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    <p>You are right that integer and floating point arithmetic costs are different but not as much as one could assume. It highly depends on which processor unit is used for the computation. Particularly for Intel processors, you can find helpful information in the "Optimization Reference Manual" available at <a href="http://www.intel.com/products/processor/manuals/" rel="nofollow">http://www.intel.com/products/processor/manuals/</a>. Appendix C lists instruction latencies for all instructions.</p> <p>To your specific question, if computation time for matrix multiplication is depending on whether the entries of the two matrices contain identical or random values, the answer is "no". If you look at the amount and sequence of instructions as well as the memory access pattern at computation runtime, it is all the same in both cases. The compiler usually also can't take any advantage of the fact that the matrices are all made up of the same entries because matrix multiplication needs to cover all possible cases. (Ok, unless you pack everything - filling the matrix entries and the multiplication itself - in one function and rule out all side effects like aliasing, then a very very smart compiler could probably make something out of it, but we are not talking about that, right?)</p> <p>Also, the size in digits (I assume you are referring to decimal digits) does not matter. Every matrix entry is represented by <strong>all</strong> of its 32 bits in case of single precision floating point numbers (or 64 bits in case of double precision).</p>
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    1. USChristoph Freundl
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    1. COYou're probably right, but there are a few caveats. Very small (denormal) numbers can be very slow to handle, like 100 times slower; and such numbers aren't actually that uncommon particularly in iterative optimization problems since these sometimes converge (some coefficients) to zero. Similarly, operatiosn on NaN and Inf values aren't always as fast as operations on other values. Also, some operations (though not, I believe, multiplication and addition) are value-dependent in their implementation on the actual chips, though I don't think this effect is likely to hit anyone very often.
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      1. USEamon Nerbonne
 

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