Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 6. LAYOUT CASE STUDIES 136
Slices Util. t-PAR (s) Max freq. (Mhz)
Unpipelined/Auto 777 15% 48 61.5
Unpipelined/Placed 406 7% 19 69.7
Pipelined/Auto 777 15% 28 206.0
Pipelined/Placed 406 7% 26 152.0
Table 6.1: Results for a single adder tree
R using the timeless pre-condition and split into two repeated anti-delays within the parallel
composition using the property that D is polymorphic:
half 2 n ; [btreen(R ; D) ; D −n , btreen(R ; D) ; D −n ] ; R
The induction hypothesis can then be used to complete the proof.
Using this combinator we generate a 6-level tree of 8-bit ripple adders, producing a circuit
which adds together 64 input values. The manual placement is compared with the identical
circuit compiled without placement and using the Xilinx placement algorithm.
6.2.3 Results
Table 6.1 shows the results for placed and unplaced pipelined and unpipelined adder trees.
“Util” is the percentage of resources utilised on the device, “t-PAR” is the amount of time
required to place and route the circuit. As expected, pipelining increases the maximum clock
frequency significantly (although far from the predicted theoretical maximum of ×8). It is
also interesting to note that the manually placed design has worse performance than the
automatically placed version when the circuit is pipelined, even though the manually placed
version has been mapped into fewer Virtex slices.
We also experimented with placing multiple adder trees on the FPGA. Table 6.2 illustrates
the results for an FPGA loaded with 7 of the adder trees. The difference in the resources used
by the placed and unplaced descriptions is very significant, and possibly partially responsible
for the fact that the placed version now exhibits significantly higher performance than the
unplaced version regardless of pipelining.
The difference in the number of slices used is quite interesting. It implies that the process
of packing primitives into slices automatically does so much less densely than the manual