TruncatedDivision

Division and modulus for computer scientists taken from https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/divmodnote-letter.pdf

For two numbers x (dividend) and y (divisor) on an ordered ring with y != 0, there exists a pair of numbers q (quotient) and r (remainder) such that these laws are satisfied:

(1) q is an integer (2) x = y * q + r (division rule) (3) |r| < |y|, (4t) r = 0 or sign(r) = sign(x), (4f) r = 0 or sign(r) = sign(y).

where sign is the sign function, and the absolute value function |x| is defined as |x| = x if x >=0, and |x| = -x otherwise.

We define functions tmod and tquot such that: q = tquot(x, y) and r = tmod(x, y) obey rule (4t), (which truncates effectively towards zero) and functions fmod and fquot such that: q = fquot(x, y) and r = fmod(x, y) obey rule (4f) (which floors the quotient and effectively rounds towards negative infinity).

Law (4t) corresponds to ISO C99 and Haskell's quot/rem. Law (4f) is described by Knuth and used by Haskell, and fmod corresponds to the REM function of the IEEE floating-point standard.