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The hardest logarithm to compute

Suppose you want to compute the natural logarithms of every floating point number, correctly truncated to a floating point result. Here by floating point number we mean an IEEE standard 64-bit float, what C calls a double. Which logarithm is hardest to compute?
We’ll get to the hardest logarithm shortly, but we’ll first start with a warm up problem.

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Just evaluating a polynomial: how hard could it be?

The previous post looked at an example of strange floating point behavior taking from book End of Error. This post looks at another example.
This example, by Siegfried Rump, asks us to evaluate
333.75 y6 + x2 (11 x2y2 – y6 – 121 y4 – 2) + 5.5 y8 + x/(2y)
at x = 77617 and y = 33096.
Here we evaluate Rump’s example in single, double, and quadruple precision.

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