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June 16, 2006
Less is More: Exploiting Single Precision Math in HPC
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Some of the most widely used processors for high-performance computing today demonstrate much higher performance for 32-bit floating point arithmetic (single precision) than for 64-bit floating point arithmetic (double precision). These include the AMD Opteron, the Intel Pentium, the IBM PowerPC, and the Cray X1. These architectures demonstrate approximately twice the performance for single precision execution when compared to double precision.
And although not currently widely used in HPC systems, the Cell processor has even greater advantages for 32-bit floating point execution. Its single precision performance is 10 times better than its double precision performance.
At this point you might be thinking -- so what? Everyone knows double precision rules in HPC. And while that's true, the difference in performance between single precision and double precision is a tempting target for people who want to squeeze more computational power out of their hardware.
Apparently it was too tempting to ignore. Jack Dongarra and his fellow researchers at the Innovative Computing Laboratory (ICL) at the University of Tennessee have devised algorithms which use single precision arithmetic to do double precision work. Using this method, they have demonstrated execution speedups that correspond closely with the expected single precision performance characteristics of the processors.
The overall approach of the ICL team was to use single precision math whenever possible, especially for the most compute-intensive parts of the software, and then fall back to double precision only when necessary. Most applications use double precision math for the following reasons:
(1) To minimize the accumulation of round-off error,
(2) For ill-conditioned problems that require higher precision,
(3) The 8 bit exponent defined by the IEEE floating point standard for 32-bit arithmetic will not accommodate the calculation, or
(4) There are critical sections in the code which require higher precision.
But for many calculations these restrictions don't apply, or if they do they only apply to a portion of the calculation. According to Dongarra, the types of problems where single precision optimization would be most applicable include linear systems (dense and sparse), large sparse linear system using iterative methods, and eigenvalue problems. These types of calculations apply to a wide range of applications in technical computing.
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