Let X and Y be two natural numbers 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 natural numbers, and R < Y. 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 X < Y, a previous alignment step is necessary. Assume that X is an m-digit base-B number, that is, X < Bm; then
substitute Y by Y′ = Bm.Y, so that Y′ ≥ Bm.1> X;
compute the quotient Q and the remainder R′ of the division of X by Y′, with an accuracy of p + m fractional base-B digits, that is,
The next theorem ...