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The implementation of the Fused Multiply and Add (FMA) operation has been extensively studied in the literature on standard and large precisions. We suggest re- visiting those studies for 16-bit precision. We introduce a variation of the Mixed precision FMA targeted for applications processing low precision inputs (such as machine learning). We also introduce several versions of a fixed point based...
Elementary functions from the mathematical library input and output floating-point numbers. However it is possible to implement them purely using integer/fixed-point arithmetic. This option was not attractive between 1985 and 2005, because mainstream processor hardware supported 64-bit floating-point, but only 32-bit integers. This has changed in recent years, in particular with the generalization...
A typical floating-point environment includes support for a small set of about 30 mathematical functions such as exponential, logarithm, trigonometric and hyperbolic functions. These functions are provided by mathematical software libraries (libm), typically in IEEE754 single, double and quad precision. This article suggests to replace this libm paradigm by a more general approach: the on-demand generation...
A bit heap is a data structure that holds the unevaluated sum of an arbitrary number of bits, each weighted by some power of two. Most advanced arithmetic cores can be viewed as involving one or several bit heaps. We claim here that this point of view leads to better global optimization at the algebraic level, at the circuit level, and in terms of software engineering. To demonstrate it, a generic...
Instruction Multiple-Thread (SIMT) micro-architectures implemented in Graphics Processing Units (GPUs) run fine-grained threads in lockstep by grouping them into units, referred to as warps, to amortize the cost of instruction fetch, decode and control logic over multiple execution units. As individual threads take divergent execution paths, their processing takes place sequentially, defeating part...
The floating-point fused multiply and add, computing R=AB+C with a single rounding, is now an IEEE-754 standard operator. This article investigates variants in which the addend C and the result R are of a larger format, for instance binary64 (double precision), while the multiplier inputs A and B are of a smaller format, for instance binary32 (single precision). Like the standard FMA operator, the...
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