Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 3. GENERATING PARAMETERISED LIBRARIES WITH LAYOUT 61
3.7.5 Expression Simplification
After height() and width() functions have been eliminated the result is valid Pebble ex-
pressions, however these are often significantly more complex than necessary and can be
extensively simplified - for example, maxf(j = 0..5, 3) can be simplified to just 3. The Quartz
compiler already possesses a sophisticated logic optimiser which simplifies simple logical and
arithmetic expressions such as x + y − y and also evaluates any expressions where the result
is known, such as 4 + 5 × 2.
The logic optimiser is extended to implement the evaluation of max, if, maxf and sum expres-
sion types when all parameters are known. Since we are generating parameterised Pebble
output it is often the case that functions can not be fully evaluated and in these circumstances
a range of transformations are applied to simplify expressions.
Uses of if can be simplified if the conditional test can be evaluated statically. Uses of max
are simplified to remove any values which are known to be less than or equal to another
value (remaining as a parameter). For example, max(a, b, 4, 5, a + 2) can be simplified to
max(b, 5, a + 2) since a + 2 > a and 5 > 4 always. These relationships can be determined
using the logic optimiser itself to investigate whether x > y can be simplified to true or false
for each pair x and y of parameters.
sum and maxf are more complex functions to simplify. A number of generic transformations
have been developed for expressions involving these functions and these transformations are
applied by the optimiser where possible. The transformations applied to sum are:
n < m ⇒ sum(i = m..n, f) = 0
m ≤ n ⇒ sum(i = m..n, f) = (n − m + 1) × f
i /∈ f ⇒ sum(i = m..n, f) = if(m ≤ n, (n − m + 1) × f, 0)
While similar transformations are applied to maxf:
n < m ⇒ maxf(i = m..n, f) = 0
i /∈ f ⇒ maxf(i = m..n, f) = if(m ≤ n, f, 0)