By D. G. de Figueiredo, C. S. Hönig
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Extra info for Differential Equations, Sao Paulo, Brazil, 1981: Proceedings
Example text
This would lead either to more complex FPU hardware or a system that emulated some or all of the denormalized computations in software or microcode. The former could make the FPUs more expensive to produce, while the latter could lead to greatly decreased performance of the floating-point system when denormals are generated. However, several manufacturers showed that it could be implemented in floating-point hardware, paving the way for this more accurate method to become part of the de facto (and later, official) standard.
In a sense the infinities are not pure mathematical values. Rather, they are used to represent values that fall outside of the range of valid 18 Chapter 1 Real-World Computer Number Representation exponents. 0 × 1076 ≈ ∞fp The behavior of infinity is defined by the standard as follows (the standard covers many more cases, but these are representative): ∞fp − P = ∞fp P = +0 ∞fp −P = −0 ∞fp where 0
How can the system divide the single value ∞fp and get back the original number? A look at the assembly code gives a hint. The basic steps the compiler generates are as follows: 1. 0e38 into the FPU. 2. Multiply the two loaded values and return ∞fp , keeping the result in the FPU as well. 3. 0e38 (still in the FPU), returning the correct result. The important item to note is that the result of each computation was both returned and kept in the FPU for later computation. This step is where the apparent “magic” occurs.