Design Space Exploration of Application Specific Number Formats Targeting an FPGA Implementation of SPICE

Most scientific computations use double precision floating point numbers. Recently, posits as an additional alternative have been established and are subject to ongoing research. In FPGA implementations arbitrary combinations of mantissa and exponent widths are possible. For some applications the required precision can be determined analytically without knowledge of the input data. Thus, in these cases a lower bound for the hardware effort can be given. Other applications may be more resilient to the precision of the chosen number representation. One example of such application is SPICE for circuit simulation. SPICE exhibits kind of self-healing behavior, since it detects the accumulated error and if the error gets too large, it can take recovery measures. In this case, more iterations are required, leading to more operations in total. This allows us an additional degree of freedom: We can trade lower precision and thus smaller area against the increased calculation effort. This paper develops a methodology to use these different options to find an optimal solution for each specific SPICE application scenario. It turns out that for regular IEEE-754 floating point formats a number format between single and double precision delivers the best trade off between operator size and computation time. Surprisingly, using posit based representations does not improve the overall runtime of simulations.