bug: "clean" mode has worse accuracy than regular Float64 about multifloats.jl HOT 1 CLOSED

Hey @nsajko, thanks for discovering this surprising behavior! It seems that, in this case, the extra limb computed in clean mode is not quite cancelled out in the multiplication x * inv(x), producing a number that is very slightly different from 1.0.

This behavior is counterintuitive but actually correct. The number (1/3)^100 cannot be exactly represented as a terminating decimal in base 2, so you should not expect inv(3.0^100) to be the exact multiplicative inverse of 3^100. Indeed, if we compare using BigFloat:

julia> MultiFloats.use_clean_multifloat_arithmetic(20)

julia> Float64x{13}(3.0^100) * inv(Float64x{13}(3.0^100))
1.00000000000000000 [ ... lots of zeroes ... ] 000009233803766632279

julia> setprecision(BigFloat, 1000)

julia> BigFloat(3.0^100) * BigFloat(inv(Float64x{13}(3.0^100)))
1.00000000000000000 [ ... lots of zeroes ... ] 000009233803766632279027870659735403696888168679729487436063148964080335784965191272233867971

You can see that clean mode correctly produces a more accurate result. Therefore, there is no bug here. The only issue is the false expectation that isone(x * inv(x)) hold for any x whose multiplicative inverse cannot be exactly represented.

from multifloats.jl.