Use analytic delta updates in CPU divisor sessions
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Marek Lemańczyk committed
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@@ -343,6 +343,7 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
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private sealed class DivisorScanSession : IMersenneNumberDivisorByDivisorTester.IDivisorScanSession
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{
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private readonly MersenneNumberDivisorByDivisorCpuTester _owner;
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private readonly Dictionary<ulong, PrimeDivisorState> _primeStates = new();
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private bool _disposed;
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internal DivisorScanSession(MersenneNumberDivisorByDivisorCpuTester owner)
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@@ -353,6 +354,7 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
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internal void Reset()
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{
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_disposed = false;
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_primeStates.Clear();
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}
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public void CheckDivisor(ulong divisor, in MontgomeryDivisorData divisorData, ulong divisorCycle, ReadOnlySpan<ulong> primes, Span<byte> hits)
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@@ -384,21 +386,50 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
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}
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}
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// Keep these remainder steppers in place so future updates continue reusing the previously computed residues.
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// They are critical for avoiding repeated full Montgomery exponentiation work when scanning divisors.
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var exponentStepper = new ExponentRemainderStepper(cachedData);
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if (!exponentStepper.IsValidModulus)
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var cycleStepper = new CycleRemainderStepper(divisorCycle);
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ExponentRemainderStepper exponentStepper = default;
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bool exponentStepperInitialized = false;
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byte hitValue = 0; // Reused for every prime processed in this batch.
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bool EnsureExponentStepper()
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{
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hits.Clear();
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return;
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if (exponentStepperInitialized)
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{
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return true;
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}
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exponentStepper = new ExponentRemainderStepper(cachedData);
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if (!exponentStepper.IsValidModulus)
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{
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hits.Clear();
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return false;
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}
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exponentStepperInitialized = true;
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return true;
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}
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var cycleStepper = new CycleRemainderStepper(divisorCycle);
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ulong remainder = cycleStepper.Initialize(primes[0]);
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hits[0] = remainder == 0UL
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? (exponentStepper.ComputeNextIsUnity(primes[0]) ? (byte)1 : (byte)0)
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: (byte)0;
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if (remainder == 0UL)
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{
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if (TryComputeAnalyticHit(primes[0], divisor, cachedData, out hitValue))
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{
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hits[0] = hitValue;
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}
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else
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{
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if (!EnsureExponentStepper())
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{
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return;
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}
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hits[0] = exponentStepper.ComputeNextIsUnity(primes[0]) ? (byte)1 : (byte)0;
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}
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}
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else
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{
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hits[0] = 0;
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}
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for (int i = 1; i < length; i++)
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{
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@@ -409,10 +440,197 @@ public sealed class MersenneNumberDivisorByDivisorCpuTester : IMersenneNumberDiv
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continue;
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}
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if (TryComputeAnalyticHit(primes[i], divisor, cachedData, out hitValue))
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{
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hits[i] = hitValue;
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continue;
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}
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if (!EnsureExponentStepper())
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{
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return;
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}
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hits[i] = exponentStepper.ComputeNextIsUnity(primes[i]) ? (byte)1 : (byte)0;
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}
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}
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private bool TryComputeAnalyticHit(ulong prime, ulong divisor, in MontgomeryDivisorData divisorData, out byte hit)
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{
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hit = 0;
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if (!TryGetMersenneMinusOne(prime, out ulong mersenne))
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{
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return false;
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}
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if (divisor <= 1UL)
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{
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return false;
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}
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ulong step = prime << 1;
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if (step == 0UL)
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{
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return false;
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}
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if (!_primeStates.TryGetValue(prime, out PrimeDivisorState state) || !state.HasState)
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{
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state = CreatePrimeState(step, divisor, mersenne, divisorData);
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_primeStates[prime] = state;
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hit = state.Remainder == 0UL ? (byte)1 : (byte)0;
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return true;
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}
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if (state.Step != step || divisor <= state.LastDivisor)
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{
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state = CreatePrimeState(step, divisor, mersenne, divisorData);
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_primeStates[prime] = state;
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hit = state.Remainder == 0UL ? (byte)1 : (byte)0;
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return true;
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}
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ulong difference = divisor - state.LastDivisor;
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if (difference == 0UL)
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{
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hit = state.Remainder == 0UL ? (byte)1 : (byte)0;
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return true;
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}
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if (difference % step != 0UL)
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{
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state = CreatePrimeState(step, divisor, mersenne, divisorData);
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_primeStates[prime] = state;
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hit = state.Remainder == 0UL ? (byte)1 : (byte)0;
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return true;
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}
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ulong increments = difference / step;
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ulong quotient = state.Quotient;
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ulong remainder = state.Remainder;
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ulong currentDivisor = state.LastDivisor;
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for (ulong index = 0UL; index < increments; index++)
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{
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ulong nextDivisor = currentDivisor + step;
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Int128 stepTimesQuotient = (Int128)step * (Int128)quotient;
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Int128 numerator = stepTimesQuotient - (Int128)remainder;
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Int128 divisorInt = (Int128)nextDivisor;
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Int128 deltaInt = numerator >= 0
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? (numerator + divisorInt - Int128.One) / divisorInt
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: numerator / divisorInt;
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if (deltaInt < Int128.Zero)
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{
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deltaInt = Int128.Zero;
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}
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ulong delta = (ulong)deltaInt;
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if (delta > quotient)
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{
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state = CreatePrimeState(step, divisor, mersenne, divisorData);
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_primeStates[prime] = state;
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hit = state.Remainder == 0UL ? (byte)1 : (byte)0;
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return true;
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}
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Int128 t = (Int128)remainder - stepTimesQuotient;
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Int128 nextRemainder = t + deltaInt * divisorInt;
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if (nextRemainder < Int128.Zero || nextRemainder >= divisorInt)
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{
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state = CreatePrimeState(step, divisor, mersenne, divisorData);
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_primeStates[prime] = state;
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hit = state.Remainder == 0UL ? (byte)1 : (byte)0;
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return true;
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}
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quotient -= delta;
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remainder = (ulong)nextRemainder;
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currentDivisor = nextDivisor;
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}
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ulong residue = ComputeResidueFromRemainder(remainder, divisor);
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state.LastDivisor = divisor;
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state.Quotient = quotient;
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state.Remainder = remainder;
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state.MontgomeryResidue = ToMontgomeryResidue(residue, divisorData);
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// Reuse state to persist the updated quotient and remainder tuple for the next divisor.
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_primeStates[prime] = state;
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hit = remainder == 0UL ? (byte)1 : (byte)0;
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return true;
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}
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private static PrimeDivisorState CreatePrimeState(ulong step, ulong divisor, ulong mersenne, in MontgomeryDivisorData divisorData)
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{
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PrimeDivisorState state = new()
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{
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Step = step,
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LastDivisor = divisor,
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HasState = true,
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};
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ulong quotient = Math.DivRem(mersenne, divisor, out ulong remainder);
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state.Quotient = quotient;
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state.Remainder = remainder;
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ulong residue = ComputeResidueFromRemainder(remainder, divisor);
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state.MontgomeryResidue = ToMontgomeryResidue(residue, divisorData);
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return state;
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}
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private static bool TryGetMersenneMinusOne(ulong exponent, out ulong mersenne)
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{
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if (exponent <= 63UL)
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{
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mersenne = (1UL << (int)exponent) - 1UL;
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return true;
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}
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if (exponent == 64UL)
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{
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mersenne = ulong.MaxValue;
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return true;
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}
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mersenne = 0UL;
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return false;
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}
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private static ulong ComputeResidueFromRemainder(ulong remainder, ulong divisor)
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{
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ulong residue = remainder + 1UL;
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if (residue >= divisor)
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{
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residue -= divisor;
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}
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return residue;
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}
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private static ulong ToMontgomeryResidue(ulong value, in MontgomeryDivisorData divisorData)
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{
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ulong modulus = divisorData.Modulus;
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if (modulus <= 1UL)
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{
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return 0UL;
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}
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value %= modulus;
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UInt128 shifted = (UInt128)value << 64;
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return (ulong)(shifted % modulus);
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}
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private struct PrimeDivisorState
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{
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public ulong Step;
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public ulong LastDivisor;
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public ulong Quotient;
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public ulong Remainder;
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public ulong MontgomeryResidue;
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public bool HasState;
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}
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public void Dispose()
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{
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if (_disposed)
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@@ -0,0 +1,119 @@
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# CPU `--mersenne=bydivisor` pipeline notes
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This document summarizes how the `EvenPerfectBitScanner` executable executes the
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`--mersenne=bydivisor` mode on the CPU, and how the analytic observations we
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derived for the quotients
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\[
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Q_k = \frac{N}{d_k}, \qquad d_k = 2pk + 1, \qquad N = 2^p - 1,
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\]
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can help us identify concrete optimization opportunities.
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## High-level control flow
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* `Program.Main` wires `_byDivisorTester` to a CPU implementation whenever the
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CLI switches resolve to `--mersenne=bydivisor` and
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`--mersenne-device=cpu`. The CLI also forces a block size of 1 so that the
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pipeline pushes exponents to the tester one at a time.
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(`Program.cs`, lines 459-520.)
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* When the mode starts, `RunByDivisorMode` delegates to
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`MersenneNumberDivisorByDivisorTester.Run`, passing the candidate list from
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`--filter-p`, the CPU tester instance and the optional replay of previous
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outcomes. (`Program.cs`, lines 809-826.)
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* `MersenneNumberDivisorByDivisorTester.Run` sorts and deduplicates the
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candidate exponents, drops the ones already confirmed in prior runs, validates
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primality with the `Open.Numeric.Primes` enumerator and configures the tester
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with the largest remaining exponent. (`MersenneNumberDivisorByDivisorTester.cs`,
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lines 14-162.)
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* For each confirmed prime exponent, the CPU tester computes the largest divisor
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it needs to probe through `ComputeAllowedMaxDivisor`. If the range is non-empty
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it enters the `CheckDivisors` loop. (`MersenneNumberDivisorByDivisorCpuTester.cs`,
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lines 83-110 and 161-266.)
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## Inner loop behaviour (`CheckDivisors`)
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The hot path is the `while (divisor <= limit)` loop in
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`MersenneNumberDivisorByDivisorCpuTester.CheckDivisors`.
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1. The loop enumerates divisor candidates of the form `d_k = 2pk + 1` by
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incrementing `divisor` by `step = 2p` every iteration. (`MersenneNumberDivisorByDivisorCpuTester.cs`,
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lines 177-256.)
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2. Small modulus filters track `d_k (mod 3·5·7·8·10·11)` to discard impossible
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candidates without touching the divisor-cycle cache. (`MersenneNumberDivisorByDivisorCpuTester.cs`,
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lines 195-263.)
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3. For every admissible divisor the tester fetches its Montgomery data and the
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cached cycle length. When the cache does not contain an entry the
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implementation recomputes the cycle by factoring the divisor. (`MersenneNumberDivisorByDivisorCpuTester.cs`,
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lines 223-254.)
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4. A cycle equal to the tested exponent proves divisibility, so the loop aborts
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as soon as one hit is found. (`MersenneNumberDivisorByDivisorCpuTester.cs`,
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lines 241-247.)
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`DivisorScanSession.CheckDivisor` follows the same `d_k` progression when a
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batch API is used. It keeps a running Montgomery residue via
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`ExponentRemainderStepper`, but the stepper currently recomputes the delta by
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calling a full `Pow2MontgomeryModWindowed` whenever the exponent jumps.
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(`MersenneNumberDivisorByDivisorCpuTester.cs`, lines 358-405 and
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`ExponentRemainderStepper.cs`, lines 26-95.)
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## Where the analytic model fits
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The formulas we established for the quotient sequence
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\[
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\frac{Q_{k+1}}{Q_k} = \frac{2pk + 1}{2p(k+1) + 1}, \qquad
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Q_k - Q_{k+1} = \frac{N·2p}{(2pk + 1)(2p(k+1) + 1)}
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\]
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tell us that the successive quotients and remainders shrink smoothly and that the
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per-step update depends only on the previous `(Q_k, r_k)` state and the next
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`d_{k+1}`. This lines up with several TODO comments that highlight missing
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incremental updates:
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* `ExponentRemainderStepper` already caches the previous exponent and a running
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Montgomery residue, but it falls back to a full modular exponentiation to
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compute the delta between consecutive divisors. (`ExponentRemainderStepper.cs`,
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lines 49-72.)
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* `CheckDivisors` rebuilds the Montgomery data for every hit candidate even
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though the sequence of `d_k` values is strictly monotone.
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The quotient identity gives us a way to compute `Δ_k = Q_k - Q_{k+1}` by reading
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`Q_k` and the inexpensive division `(2pQ_k - r_k) / d_{k+1}`. Once we know `Δ_k`
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we can update
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\[
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Q_{k+1} = Q_k - Δ_k, \qquad r_{k+1} = r_k - 2p·Q_k + Δ_k · d_{k+1},
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\]
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without ever evaluating `2^p mod d_{k+1}` from scratch. Incorporating this into
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`ExponentRemainderStepper` would let the CPU path reuse a single Montgomery state
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across the entire scan and swap the current delta exponentiation (which costs a
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full modular ladder) for one multiply-high with a precomputed reciprocal of
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`d_{k+1}`. The loop already maintains `d_k` and `step`, so the extra bookkeeping
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fits naturally.
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## Optimization ideas grounded in the code
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1. **Delta-driven Montgomery updates.** Extend `DivisorScanSession` so it tracks
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`(Q_k, r_k)` alongside the Montgomery residue and uses the analytic delta
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formula to derive the next residue. This replaces the per-step powmod in
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`ExponentRemainderStepper.ComputeNext` with a handful of 64-bit operations.
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(`MersenneNumberDivisorByDivisorCpuTester.cs`, lines 358-405;
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`ExponentRemainderStepper.cs`, lines 49-72.)
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2. **Bit-range skipping.** Because `Q_{k+1} / Q_k` is exactly the rational
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`(2pk + 1)/(2p(k+1) + 1)`, we can predict when `⌊log₂ Q_k⌋` loses a bit and
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jump the loop counter to the first `k` that crosses a target quotient bound
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instead of checking each divisor. This is compatible with the existing
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`allowedMax` guard because it only modifies how we arrive at the stopping
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divisor. (`MersenneNumberDivisorByDivisorCpuTester.cs`, lines 161-266.)
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3. **Reuse Montgomery metadata.** The loop already calls `MontgomeryDivisorDataCache.Get`
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for each divisor, but the cache lookup is repeated even though the next
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divisor is a small increment away. Persisting the previous `MontgomeryDivisorData`
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together with the `(Q_k, r_k)` tuple lets us skip redundant cache reads when
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the divisor stays within the hot range. (`MersenneNumberDivisorByDivisorCpuTester.cs`,
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lines 223-256.)
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These adjustments directly leverage the quotient progression we analysed and
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align with the existing TODO list, giving us a clear path to reduce the number of
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modular exponentiations and cache lookups in the CPU by-divisor path.
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