How is factorial(BigInt) so fast?

As an experiment, I copy/pasted Julia’s factorial() function and renamed it. But when I use it to evaluate a bigint, it is way slower than Julia’s factorial(BigInt). If the base function uses the same code, then why is the base function faster and is there a way I can achieve those speeds?

I’m guessing Julia’s either uses a dictionary lookup or somehow avoids most of the performance loss of bigints. The latter would be a useful trick to know.

I understand that creating a bigint longer than 9 digits is about 10x slower than creating a smaller bigint, and that another slowdown happens at 20 digits. Therefore, hypothetically if you evaluate BigInt(15)! manually, it’s faster to do

BigInt(15*14*13*12*11) * BigInt(10*9*8*7*6*5*4*3*2)

than it is to do

BigInt(15*14*13*12*11*10*9*8*7*6) * BigInt(5*4*3*2)

because (15 P 5) and 10! are both under 10 digits whereas (15 P 10) is not.

In fact, I know of no faster way to manually evaluate BigInt(15)! than to split it into two BigInts where each is smaller than 1 billion. It’s faster than BigInt(1514…*2), or BigInt(15)BigInt(14)…*BigInt(2), or any conceivable possibility.

However, even that is twice as slow as factorial(BigInt(15)) according to @btime. Julia has a trick up its sleeve and I want to know about it :slight_smile:


You copied the wrong function

julia> @which factorial(BigInt(1))
factorial(x::BigInt) in Base.GMP at gmp.jl:564

julia> @which factorial(1)
factorial(n::Union{Int64, UInt64}) in Base at combinatorics.jl:27

The correct one seems to be implemented by GMP.
You might also have luck with adapting the factorial code to be in-place (BigInts are mutable I think).

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Why are there multiple versions of factorial, and is this an ambiguity that happens more often, or does it only happen with factorial?

It’s the strength of Julia by dispatching in type. The algorithm for factorial (or any other function) may be optimized for different types.


Try methods(*) in the REPL.

Right, I’m aware of the multiple dispatch of Julia, I use it all the time. But normally, if a function is extended, it was extended globally, but maybe the behavior is different in 1.0, so it should still have all the same methods available if it is reassigned to a new name, shouldnt it?

I don’t know what extended globally means but factorial is no different from *

julia> methods(factorial)
# 7 methods for generic function "factorial":
[1] factorial(n::UInt128) in Base at combinatorics.jl:26
[2] factorial(n::Int128) in Base at combinatorics.jl:25
[3] factorial(x::BigFloat) in Base.MPFR at mpfr.jl:594
[4] factorial(x::BigInt) in Base.GMP at gmp.jl:564
[5] factorial(n::Union{Int64, UInt64}) in Base at combinatorics.jl:27
[6] factorial(n::Union{Int16, Int32, Int8, UInt16, UInt32, UInt8}) in Base at combinatorics.jl:33
[7] factorial(n::Integer) in Base at intfuncs.jl:820

Well, if I do g = factorial and take methods of that, then all the definitions are retained. So, it behaves as I expected. I figured thats what OP meant by renaming the function.

No, they copied the source code of one method.

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Mystery solved, thanks!