962 lines
32 KiB
C#
962 lines
32 KiB
C#
using System;
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using System.Buffers;
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using System.Numerics;
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using System.Runtime.CompilerServices;
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using ILGPU.Algorithms;
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using PerfectNumbers.Core.Gpu;
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namespace PerfectNumbers.Core;
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public static class ULongExtensions
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{
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private const int Pow2WindowSize = 8;
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private const ulong Pow2WindowFallbackThreshold = 32UL;
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private const int Pow2WindowOddPowerCount = 1 << (Pow2WindowSize - 1);
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public static ulong CalculateOrder(this ulong q)
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{
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if (q <= 2UL)
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{
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return 0UL;
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}
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ulong order = q - 1UL, prime, temp;
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uint[] smallPrimes = PrimesGenerator.SmallPrimes;
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ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
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int i = 0, primesLength = smallPrimes.Length;
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UInt128 q128 = q,
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cycle = MersenneDivisorCycles.GetCycle(q128);
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// TODO: When the shared cycle snapshot cannot serve this divisor, trigger an on-demand
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// GPU computation (respecting the configured device) without promoting the result into
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// the cache so the order calculator still benefits from cycle stepping while keeping the
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// single-block memory plan intact.
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for (; i < primesLength; i++)
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{
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if (smallPrimesPow2[i] > order)
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{
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break;
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}
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prime = smallPrimes[i];
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// TODO: Replace this `%` driven factor peeling with the divisor-cycle aware
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// factoring helper so large orders reuse the cached remainders highlighted in
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// the latest divisor-cycle benchmarks instead of recomputing slow modulo checks.
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while (order % prime == 0UL)
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{
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temp = order / prime;
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if (temp.PowModWithCycle(q128, cycle) == UInt128.One)
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{
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order = temp;
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}
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else
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{
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break;
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}
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}
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}
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return order;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static int GetBitLength(this ulong value)
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{
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return 64 - BitOperations.LeadingZeroCount(value);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MultiplyShiftRight(ulong value, ulong multiplier, int shift)
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{
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UInt128 product = (UInt128)value * multiplier;
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return (ulong)(product >> shift);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MultiplyShiftRightShiftFirst(ulong value, ulong multiplier, int shift)
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{
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ulong high = value >> shift;
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ulong mask = (1UL << shift) - 1UL;
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ulong low = value & mask;
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UInt128 highContribution = (UInt128)high * multiplier;
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UInt128 lowContribution = (UInt128)low * multiplier;
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UInt128 combined = highContribution + (lowContribution >> shift);
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return (ulong)combined;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong FastDiv64(this ulong value, ulong divisor, ulong mul)
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{
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ulong quotient = (ulong)(((UInt128)value * mul) >> 64);
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UInt128 remainder = (UInt128)value - ((UInt128)quotient * divisor);
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if (remainder >= divisor)
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{
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quotient++;
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}
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return quotient;
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}
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public const ulong WordBitMask = 0xFFFFUL;
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static bool IsPrimeCandidate(this ulong n)
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{
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int i = 0;
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uint[] smallPrimes = PrimesGenerator.SmallPrimes;
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ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
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int len = smallPrimes.Length;
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ulong p;
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for (; i < len; i++)
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{
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if (smallPrimesPow2[i] > n)
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{
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break;
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}
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p = smallPrimes[i];
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// TODO: Swap this modulo check for the shared small-prime cycle filter once the
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// divisor-cycle cache is mandatory, matching the PrimeTester improvements noted in
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// the CPU sieve benchmarks.
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if ((n % p) == 0UL)
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{
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return n == p;
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}
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}
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return true;
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}
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// Benchmarks (Mod5ULongBenchmarks) show the direct `% 5` is still cheaper (~0.26 ns vs 0.43 ns), so keep the modulo until a faster lookup is proven.
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// (Mod8/Mod10 stay masked because they win; Mod5 currently does not.)
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Mod10(this ulong value) => (value & 1UL) == 0UL
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? (value % 5UL) switch
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{
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0UL => 0UL,
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1UL => 6UL,
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2UL => 2UL,
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3UL => 8UL,
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_ => 4UL,
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}
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: (value % 5UL) switch
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{
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0UL => 5UL,
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1UL => 1UL,
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2UL => 7UL,
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3UL => 3UL,
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_ => 9UL,
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};
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Mod128(this ulong value) => value & 127UL;
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// Benchmarks confirm `%` beats our current Mod5/Mod3 helpers for 64-bit inputs, so leave these modulo operations in place until a superior lookup is available.
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static void Mod10_8_5_3(this ulong value, out ulong mod10, out ulong mod8, out ulong mod5, out ulong mod3)
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{
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mod8 = value & 7UL;
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// Benchmarks show `%` remains faster for the Mod5/Mod3 pair on ulong, so we keep the modulo path here for now.
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mod5 = value % 5UL;
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mod3 = value % 3UL;
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mod10 = (mod8 & 1UL) == 0UL
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? mod5 switch
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{
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0UL => 0UL,
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1UL => 6UL,
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2UL => 2UL,
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3UL => 8UL,
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_ => 4UL,
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}
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: mod5 switch
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{
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0UL => 5UL,
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1UL => 1UL,
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2UL => 7UL,
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3UL => 3UL,
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_ => 9UL,
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};
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}
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// Mod5/Mod3 lookup tables are currently slower on 64-bit operands; keep the direct modulo until benchmarks flip.
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static void Mod10_8_5_3Steps(this ulong value, out ulong step10, out ulong step8, out ulong step5, out ulong step3)
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{
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ulong mod8 = value & 7UL;
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// Same rationale: `%` wins in Mod5/Mod3 benches today, so avoid swapping until a faster lookup exists.
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ulong mod5 = value % 5UL;
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ulong mod3 = value % 3UL;
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ulong mod10 = (mod8 & 1UL) == 0UL
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? mod5 switch
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{
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0UL => 0UL,
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1UL => 6UL,
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2UL => 2UL,
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3UL => 8UL,
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_ => 4UL,
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}
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: mod5 switch
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{
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0UL => 5UL,
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1UL => 1UL,
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2UL => 7UL,
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3UL => 3UL,
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_ => 9UL,
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};
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step10 = mod10 + mod10;
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if (step10 >= 10UL)
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{
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step10 -= 10UL;
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}
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step8 = (mod8 + mod8) & 7UL;
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step5 = mod5 + mod5;
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if (step5 >= 5UL)
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{
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step5 -= 5UL;
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}
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step3 = mod3 + mod3;
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if (step3 >= 3UL)
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{
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step3 -= 3UL;
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}
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static UInt128 Mul64(this ulong a, ulong b) => ((UInt128)a.MulHigh(b) << 64) | (UInt128)(a * b);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulHigh(this ulong x, ulong y)
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{
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// TODO: Investigate replacing this manual decomposition with the UInt128-based implementation
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// for CPU callers; the latest benchmarks show the intrinsic path is an order of magnitude
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// faster, while GPU code can keep using GpuUInt128.MulHigh.
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ulong xLow = (uint)x;
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ulong xHigh = x >> 32;
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ulong yLow = (uint)y;
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ulong yHigh = y >> 32;
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ulong w1 = xLow * yHigh;
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ulong w2 = xHigh * yLow;
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ulong w3 = xLow * yLow;
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// Matching the layout used in GpuUInt128.MulHigh: introducing the
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// intermediate result looks like one extra store, but it lets RyuJIT keep
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// the accumulated high word entirely in registers. Without this explicit
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// local the JIT spills the partial sum, which is where the performance
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// regression in the benchmarks came from.
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ulong result = (xHigh * yHigh) + (w1 >> 32) + (w2 >> 32);
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result += ((w3 >> 32) + (uint)w1 + (uint)w2) >> 32;
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return result;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulHighGpuCompatible(this ulong x, ulong y)
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{
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GpuUInt128 product = new(x);
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product.Mul64(new GpuUInt128(y));
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return product.High;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong ModPow64(this ulong value, ulong exponent, ulong modulus)
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{
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ulong result = 1UL;
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// TODO: Replace this `%` with the Montgomery folding helper highlighted in MulMod64Benchmarks so the
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// modular exponentiation avoids the slow integer division before the ladder even starts.
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value %= modulus;
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while (exponent != 0UL)
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{
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if ((exponent & 1UL) != 0UL)
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{
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result = MulMod64(result, value, modulus);
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}
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value = MulMod64(value, value, modulus);
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exponent >>= 1;
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}
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return result;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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// TODO: Replace this fallback with the UInt128 Montgomery helper measured fastest in
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// MulMod64Benchmarks so CPU callers stop paying for triple modulo operations.
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public static ulong MulMod64(this ulong a, ulong b, ulong modulus) => (ulong)(UInt128)(((a % modulus) * (b % modulus)) % modulus);
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulMod64GpuCompatible(this ulong a, ulong b, ulong modulus)
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{
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// TODO: Remove this GPU-compatible shim from production once callers migrate to MulMod64,
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// which the benchmarks show is roughly 6-7× faster on dense 64-bit inputs.
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GpuUInt128 state = new(a % modulus);
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return state.MulMod(b, modulus);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MulMod64GpuCompatibleDeferred(this ulong a, ulong b, ulong modulus)
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{
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// TODO: Move this deferred helper to the benchmark suite; the baseline MulMod64 avoids the
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// 5-40× slowdown seen across real-world operand distributions.
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GpuUInt128 state = new(a);
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return state.MulModWithNativeModulo(b, modulus);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong MontgomeryMultiply(this ulong a, ulong b, ulong modulus, ulong nPrime)
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{
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ulong tLow = unchecked(a * b);
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ulong m = unchecked(tLow * nPrime);
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ulong mTimesModulusLow = unchecked(m * modulus);
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ulong result = unchecked(a.MulHigh(b) + m.MulHigh(modulus) + (unchecked(tLow + mTimesModulusLow) < tLow ? 1UL : 0UL));
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if (result >= modulus)
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{
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result -= modulus;
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}
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return result;
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Pow2MontgomeryModWindowedCpu(this ulong exponent, in MontgomeryDivisorData divisor, bool keepMontgomery)
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{
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ulong modulus = divisor.Modulus;
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if (exponent == 0UL)
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{
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return keepMontgomery ? divisor.MontgomeryOne : 1UL % modulus;
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}
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if (exponent <= Pow2WindowFallbackThreshold)
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{
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return Pow2MontgomeryModSingleBit(exponent, divisor, keepMontgomery);
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}
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int bitLength = GetPortableBitLength(exponent);
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int windowSize = GetWindowSize(bitLength);
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int oddPowerCount = 1 << (windowSize - 1);
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ulong result = divisor.MontgomeryOne;
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ulong nPrime = divisor.NPrime;
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Span<ulong> oddPowerStorage = oddPowerCount <= PerfectNumberConstants.MaxOddPowersCount ? stackalloc ulong[PerfectNumberConstants.MaxOddPowersCount] : default;
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ulong[]? pooledOddPowers = null;
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if (oddPowerCount < PerfectNumberConstants.PooledArrayThreshold)
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{
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oddPowerStorage = new ulong[oddPowerCount];
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}
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else
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{
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pooledOddPowers = ArrayPool<ulong>.Shared.Rent(oddPowerCount);
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oddPowerStorage = pooledOddPowers;
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}
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Span<ulong> oddPowers = oddPowerStorage[..oddPowerCount];
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InitializeMontgomeryOddPowersCpu(divisor, modulus, nPrime, oddPowers);
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int index = bitLength - 1;
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while (index >= 0)
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{
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if (((exponent >> index) & 1UL) == 0UL)
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{
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result = result.MontgomeryMultiply(result, modulus, nPrime);
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index--;
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continue;
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}
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int windowStart = index - windowSize + 1;
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if (windowStart < 0)
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{
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windowStart = 0;
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}
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while (((exponent >> windowStart) & 1UL) == 0UL)
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{
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windowStart++;
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}
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int windowLength = index - windowStart + 1;
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for (int square = 0; square < windowLength; square++)
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{
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result = result.MontgomeryMultiply(result, modulus, nPrime);
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}
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ulong mask = (1UL << windowLength) - 1UL;
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ulong windowValue = (exponent >> windowStart) & mask;
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int tableIndex = (int)((windowValue - 1UL) >> 1);
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ulong multiplier = oddPowers[tableIndex];
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result = result.MontgomeryMultiply(multiplier, modulus, nPrime);
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index = windowStart - 1;
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}
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if (pooledOddPowers is not null)
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{
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ArrayPool<ulong>.Shared.Return(pooledOddPowers);
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}
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if (keepMontgomery)
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{
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return result;
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}
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return result.MontgomeryMultiply(1UL, modulus, nPrime);
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}
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
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public static ulong Pow2MontgomeryModWindowedGpu(this ulong exponent, in MontgomeryDivisorData divisor, bool keepMontgomery)
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{
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ulong modulus = divisor.Modulus;
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if (exponent == 0UL)
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{
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return keepMontgomery ? divisor.MontgomeryOne : 1UL % modulus;
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}
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if (exponent <= Pow2WindowFallbackThreshold)
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{
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return Pow2MontgomeryModSingleBit(exponent, divisor, keepMontgomery);
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}
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int bitLength = GetPortableBitLength(exponent);
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int windowSize = GetWindowSize(bitLength);
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int oddPowerCount = 1 << (windowSize - 1);
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ulong result = divisor.MontgomeryOne;
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ulong nPrime = divisor.NPrime;
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ulong[] oddPowers = InitializeMontgomeryOddPowersGpu(divisor, modulus, nPrime, oddPowerCount);
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int index = bitLength - 1;
|
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while (index >= 0)
|
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{
|
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if (((exponent >> index) & 1UL) == 0UL)
|
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{
|
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result = result.MontgomeryMultiply(result, modulus, nPrime);
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index--;
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continue;
|
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}
|
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int windowStart = index - windowSize + 1;
|
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if (windowStart < 0)
|
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{
|
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windowStart = 0;
|
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}
|
||
|
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while (((exponent >> windowStart) & 1UL) == 0UL)
|
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{
|
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windowStart++;
|
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}
|
||
|
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int windowLength = index - windowStart + 1;
|
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for (int square = 0; square < windowLength; square++)
|
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{
|
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result = result.MontgomeryMultiply(result, modulus, nPrime);
|
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}
|
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|
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ulong mask = (1UL << windowLength) - 1UL;
|
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ulong windowValue = (exponent >> windowStart) & mask;
|
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int tableIndex = (int)((windowValue - 1UL) >> 1);
|
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ulong multiplier = oddPowers[tableIndex];
|
||
result = result.MontgomeryMultiply(multiplier, modulus, nPrime);
|
||
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index = windowStart - 1;
|
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}
|
||
|
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if (keepMontgomery)
|
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{
|
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return result;
|
||
}
|
||
|
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return result.MontgomeryMultiply(1UL, modulus, nPrime);
|
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}
|
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[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
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private static ulong Pow2MontgomeryModSingleBit(ulong exponent, in MontgomeryDivisorData divisor, bool keepMontgomery)
|
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{
|
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ulong modulus = divisor.Modulus;
|
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ulong nPrime = divisor.NPrime;
|
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ulong result = divisor.MontgomeryOne;
|
||
ulong baseVal = divisor.MontgomeryTwo;
|
||
ulong remainingExponent = exponent;
|
||
|
||
while (remainingExponent != 0UL)
|
||
{
|
||
if ((remainingExponent & 1UL) != 0UL)
|
||
{
|
||
result = result.MontgomeryMultiply(baseVal, modulus, nPrime);
|
||
}
|
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remainingExponent >>= 1;
|
||
if (remainingExponent == 0UL)
|
||
{
|
||
break;
|
||
}
|
||
|
||
baseVal = baseVal.MontgomeryMultiply(baseVal, modulus, nPrime);
|
||
}
|
||
|
||
return keepMontgomery ? result : result.MontgomeryMultiply(1UL, modulus, nPrime);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static int GetPortableBitLength(ulong value)
|
||
{
|
||
if (value == 0UL)
|
||
{
|
||
return 0;
|
||
}
|
||
|
||
return 64 - XMath.LeadingZeroCount(value);
|
||
}
|
||
|
||
private static int GetWindowSize(int bitLength)
|
||
{
|
||
if (bitLength <= 6)
|
||
{
|
||
return Math.Max(bitLength, 1);
|
||
}
|
||
|
||
if (bitLength <= 23)
|
||
{
|
||
return 4;
|
||
}
|
||
|
||
if (bitLength <= 79)
|
||
{
|
||
return 5;
|
||
}
|
||
|
||
if (bitLength <= 239)
|
||
{
|
||
return 6;
|
||
}
|
||
|
||
if (bitLength <= 671)
|
||
{
|
||
return 7;
|
||
}
|
||
|
||
return Pow2WindowSize;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static void InitializeMontgomeryOddPowersCpu(in MontgomeryDivisorData divisor, ulong modulus, ulong nPrime, Span<ulong> destination)
|
||
{
|
||
int oddPowerCount = destination.Length;
|
||
if (oddPowerCount == 0)
|
||
{
|
||
return;
|
||
}
|
||
|
||
bool computedOnGpu = oddPowerCount <= PerfectNumberConstants.MaxOddPowersCount
|
||
&& MontgomeryOddPowerGpu.TryCompute(divisor, oddPowerCount, destination);
|
||
if (computedOnGpu)
|
||
{
|
||
return;
|
||
}
|
||
|
||
destination[0] = divisor.MontgomeryTwo;
|
||
if (oddPowerCount == 1)
|
||
{
|
||
return;
|
||
}
|
||
|
||
ulong square = divisor.MontgomeryTwoSquared;
|
||
ulong current = destination[0];
|
||
for (int i = 1; i < oddPowerCount; i++)
|
||
{
|
||
current = current.MontgomeryMultiply(square, modulus, nPrime);
|
||
destination[i] = current;
|
||
}
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
private static ulong[] InitializeMontgomeryOddPowersGpu(in MontgomeryDivisorData divisor, ulong modulus, ulong nPrime, int oddPowerCount)
|
||
{
|
||
ulong[] oddPowers = new ulong[PerfectNumberConstants.MaxOddPowersCount];
|
||
oddPowers[0] = divisor.MontgomeryTwo;
|
||
if (oddPowerCount == 1)
|
||
{
|
||
return oddPowers;
|
||
}
|
||
|
||
ulong square = divisor.MontgomeryTwoSquared;
|
||
for (int i = 1; i < oddPowerCount; i++)
|
||
{
|
||
ulong previous = oddPowers[i - 1];
|
||
oddPowers[i] = previous.MontgomeryMultiply(square, modulus, nPrime);
|
||
}
|
||
|
||
return oddPowers;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Pow2MontgomeryModWithCycleCpu(this ulong exponent, ulong cycleLength, in MontgomeryDivisorData divisor)
|
||
{
|
||
ulong rotationCount = exponent % cycleLength;
|
||
return Pow2MontgomeryModWindowedCpu(rotationCount, divisor, keepMontgomery: false);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Pow2MontgomeryModFromCycleRemainderCpu(this ulong reducedExponent, in MontgomeryDivisorData divisor)
|
||
{
|
||
return Pow2MontgomeryModWindowedCpu(reducedExponent, divisor, keepMontgomery: false);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Pow2MontgomeryModWithCycleGpu(this ulong exponent, ulong cycleLength, in MontgomeryDivisorData divisor)
|
||
{
|
||
ulong rotationCount = exponent % cycleLength;
|
||
return Pow2MontgomeryModWindowedGpu(rotationCount, divisor, keepMontgomery: false);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static ulong Pow2MontgomeryModFromCycleRemainderGpu(this ulong reducedExponent, in MontgomeryDivisorData divisor)
|
||
{
|
||
return Pow2MontgomeryModWindowedGpu(reducedExponent, divisor, keepMontgomery: false);
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static UInt128 PowMod(this ulong exponent, UInt128 modulus)
|
||
{
|
||
UInt128 result = UInt128.One;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Port this scalar PowMod fallback to the ProcessEightBitWindows helper so CPU callers get the
|
||
// eight-bit window wins measured against the classic square-and-subtract implementation.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == 0UL)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == UInt128.One)
|
||
return UInt128.Zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < 64 || modulus < 4)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - 1)) == 0)
|
||
{
|
||
result = UInt128.One << (int)(exponent & 127);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing exponentLoopIndex to iterate again over the exponent range for the general-case accumulation.
|
||
exponentLoopIndex = 0UL;
|
||
// Reusing result after resetting it for the general modulus accumulation phase.
|
||
result = UInt128.One;
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using a known cycle length.
|
||
/// </summary>
|
||
public static UInt128 PowModWithCycle(this ulong exponent, UInt128 modulus, ulong cycleLength)
|
||
{
|
||
UInt128 result = UInt128.One;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Wire this cycle-aware overload into the ProcessEightBitWindows helper so the reduced exponent path
|
||
// inherits the faster windowed pow2 routine highlighted in the Pow2Montgomery benchmarks.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == 0UL)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == UInt128.One)
|
||
return UInt128.Zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < 64 || modulus < 4)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - 1)) == 0)
|
||
{
|
||
result = UInt128.One << (int)(exponent & 127);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing exponentLoopIndex to iterate over the rotation count for the cycle-aware path.
|
||
exponentLoopIndex = 0UL;
|
||
// Reusing result after resetting it for the rotation-accumulation pass.
|
||
result = UInt128.One;
|
||
// TODO: Replace this modulo with the cached cycle remainder produced by the divisor-cycle cache so PowModWithCycle avoids
|
||
// repeated `%` work, matching the ProcessEightBitWindows wins captured in Pow2MontgomeryModCycleComputationBenchmarks.
|
||
ulong rotationCount = exponent % cycleLength;
|
||
for (; exponentLoopIndex < rotationCount; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using a known cycle length.
|
||
/// </summary>
|
||
public static UInt128 PowModWithCycle(this ulong exponent, UInt128 modulus, UInt128 cycleLength)
|
||
{
|
||
UInt128 result = UInt128.One;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Replace this UInt128-cycle overload with the ProcessEightBitWindows helper so large-exponent CPU scans
|
||
// reuse the faster windowed pow2 ladder instead of the manual rotation loop measured to lag behind in benchmarks.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == UInt128.Zero)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == UInt128.One)
|
||
return UInt128.Zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < UInt128Numbers.SixtyFour || modulus < UInt128Numbers.Four)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - UInt128.One)) == UInt128.Zero)
|
||
{
|
||
result = UInt128.One << (int)(exponent & UInt128Numbers.OneHundredTwentySeven);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing result after resetting it for the rotation-driven accumulation phase.
|
||
result = UInt128.One;
|
||
// TODO: Swap this modulo with the upcoming UInt128 cycle remainder helper so large-exponent scans reuse cached
|
||
// reductions instead of recomputing `%` for every lookup, as demonstrated in Pow2MontgomeryModCycleComputationBenchmarks.
|
||
UInt128 rotationCount = exponent % cycleLength;
|
||
UInt128 rotationIndex = UInt128.Zero;
|
||
while (rotationIndex < rotationCount)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
|
||
rotationIndex += UInt128.One;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using a known cycle length.
|
||
/// </summary>
|
||
public static UInt128 PowModWithCycle(this UInt128 exponent, UInt128 modulus, UInt128 cycleLength)
|
||
{
|
||
UInt128 one = UInt128.One,
|
||
result = one,
|
||
zero = UInt128.Zero;
|
||
ulong exponentLoopIndex = 0UL;
|
||
|
||
// TODO: Migrate this UInt128 exponent overload to ProcessEightBitWindows so the large-cycle reductions drop the
|
||
// slow manual loop that underperforms the windowed pow2 helper in the Pow2 benchmark suite.
|
||
// Return 1 because 2^0 = 1
|
||
if (exponent == zero)
|
||
return result;
|
||
|
||
// Any number mod 1 is 0
|
||
if (modulus == one)
|
||
return zero;
|
||
|
||
// For small exponents, do classic method
|
||
if (exponent < UInt128Numbers.SixtyFour || modulus < UInt128Numbers.Four)
|
||
{
|
||
for (; exponentLoopIndex < exponent; exponentLoopIndex++)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
// Special case: if modulus is a power of two, use bitmasking for efficiency and correctness
|
||
if ((modulus & (modulus - one)) == zero)
|
||
{
|
||
result = one << (int)(exponent & UInt128Numbers.OneHundredTwentySeven);
|
||
return result & (modulus - 1);
|
||
}
|
||
|
||
// Reusing result after resetting it for the rotation-driven accumulation phase.
|
||
result = one;
|
||
// TODO: Swap this modulo with the shared UInt128 cycle remainder helper once available so CRT powmods reuse cached
|
||
// reductions in the windowed ladder, avoiding the `%` cost highlighted in Pow2MontgomeryModCycleComputationBenchmarks.
|
||
UInt128 rotationCount = exponent % cycleLength;
|
||
|
||
// We're reusing "zero" as rotation index for just a little better performance
|
||
while (zero < rotationCount)
|
||
{
|
||
result <<= 1;
|
||
if (result >= modulus)
|
||
result -= modulus;
|
||
|
||
zero += one;
|
||
}
|
||
|
||
return result;
|
||
}
|
||
|
||
/// <summary>
|
||
/// Computes 2^exponent mod modulus using iterative CRT composition from mod 10 up to modulus.
|
||
/// Only for modulus >= 10 and reasonable size.
|
||
/// </summary>
|
||
public static UInt128 PowModCrt(this ulong exponent, UInt128 modulus, MersenneDivisorCycles cycles)
|
||
{
|
||
if (modulus < 10)
|
||
return PowMod(exponent, modulus); // fallback to classic
|
||
|
||
// Use cycle length 4 for mod 10
|
||
UInt128 currentModulus = 10,
|
||
cycle,
|
||
modulusCandidate = 11,
|
||
remainderForCandidate,
|
||
result = PowModWithCycle(exponent, 10, 4),
|
||
zero = UInt128.Zero;
|
||
|
||
for (; modulusCandidate <= modulus; modulusCandidate++)
|
||
{
|
||
cycle = MersenneDivisorCycles.GetCycle(modulusCandidate);
|
||
remainderForCandidate = cycle > zero
|
||
? PowModWithCycle(exponent, modulusCandidate, cycle)
|
||
: PowMod(exponent, modulusCandidate);
|
||
|
||
// Solve x ≡ result mod currentModulus
|
||
// x ≡ remM mod m
|
||
// Find x mod (currentModulus * m)
|
||
// Since currentModulus and m are coprime, use CRT:
|
||
// x = result + currentModulus * t, where t ≡ (remM - result) * inv(currentModulus, m) mod m
|
||
|
||
// TODO: Replace this `% modulusCandidate` with the cached residue helper derived from Mod10_8_5_3Benchmarks so CRT
|
||
// composition avoids repeated modulo divisions when combining residues for large divisor sets.
|
||
result += currentModulus * ((remainderForCandidate + modulusCandidate - (result % modulusCandidate)) * ModInverse(currentModulus, modulusCandidate) % modulusCandidate);
|
||
currentModulus *= modulusCandidate;
|
||
|
||
if (currentModulus >= modulus)
|
||
break;
|
||
}
|
||
|
||
// TODO: Swap this final `% modulus` with the pooled remainder cache so the CRT result write-back avoids one more division,
|
||
// aligning with the optimizations captured in Mod10_8_5_3Benchmarks.
|
||
return result % modulus;
|
||
}
|
||
|
||
// Helper: modular inverse (extended Euclidean algorithm)
|
||
private static UInt128 ModInverse(UInt128 a, UInt128 m)
|
||
{
|
||
UInt128 m0 = m,
|
||
originalA,
|
||
originalM,
|
||
temp,
|
||
x0 = 0,
|
||
x1 = 1;
|
||
|
||
if (m == 1)
|
||
{
|
||
return 0;
|
||
}
|
||
|
||
while (a > 1)
|
||
{
|
||
originalA = a;
|
||
originalM = m;
|
||
m = originalA % originalM;
|
||
a = originalM;
|
||
temp = x0;
|
||
x0 = x1 - (originalA / originalM) * x0;
|
||
x1 = temp;
|
||
}
|
||
|
||
if (x1 < 0)
|
||
{
|
||
x1 += m0;
|
||
}
|
||
|
||
return x1;
|
||
}
|
||
|
||
[MethodImpl(MethodImplOptions.AggressiveInlining)]
|
||
public static bool SharesFactorWithExponentMinusOne(this ulong exponent)
|
||
{
|
||
ulong prime, value = exponent - 1UL;
|
||
value >>= BitOperations.TrailingZeroCount(value);
|
||
uint[] smallPrimes = PrimesGenerator.SmallPrimes;
|
||
ulong[] smallPrimesPow2 = PrimesGenerator.SmallPrimesPow2;
|
||
int i = 1, smallPrimesLength = smallPrimes.Length;
|
||
|
||
for (; i < smallPrimesLength && smallPrimesPow2[i] <= value; i++)
|
||
{
|
||
prime = smallPrimes[i];
|
||
if (value % prime != 0UL)
|
||
{
|
||
continue;
|
||
}
|
||
|
||
if (exponent % prime.CalculateOrder() == 0UL)
|
||
{
|
||
return true;
|
||
}
|
||
|
||
do
|
||
{
|
||
value /= prime;
|
||
}
|
||
while (value % prime == 0UL);
|
||
}
|
||
|
||
if (value > 1UL && exponent % value.CalculateOrder() == 0UL)
|
||
{
|
||
return true;
|
||
}
|
||
|
||
return false;
|
||
}
|
||
}
|