fun fromShort(short: Short): ULongArray = ulongArrayOf(short.toInt().absoluteValue.toULong())
override fun fromByte(byte: Byte): ULongArray = ulongArrayOf(byte.toInt().absoluteValue.toULong())
override fun toByteArray(operand: ULongArray, sign: Sign): Array {
return BigInteger32Arithmetic.toByteArray(convertTo32BitRepresentation(operand), sign)
}
override fun fromByteArray(byteArray: Array): Pair {
val result = BigInteger32Arithmetic.fromByteArray(byteArray)
return Pair(convertFrom32BitRepresentation(result.first), result.second)
}
override fun fromUByteArray(uByteArray: Array, endianness: Endianness): Pair {
val result = BigInteger32Arithmetic.fromUByteArray(uByteArray, endianness)
return Pair(convertFrom32BitRepresentation(result.first), result.second)
}
override fun toUByteArray(operand: ULongArray, endianness: Endianness): Array {
val result = BigInteger32Arithmetic.toUByteArray(convertTo32BitRepresentation(operand), endianness)
return result
}
private fun debugOperandsCheck(first: ULongArray, second: ULongArray) {
if (debugOperandSize && (first.isEmpty() || second.isEmpty())) {
throw RuntimeException("Empty operands")
}
}
// ------------- Useful constants --------------
val powersOf10 = arrayOf(
ulongArrayOf(1UL),
ulongArrayOf(10UL),
ulongArrayOf(100UL),
ulongArrayOf(1000UL),
ulongArrayOf(10000UL),
ulongArrayOf(100000UL),
ulongArrayOf(1000000UL),
ulongArrayOf(10000000UL),
ulongArrayOf(100000000UL),
ulongArrayOf(1000000000UL),
ulongArrayOf(10000000000UL),
ulongArrayOf(100000000000UL),
ulongArrayOf(1000000000000UL),
ulongArrayOf(10000000000000UL),
ulongArrayOf(100000000000000UL),
ulongArrayOf(1000000000000000UL),
ulongArrayOf(10000000000000000UL),
ulongArrayOf(100000000000000000UL),
ulongArrayOf(1000000000000000000UL),
ulongArrayOf(776627963145224192UL, 1UL),
ulongArrayOf(7766279631452241920UL, 10UL),
ulongArrayOf(3875820019684212736UL, 108UL),
ulongArrayOf(1864712049423024128UL, 1084UL),
ulongArrayOf(200376420520689664UL, 10842UL),
ulongArrayOf(2003764205206896640UL, 108420UL),
ulongArrayOf(1590897978359414784UL, 1084202UL),
ulongArrayOf(6685607746739372032UL, 10842021UL),
ulongArrayOf(2292473209410289664UL, 108420217UL),
ulongArrayOf(4477988020393345024UL, 1084202172UL),
ulongArrayOf(7886392056514347008UL, 10842021724UL),
ulongArrayOf(5076944270305263616UL, 108420217248UL),
ulongArrayOf(4652582518778757120UL, 1084202172485UL),
ulongArrayOf(408965003513692160UL, 10842021724855UL),
ulongArrayOf(4089650035136921600UL, 108420217248550UL),
ulongArrayOf(4003012203950112768UL, 1084202172485504UL),
ulongArrayOf(3136633892082024448UL, 10842021724855044UL),
ulongArrayOf(3696222810255917056UL, 108420217248550443UL),
ulongArrayOf(68739955140067328UL, 1084202172485504434UL),
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