Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems

6.2 INTEGERS

6.2.1 General Algorithm

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, −B^m−1 ≤ X < B^m−1; then

substitute Y by Y′ = B^m−1.Y, so that Y′ ≥ B^m−1.1 > X and − Y′ ≤ − B^m−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,

that is,

Comment 6.2 ...