Why Doesn't A*B == B*A?

[Andrew Maurer]

I am working on some code that calculates percentiles and am running into issues in Mata. Digging into it, I found that the mod() operator wasn't acting as expected in either Mata or Stata. Going one level deeper, I found that numerical precision problems are the issue, that arise even when applied to relatively small numbers.

Initially, I was looking at a case where A/(B/C) != A/B*C.

// 75th percentile sometimes fails
// eg mod(588,784/20)
di mod(588,39.2)
di 588/39.2 // This should be exactly 15
di 588-39.2*floor(588/39.2) // Returns 39.2. What!?
di 588-39.2*floor(float(588/39.2))==0 // Float solution returns 0. So it's a precision issue...
di 15 == 588/39.2 // False
di 15 == 588/(784/20) // False
di 15 == 588/784*20 // True... Weird!!

So the solution could be to manually write out a revised definition of mod(x,y) each time I wan't to use mod(). Ie define
mod(x,y) = x-y*floor(float(x/y))

Then I started seeing another case, where A*B/C != A/C*B

// 55th percentile sometimes fails
// eg 3916-float(7120*55/100)
scalar x = 3916
scalar y = 7120*11/20 // This should be exactly 3916
di mod(x,y) // Returns 3916... Wrong!
di x-y*floor(float(x/y)) // Float solution from above doesn't work! Returns -4.547e-13 (nonzero)
di x-y*floor(float(x/y))==0 // False (implied by above)
di x-float(y*floor(float(x/y)))==0 // True. Gah, finally! Man is this ever a kludge
di 3916 == 7120*55/100 // False
di 3916 == 7120/100*55 // True... Weird!!

So define yet a new version of the mod(x,y)
mod(x,y) = x-float(y*floor(float(x/y)))

What is causing this?

Thanks,
Andrew Maurer

[Joseph Coveney]

help operator

It's not that A * B != B * A. It's that division has precedence over multiplication. So that in 7120 * 11 / 20, 11 / 20 gets done first, and the quotient cannot be represented with infinite precision in digital computers.