I was using BigFloats to compute the Harmonic series (naïve implementation) with arbitrary precision, and it was brought to my attention a bug in my code: in particular, the difference between BigFloat(1/i) and 1/BigFloat(i).

Indeed, when I use BigFloat(1/10^10) I get: 1.0000000000000000364321973154977415791655470655996396089904010295867919921875e-10
While using 1/BigFloat(10^10) correctly yields: 1.000000000000000000000000000000000000000000000000000000000000000000000000000003e-10

I was then left wondering, why this behavior? How can I learn more about properly using arbitrary precision arithmetic?

You want to convert to “big” as early as possible—or more precisely, before any loss of precision. In this case, 10^10 is precise in Int but 1/10^10 produces a Float64 which has already lost precision. When you do big(1/10^10) what you get is that already-low-precision Float64 value at higher precision, which is not what you want. When you do 1/big(10^10) you do BigFloat division instead of Float64 division, which is what you want.

I’m a bit surprised that Julia doesn’t have a @big macro that would turn @big 1/10^10 into big(1)/big(10)^big(10). That’d be an easy way of avoiding accidental loss of precision.

I think the best approach (in general) is to first write something that works, then make it fast. Just converting all leaf-nodes to big is something that can be done in a few lines of code and should be relatively easy to make bug-free.

For the “make it fast” part, one could perhaps introduce a LazyBig{T} type which is simply a thin wrapper indicating that the result of computations should be BigInt or BigFloat unless it fits exactly in some smaller type. The return type of *(::LazyBig{Int}, ::LazyBig{Int}) could be Union{LazyBig{Int128}, LazyBig{Int}}), which might make taking the power by squaring relatively fast in the 10^10 case.

As I understand it, @changeprecision only targets literals. I think it’d be useful to have a macro that wraps all leaf nodes, so that expressions like @big x/y, where x and y are variables, would also work.

(The wrapper function would of course have to pass non-numbers through unharmed.)

I don’t know how workable this would be—consider the typemax example. It would wrap Int64 in a big call: if that returns BigInt then the typemax call fails; if it returns Int64 then it has no effect. It seems like the best one could do is also replace arithmetic operations and functions with versions that transform their arguments to higher precision first.

The use of the @big macro would be to ensure that a certain expression is evaluated at high precision, not to change the default precision for a block of code like @changeprecision. Given your typemax example, I’ll say the wrapper should be applied to every node. Not just leaves.

So @big 1/typemax(Int64) would become w(w(1)/w(typemax(w(Int64)))) where w is the wrapper defined something like:

w(x) = x
w(x::Number) = big(x)

After w is evaluated, we’d have big(big(1)/big(typemax(Int64))), which I’d say is what we want.

Here’s an example of a wrapper type that will automatically decide whether to convert to BigInt before multiplication (enabling power by squaring).

It’s significantly faster than BigInt for the specific case 10^10. (This calculation takes 20 nanoseconds on my system, which is amazing given that I didn’t modify the power_by_squaring function at all, and the multiplication code is not even type stable.)

struct LazyBigInt{T} <: Integer
x::T
end
# Conservative bound on when to move from Int64 to higher precision
const cutoff64 = 0.5 * Float64(typemax(Int))
function Base.:*(a::LazyBigInt{Int}, b::LazyBigInt{Int})
rf = Float64(a.x) * Float64(b.x)
-cutoff64 <= rf <= cutoff64 ? LazyBigInt(a.x * b.x) : big(a.x) * b.x
end
Base.promote_type(::Type{<:LazyBigInt}, ::Type{BigInt}) = BigInt
Base.promote_type(::Type{<:LazyBigInt{T}}, ::Type{T}) where {T} = LazyBigInt{T}
Base.BigInt(x::LazyBigInt) = BigInt(x.x)
@assert LazyBigInt(10)^10 == 10000000000
@assert LazyBigInt(10)^40 == 10000000000000000000000000000000000000000
using BenchmarkTools
@benchmark $(BigInt(10))^10
@benchmark $(LazyBigInt(10))^10

Of course, this is just an example. It would take some effort to design proper heuristics for when to switch to higher precision for each type of operation.