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Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems by Gustavo D. Sutter, Gery J.A. Bioul, Jean-Pierre Deschamps

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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:

image

where Q and R are integers, − YR < Y and sign(R) = sign(X). In other words,

image

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,

image

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

image

In the general case, to ensure that −YX < Y, a previous alignment step is necessary. Assume that X is an m-digit reduced B's complement number, that is, −Bm−1X < 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,

image

that is,

Comment 6.2 ...

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