Let X be an integer and Y a natural number with Y > 0. Define Q and R respectively as the quotient and the remainder of the division of X by Y, with an accuracy of p fractional base-B digits:
where Q and R are integers, − Y ≤ R < Y and sign(R) = sign(X). In other words,
so that the unit in the least significant position (ulp) of Q.B−p and R.B−p is equal to B−p. In the particular case where p = 0, that is,
Q and R are the quotient and the remainder of the integer division of X by Y.
The basic algorithm applies to operands X and Y such that
In the general case, to ensure that −Y ≤ X < Y, a previous alignment step is necessary. Assume that X is an m-digit reduced B's complement number, that is, −Bm−1 ≤ X < Bm−1; then
substitute Y by Y′ = Bm−1.Y, so that Y′ ≥ Bm−1.1 > X and − Y′ ≤ − Bm−1.1 ≤ X; compute the quotient Q and the remainder R′ of the division of X by Y′, with an accuracy of p + m − 1 fractional base-B digits, that is,
Comment 6.2 ...