Synthesis of Arithmetic Circuits: FPGA, ASIC and Embedded Systems

16.2 ARITHMETIC OPERATIONS

First analyze the main arithmetic operations and generate the corresponding computation algorithms.

16.2.1 Addition of Positive Numbers

Given two positive floating-point numbers s₁.B^e1 and s₂.B^e2 their sum s.B^e is computed as follows.

Assume that e₁ is greater than or equal to e₂; then (alignment) the sum of s₁.B^e1 and s₂.B^e2 can be expressed in the form s.B^e, where

The value of s belongs to the interval

so that s could be greater than or equal to B. If it is the case, that is, if

then (normalization) substitute s by s/B, and e by e + 1, so that the value of s.B^e is the same as before, and the new value of s satisfies

The significands s₁ and s₂ of the operands are multiples of ulp. If e₁ is greater than e₂, the value of s could no longer be a multiple of ulp and some rounding function should be applied to s. Assume that

s′ and s″ being two successive multiples of ulp. Then the rounding function associates to s either s′ or s″, according to some rounding strategy. ...