Optimize BigInteger.Multiply by Toom-Cook multiplication

[kzrnm]

https://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication

I updated BigInteger multiplication to use the Toom-3 algorithm.

The current Karatsuba algorithm has a time complexity of $O(n^{\log_2{3}}) \simeq O(n^{1.58})$, which is expected to improve to $O(n^{\log_3{5}}) \simeq O(n^{1.46})$ resulting in better performance.

in other languages:

• OpenJDK
• JavaScript-V8

About the Implementation

• Merged SquareThreshold and MultiplyKaratsubaThreshold.
  • Since both had the same value, this improves testability.
• Added {[MethodImpl(MethodImplOptions.AggressiveInlining)] to avoid stack consumption when determining the algorithm.
• In some cases, the Toom-2.5 algorithm is used.

Why MultiplyToom3Threshold is 256?

Based on the benchmark results, I decided to set MultiplyToom3Threshold to 256.

Benchmark

When the number of digits is small, the preprocessing for algorithm selection is relatively high, leading to a slight regression—for example, a computation that previously took 19 μs now takes 20 μs.

However, as the number of digits increases, performance improves; for instance, a multiplication that used to take 750 μs is now completed in 690 μs.

Code

using BenchmarkDotNet.Attributes;
using BenchmarkDotNet.Running;
using System.Numerics;
using System.Runtime.InteropServices;

[MemoryDiagnoser(false)]
[HideColumns("Job", "Error", "StdDev", "Median", "RatioSD")]
public class MultiplySomeSizeTests
{
    public IEnumerable<object> GetMultiplyArgs()
    {
        var rnd = new Random(227);
        var bytes = new byte[1000000];
        var lengths = new int[] { 100, 500, 1000, 10000, 100000, 1000000 };
        for (int i = lengths.Length - 1; i >= 0; i--)
        {
            var largeLength = lengths[i];
            var large = Make(largeLength);
            foreach (var p in new double[] { 0.999999999999999, 0.75, 0.5, 0.25 })
            {
                var smallLength = (int)(p * lengths[i]);
                var small = Make(smallLength);
                yield return new Data($"{largeLength:D7}-{smallLength:D7}", large, small);
            }

            yield return new Data($"Square{largeLength:D7}", large, large);
        }
        BigInteger Make(int length)
        {
            var b = bytes.AsSpan().Slice(0, length);
            rnd.NextBytes(b);
            return new BigInteger(b, isUnsigned: true);
        }
    }

    public record Data(string Name, BigInteger Large, BigInteger Small)
    {
        public override string ToString() => Name;
    }

    [Benchmark]
    [ArgumentsSource(nameof(GetMultiplyArgs))]
    public BigInteger Multiply(Data data)
    {
        return data.Large * data.Small;
    }
}

BenchmarkDotNet v0.13.12, Windows 11 (10.0.26100.3194)
13th Gen Intel Core i5-13500, 1 CPU, 20 logical and 14 physical cores
.NET SDK 10.0.100-alpha.1.25077.2
  [Host]   : .NET 10.0.0 (10.0.25.7313), X64 RyuJIT AVX2
  ShortRun : .NET 10.0.0 (42.42.42.42424), X64 RyuJIT AVX2

Job=ShortRun  IterationCount=3  LaunchCount=1  
WarmupCount=3  

Method	Toolchain	data	Mean	Ratio	Gen0	Gen1	Gen2	Allocated	Alloc Ratio
Multiply	\main\corerun.exe	0000100-0000025	132.2 ns	1.00	0.0119	-	-	152 B	1.00
Multiply	\pr0128\corerun.exe	0000100-0000025	132.6 ns	1.00	0.0119	-	-	152 B	1.00
Multiply	\pr0256\corerun.exe	0000100-0000025	134.3 ns	1.02	0.0119	-	-	152 B	1.00
Multiply	\pr0512\corerun.exe	0000100-0000025	133.3 ns	1.01	0.0119	-	-	152 B	1.00
Multiply	\pr1024\corerun.exe	0000100-0000025	133.2 ns	1.01	0.0119	-	-	152 B	1.00
Multiply	\main\corerun.exe	0000100-0000050	223.6 ns	1.00	0.0138	-	-	176 B	1.00
Multiply	\pr0128\corerun.exe	0000100-0000050	241.7 ns	1.08	0.0138	-	-	176 B	1.00
Multiply	\pr0256\corerun.exe	0000100-0000050	232.8 ns	1.04	0.0138	-	-	176 B	1.00
Multiply	\pr0512\corerun.exe	0000100-0000050	231.1 ns	1.03	0.0138	-	-	176 B	1.00
Multiply	\pr1024\corerun.exe	0000100-0000050	228.1 ns	1.02	0.0138	-	-	176 B	1.00
Multiply	\main\corerun.exe	0000100-0000075	322.2 ns	1.00	0.0157	-	-	200 B	1.00
Multiply	\pr0128\corerun.exe	0000100-0000075	322.2 ns	1.00	0.0157	-	-	200 B	1.00
Multiply	\pr0256\corerun.exe	0000100-0000075	336.5 ns	1.04	0.0157	-	-	200 B	1.00
Multiply	\pr0512\corerun.exe	0000100-0000075	323.3 ns	1.00	0.0157	-	-	200 B	1.00
Multiply	\pr1024\corerun.exe	0000100-0000075	324.6 ns	1.01	0.0157	-	-	200 B	1.00
Multiply	\main\corerun.exe	0000100-0000099	399.4 ns	1.00	0.0176	-	-	224 B	1.00
Multiply	\pr0128\corerun.exe	0000100-0000099	445.1 ns	1.11	0.0176	-	-	224 B	1.00
Multiply	\pr0256\corerun.exe	0000100-0000099	436.7 ns	1.09	0.0176	-	-	224 B	1.00
Multiply	\pr0512\corerun.exe	0000100-0000099	407.8 ns	1.02	0.0176	-	-	224 B	1.00
Multiply	\pr1024\corerun.exe	0000100-0000099	438.0 ns	1.10	0.0176	-	-	224 B	1.00
Multiply	\main\corerun.exe	0000500-0000125	2,540.3 ns	1.00	0.0496	-	-	656 B	1.00
Multiply	\pr0128\corerun.exe	0000500-0000125	2,650.3 ns	1.04	0.0496	-	-	656 B	1.00
Multiply	\pr0256\corerun.exe	0000500-0000125	2,621.6 ns	1.03	0.0496	-	-	656 B	1.00
Multiply	\pr0512\corerun.exe	0000500-0000125	2,648.0 ns	1.04	0.0496	-	-	656 B	1.00
Multiply	\pr1024\corerun.exe	0000500-0000125	2,644.0 ns	1.04	0.0496	-	-	656 B	1.00
Multiply	\main\corerun.exe	0000500-0000250	3,945.4 ns	1.00	0.0610	-	-	776 B	1.00
Multiply	\pr0128\corerun.exe	0000500-0000250	4,185.3 ns	1.06	0.0610	-	-	776 B	1.00
Multiply	\pr0256\corerun.exe	0000500-0000250	4,546.5 ns	1.15	0.0610	-	-	776 B	1.00
Multiply	\pr0512\corerun.exe	0000500-0000250	4,777.1 ns	1.21	0.0610	-	-	776 B	1.00
Multiply	\pr1024\corerun.exe	0000500-0000250	4,604.5 ns	1.17	0.0610	-	-	776 B	1.00
Multiply	\main\corerun.exe	0000500-0000375	5,344.4 ns	1.00	0.0687	-	-	904 B	1.00
Multiply	\pr0128\corerun.exe	0000500-0000375	6,243.9 ns	1.17	0.0687	-	-	904 B	1.00
Multiply	\pr0256\corerun.exe	0000500-0000375	6,226.8 ns	1.17	0.0687	-	-	904 B	1.00
Multiply	\pr0512\corerun.exe	0000500-0000375	6,180.4 ns	1.16	0.0687	-	-	904 B	1.00
Multiply	\pr1024\corerun.exe	0000500-0000375	5,654.4 ns	1.06	0.0687	-	-	904 B	1.00
Multiply	\main\corerun.exe	0000500-0000499	6,060.4 ns	1.00	0.0763	-	-	1024 B	1.00
Multiply	\pr0128\corerun.exe	0000500-0000499	7,032.1 ns	1.16	0.0763	-	-	1024 B	1.00
Multiply	\pr0256\corerun.exe	0000500-0000499	6,844.1 ns	1.13	0.0763	-	-	1024 B	1.00
Multiply	\pr0512\corerun.exe	0000500-0000499	6,821.3 ns	1.13	0.0763	-	-	1024 B	1.00
Multiply	\pr1024\corerun.exe	0000500-0000499	6,821.3 ns	1.13	0.0763	-	-	1024 B	1.00
Multiply	\main\corerun.exe	0001000-0000250	7,894.7 ns	1.00	0.0916	-	-	1280 B	1.00
Multiply	\pr0128\corerun.exe	0001000-0000250	9,196.4 ns	1.16	0.0916	-	-	1280 B	1.00
Multiply	\pr0256\corerun.exe	0001000-0000250	9,301.5 ns	1.18	0.0916	-	-	1280 B	1.00
Multiply	\pr0512\corerun.exe	0001000-0000250	9,215.9 ns	1.17	0.0916	-	-	1280 B	1.00
Multiply	\pr1024\corerun.exe	0001000-0000250	9,514.8 ns	1.21	0.0916	-	-	1280 B	1.00
Multiply	\main\corerun.exe	0001000-0000500	12,588.0 ns	1.00	0.1068	-	-	1528 B	1.00
Multiply	\pr0128\corerun.exe	0001000-0000500	13,960.0 ns	1.11	0.1068	-	-	1528 B	1.00
Multiply	\pr0256\corerun.exe	0001000-0000500	13,931.5 ns	1.11	0.1068	-	-	1528 B	1.00
Multiply	\pr0512\corerun.exe	0001000-0000500	14,277.4 ns	1.13	0.1068	-	-	1528 B	1.00
Multiply	\pr1024\corerun.exe	0001000-0000500	13,925.9 ns	1.11	0.1068	-	-	1528 B	1.00
Multiply	\main\corerun.exe	0001000-0000750	16,363.0 ns	1.00	0.1221	-	-	1776 B	1.00
Multiply	\pr0128\corerun.exe	0001000-0000750	17,010.7 ns	1.04	0.1221	-	-	1776 B	1.00
Multiply	\pr0256\corerun.exe	0001000-0000750	17,728.1 ns	1.08	0.1221	-	-	1776 B	1.00
Multiply	\pr0512\corerun.exe	0001000-0000750	17,987.1 ns	1.10	0.1221	-	-	1776 B	1.00
Multiply	\pr1024\corerun.exe	0001000-0000750	17,786.7 ns	1.09	0.1221	-	-	1776 B	1.00
Multiply	\main\corerun.exe	0001000-0000999	19,115.1 ns	1.00	0.1526	-	-	2024 B	1.00
Multiply	\pr0128\corerun.exe	0001000-0000999	19,084.0 ns	1.00	0.1526	-	-	2024 B	1.00
Multiply	\pr0256\corerun.exe	0001000-0000999	19,898.4 ns	1.04	0.1526	-	-	2024 B	1.00
Multiply	\pr0512\corerun.exe	0001000-0000999	20,120.3 ns	1.05	0.1526	-	-	2024 B	1.00
Multiply	\pr1024\corerun.exe	0001000-0000999	20,073.3 ns	1.05	0.1526	-	-	2024 B	1.00
Multiply	\main\corerun.exe	0010000-0002500	331,491.7 ns	1.00	0.9766	-	-	12528 B	1.00
Multiply	\pr0128\corerun.exe	0010000-0002500	345,360.3 ns	1.04	0.9766	-	-	12528 B	1.00
Multiply	\pr0256\corerun.exe	0010000-0002500	331,238.7 ns	1.00	0.9766	-	-	12528 B	1.00
Multiply	\pr0512\corerun.exe	0010000-0002500	328,670.5 ns	0.99	0.9766	-	-	12528 B	1.00
Multiply	\pr1024\corerun.exe	0010000-0002500	346,899.5 ns	1.05	0.9766	-	-	12528 B	1.00
Multiply	\main\corerun.exe	0010000-0005000	505,027.1 ns	1.00	0.9766	-	-	15025 B	1.00
Multiply	\pr0128\corerun.exe	0010000-0005000	476,146.5 ns	0.94	0.9766	-	-	15024 B	1.00
Multiply	\pr0256\corerun.exe	0010000-0005000	450,121.5 ns	0.89	0.9766	-	-	15024 B	1.00
Multiply	\pr0512\corerun.exe	0010000-0005000	486,532.8 ns	0.96	0.9766	-	-	15024 B	1.00
Multiply	\pr1024\corerun.exe	0010000-0005000	490,601.3 ns	0.97	0.9766	-	-	15024 B	1.00
Multiply	\main\corerun.exe	0010000-0007500	672,133.9 ns	1.00	0.9766	-	-	17529 B	1.00
Multiply	\pr0128\corerun.exe	0010000-0007500	639,168.1 ns	0.95	0.9766	-	-	17529 B	1.00
Multiply	\pr0256\corerun.exe	0010000-0007500	593,343.2 ns	0.88	0.9766	-	-	17529 B	1.00
Multiply	\pr0512\corerun.exe	0010000-0007500	588,610.1 ns	0.88	0.9766	-	-	17528 B	1.00
Multiply	\pr1024\corerun.exe	0010000-0007500	655,762.0 ns	0.98	0.9766	-	-	17528 B	1.00
Multiply	\main\corerun.exe	0010000-0009999	752,860.1 ns	1.00	0.9766	-	-	20025 B	1.00
Multiply	\pr0128\corerun.exe	0010000-0009999	655,242.7 ns	0.87	0.9766	-	-	20025 B	1.00
Multiply	\pr0256\corerun.exe	0010000-0009999	690,172.2 ns	0.92	0.9766	-	-	20025 B	1.00
Multiply	\pr0512\corerun.exe	0010000-0009999	662,638.8 ns	0.88	0.9766	-	-	20024 B	1.00
Multiply	\pr1024\corerun.exe	0010000-0009999	739,939.7 ns	0.98	0.9766	-	-	20025 B	1.00
Multiply	\main\corerun.exe	0100000-0025000	13,501,335.9 ns	1.00	31.2500	31.2500	31.2500	125052 B	1.00
Multiply	\pr0128\corerun.exe	0100000-0025000	10,430,180.2 ns	0.77	31.2500	31.2500	31.2500	125056 B	1.00
Multiply	\pr0256\corerun.exe	0100000-0025000	10,917,271.4 ns	0.81	31.2500	31.2500	31.2500	125052 B	1.00
Multiply	\pr0512\corerun.exe	0100000-0025000	10,978,839.6 ns	0.81	31.2500	31.2500	31.2500	125052 B	1.00
Multiply	\pr1024\corerun.exe	0100000-0025000	11,131,788.0 ns	0.82	31.2500	31.2500	31.2500	125052 B	1.00
Multiply	\main\corerun.exe	0100000-0050000	19,529,617.7 ns	1.00	31.2500	31.2500	31.2500	150068 B	1.00
Multiply	\pr0128\corerun.exe	0100000-0050000	15,097,175.5 ns	0.77	46.8750	46.8750	46.8750	150062 B	1.00
Multiply	\pr0256\corerun.exe	0100000-0050000	14,365,115.1 ns	0.74	46.8750	46.8750	46.8750	150062 B	1.00
Multiply	\pr0512\corerun.exe	0100000-0050000	15,798,947.9 ns	0.81	31.2500	31.2500	31.2500	150068 B	1.00
Multiply	\pr1024\corerun.exe	0100000-0050000	15,769,881.2 ns	0.81	31.2500	31.2500	31.2500	150068 B	1.00
Multiply	\main\corerun.exe	0100000-0075000	26,245,361.5 ns	1.00	31.2500	31.2500	31.2500	175068 B	1.00
Multiply	\pr0128\corerun.exe	0100000-0075000	20,914,966.7 ns	0.80	31.2500	31.2500	31.2500	175067 B	1.00
Multiply	\pr0256\corerun.exe	0100000-0075000	18,830,658.3 ns	0.72	31.2500	31.2500	31.2500	175067 B	1.00
Multiply	\pr0512\corerun.exe	0100000-0075000	18,699,003.1 ns	0.71	31.2500	31.2500	31.2500	175058 B	1.00
Multiply	\pr1024\corerun.exe	0100000-0075000	21,151,182.3 ns	0.81	31.2500	31.2500	31.2500	175058 B	1.00
Multiply	\main\corerun.exe	0100000-0099999	29,509,821.9 ns	1.00	31.2500	31.2500	31.2500	200068 B	1.00
Multiply	\pr0128\corerun.exe	0100000-0099999	20,654,077.1 ns	0.70	31.2500	31.2500	31.2500	200067 B	1.00
Multiply	\pr0256\corerun.exe	0100000-0099999	21,727,724.0 ns	0.74	31.2500	31.2500	31.2500	200058 B	1.00
Multiply	\pr0512\corerun.exe	0100000-0099999	20,679,755.2 ns	0.70	31.2500	31.2500	31.2500	200058 B	1.00
Multiply	\pr1024\corerun.exe	0100000-0099999	23,118,738.5 ns	0.78	31.2500	31.2500	31.2500	200067 B	1.00
Multiply	\main\corerun.exe	1000000-0250000	509,747,866.7 ns	1.00	-	-	-	1250760 B	1.00
Multiply	\pr0128\corerun.exe	1000000-0250000	321,428,916.7 ns	0.63	-	-	-	1250392 B	1.00
Multiply	\pr0256\corerun.exe	1000000-0250000	314,692,916.7 ns	0.62	-	-	-	1250392 B	1.00
Multiply	\pr0512\corerun.exe	1000000-0250000	316,530,200.0 ns	0.62	-	-	-	1250472 B	1.00
Multiply	\pr1024\corerun.exe	1000000-0250000	354,507,166.7 ns	0.70	-	-	-	1250472 B	1.00
Multiply	\main\corerun.exe	1000000-0500000	778,175,866.7 ns	1.00	-	-	-	1500472 B	1.00
Multiply	\pr0128\corerun.exe	1000000-0500000	453,160,500.0 ns	0.58	-	-	-	1500472 B	1.00
Multiply	\pr0256\corerun.exe	1000000-0500000	433,646,666.7 ns	0.56	-	-	-	1500760 B	1.00
Multiply	\pr0512\corerun.exe	1000000-0500000	438,128,166.7 ns	0.56	-	-	-	1500472 B	1.00
Multiply	\pr1024\corerun.exe	1000000-0500000	467,433,166.7 ns	0.60	-	-	-	1500760 B	1.00
Multiply	\main\corerun.exe	1000000-0750000	1,016,370,266.7 ns	1.00	-	-	-	1750760 B	1.00
Multiply	\pr0128\corerun.exe	1000000-0750000	605,555,000.0 ns	0.60	-	-	-	1750472 B	1.00
Multiply	\pr0256\corerun.exe	1000000-0750000	562,233,733.3 ns	0.55	-	-	-	1750472 B	1.00
Multiply	\pr0512\corerun.exe	1000000-0750000	553,515,400.0 ns	0.54	-	-	-	1750472 B	1.00
Multiply	\pr1024\corerun.exe	1000000-0750000	564,205,566.7 ns	0.56	-	-	-	1750472 B	1.00
Multiply	\main\corerun.exe	1000000-0999999	1,194,190,333.3 ns	1.00	-	-	-	2000472 B	1.00
Multiply	\pr0128\corerun.exe	1000000-0999999	621,246,233.3 ns	0.52	-	-	-	2000760 B	1.00
Multiply	\pr0256\corerun.exe	1000000-0999999	617,586,733.3 ns	0.52	-	-	-	2000472 B	1.00
Multiply	\pr0512\corerun.exe	1000000-0999999	613,818,600.0 ns	0.51	-	-	-	2000760 B	1.00
Multiply	\pr1024\corerun.exe	1000000-0999999	613,492,666.7 ns	0.51	-	-	-	2000088 B	1.00
Multiply	\main\corerun.exe	Square0000100	275.5 ns	1.00	0.0176	-	-	224 B	1.00
Multiply	\pr0128\corerun.exe	Square0000100	275.3 ns	1.00	0.0176	-	-	224 B	1.00
Multiply	\pr0256\corerun.exe	Square0000100	280.0 ns	1.02	0.0176	-	-	224 B	1.00
Multiply	\pr0512\corerun.exe	Square0000100	277.2 ns	1.01	0.0176	-	-	224 B	1.00
Multiply	\pr1024\corerun.exe	Square0000100	276.6 ns	1.00	0.0176	-	-	224 B	1.00
Multiply	\main\corerun.exe	Square0000500	4,054.4 ns	1.00	0.0763	-	-	1024 B	1.00
Multiply	\pr0128\corerun.exe	Square0000500	4,121.7 ns	1.02	0.0763	-	-	1024 B	1.00
Multiply	\pr0256\corerun.exe	Square0000500	4,123.9 ns	1.02	0.0763	-	-	1024 B	1.00
Multiply	\pr0512\corerun.exe	Square0000500	4,142.7 ns	1.02	0.0763	-	-	1024 B	1.00
Multiply	\pr1024\corerun.exe	Square0000500	4,175.9 ns	1.03	0.0763	-	-	1024 B	1.00
Multiply	\main\corerun.exe	Square0001000	12,664.4 ns	1.00	0.1526	-	-	2024 B	1.00
Multiply	\pr0128\corerun.exe	Square0001000	12,840.2 ns	1.01	0.1526	-	-	2024 B	1.00
Multiply	\pr0256\corerun.exe	Square0001000	13,198.4 ns	1.04	0.1526	-	-	2024 B	1.00
Multiply	\pr0512\corerun.exe	Square0001000	13,049.2 ns	1.03	0.1526	-	-	2024 B	1.00
Multiply	\pr1024\corerun.exe	Square0001000	13,170.3 ns	1.04	0.1526	-	-	2024 B	1.00
Multiply	\main\corerun.exe	Square0010000	507,761.8 ns	1.00	0.9766	-	-	20025 B	1.00
Multiply	\pr0128\corerun.exe	Square0010000	449,839.6 ns	0.89	1.4648	-	-	20024 B	1.00
Multiply	\pr0256\corerun.exe	Square0010000	446,288.1 ns	0.88	1.4648	-	-	20024 B	1.00
Multiply	\pr0512\corerun.exe	Square0010000	433,092.4 ns	0.85	1.4648	-	-	20024 B	1.00
Multiply	\pr1024\corerun.exe	Square0010000	491,795.1 ns	0.97	0.9766	-	-	20024 B	1.00
Multiply	\main\corerun.exe	Square0100000	20,070,908.3 ns	1.00	31.2500	31.2500	31.2500	200059 B	1.00
Multiply	\pr0128\corerun.exe	Square0100000	14,622,159.4 ns	0.73	46.8750	46.8750	46.8750	200066 B	1.00
Multiply	\pr0256\corerun.exe	Square0100000	14,373,196.9 ns	0.72	46.8750	46.8750	46.8750	200062 B	1.00
Multiply	\pr0512\corerun.exe	Square0100000	14,284,324.5 ns	0.71	46.8750	46.8750	46.8750	200066 B	1.00
Multiply	\pr1024\corerun.exe	Square0100000	15,955,562.5 ns	0.79	31.2500	31.2500	31.2500	200068 B	1.00
Multiply	\main\corerun.exe	Square1000000	780,524,800.0 ns	1.00	-	-	-	2000472 B	1.00
Multiply	\pr0128\corerun.exe	Square1000000	445,885,933.3 ns	0.57	-	-	-	2000472 B	1.00
Multiply	\pr0256\corerun.exe	Square1000000	440,426,600.0 ns	0.56	-	-	-	2000760 B	1.00
Multiply	\pr0512\corerun.exe	Square1000000	428,211,366.7 ns	0.55	-	-	-	2000472 B	1.00
Multiply	\pr1024\corerun.exe	Square1000000	430,626,800.0 ns	0.55	-	-	-	2000760 B	1.00

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