Optimize BigInteger.ToString for large decimal string

[kzrnm]

This PR is a counterpart to #55121. divide-and-conquer algorithm

Number.FormatBigInteger() can run in $D(n)log(N)$ time using the Divide and Conquer algorithm, where $D(n)$ represents the computational complexity of BigInteger division.

The computational complexity of division have been improved by #96895.

---

Benchmark

Code

using BenchmarkDotNet.Attributes;
using BenchmarkDotNet.Configs;
using System.Numerics;

[DisassemblyDiagnoser]
[GroupBenchmarksBy(BenchmarkLogicalGroupRule.ByMethod)]
public class ToStringTest
{
    [Params(100, 1000, 10000, 100000)]
    public int N;

    BigInteger b;

    [GlobalSetup]
    public void Setup()
    {
        b = BigInteger.Parse(new string('9', N));
    }

    [Benchmark] public string DecimalString() => b.ToString();
}

BenchmarkDotNet v0.13.12, Windows 11 (10.0.26100.3037)
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	N	Mean	Error	StdDev	Ratio	RatioSD	Code Size	Gen0	Gen1	Gen2	Allocated	Alloc Ratio
DecimalString	\main\corerun.exe	100	162.6 ns	72.78 ns	3.99 ns	1.00	0.00	3,976 B	0.0176	-	-	224 B	1.00
DecimalString	\pr\corerun.exe	100	162.2 ns	80.65 ns	4.42 ns	1.00	0.05	3,836 B	0.0176	-	-	224 B	1.00
DecimalString	\main\corerun.exe	1000	10,087.1 ns	1,262.45 ns	69.20 ns	1.00	0.00	3,801 B	0.1526	-	-	2024 B	1.00
DecimalString	\pr\corerun.exe	1000	6,181.8 ns	877.68 ns	48.11 ns	0.61	0.01	3,757 B	0.1602	-	-	2024 B	1.00
DecimalString	\main\corerun.exe	10000	1,042,496.9 ns	207,485.10 ns	11,372.96 ns	1.00	0.00	3,801 B	-	-	-	20026 B	1.00
DecimalString	\pr\corerun.exe	10000	336,682.2 ns	132,151.44 ns	7,243.67 ns	0.32	0.01	3,758 B	1.4648	-	-	20025 B	1.00
DecimalString	\main\corerun.exe	100000	100,113,761.1 ns	18,888,052.58 ns	1,035,317.90 ns	1.00	0.00	3,800 B	-	-	-	200155 B	1.00
DecimalString	\pr\corerun.exe	100000	11,684,286.5 ns	508,446.11 ns	27,869.65 ns	0.12	0.00	3,757 B	46.8750	46.8750	46.8750	200088 B	1.00

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[Copilot]

Copilot reviewed 2 out of 2 changed files in this pull request and generated no comments.

Comments suppressed due to low confidence (3)

src/libraries/System.Runtime.Numerics/src/System/Number.BigInteger.cs:797

• [nitpick] The variable name 'digitRatio' is ambiguous. It should be renamed to something more descriptive like 'logBase2ToBase1E9Ratio'.

const double digitRatio = 1.070328873472;

src/libraries/System.Runtime.Numerics/src/System/Number.BigInteger.cs:933

• [nitpick] The method name 'BigIntegerToBase1E9' could be more descriptive. Consider renaming it to 'ConvertBigIntegerToBase1E9'.

private static void BigIntegerToBase1E9(ReadOnlySpan<uint> bits, Span<uint> base1E9Buffer, out int leadingWritten)

src/libraries/System.Runtime.Numerics/tests/BigInteger/BigIntegerToStringTests.cs:13

• The variable 'result' is initialized to 'null' but is not explicitly set in the 'catch' block, which could lead to a 'NullReferenceException'. Ensure 'result' is properly handled in the 'catch' block.

string result = null;