Re: Log2 seems inefficient

       Hi,

> With default settings the worst case error in nearest rounding mode is 0.547 ULP 
> with inlined fma and fma contraction.
> It uses a 1 KB lookup table,
> ...
> Note that the math.h header defines log2(x) to be log(x)/Ln2, this is
> not changed, so the new code is only used if that macro is suppressed."
>
> Paul Zimmerman regularly posts library math function comparisons on the newlib 
> and comparable library mailing lists, available at:
> 
> 	https://members.loria.fr/PZimmermann/papers/accuracy.pdf
> 
> that should give you an idea of the current state of the functions, if he is 
> even aware of the issue.

indeed, the known worst case error for Newlib log2 is 2.06 ulp:

log2 0 -1 0x1.68d778f076021p+0 [2] [2.06] 2.05526 2.055255219183752
libm gives 0x1.fb1b88680e7adp-2
mpfr gives 0x1.fb1b88680e7abp-2

The 0.547 value above could match the GNU libc worst error:

log2 0 -1 0x1.1406d79e1b574p+0 [1] [0.548] 0.547538 0.5475371438132311
libm gives 0x1.bd16a11d3404ep-4
mpfr gives 0x1.bd16a11d3404fp-4

If your goal is accuracy, you can try the CORE-MATH implementation:

https://gitlab.inria.fr/core-math/core-math/-/blob/master/src/binary64/log2/log2.c

It takes 19 cycles on average on an Intel(R) Core(TM) i7-8700, and is
correctly rounded (for any rounding mode).

Paul



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