RFR: 8189107 - AARCH64: create intrinsic for pow

[Andrew Dinn]

On 13/09/18 15:35, Dmitrij Pochepko wrote:
> Other comments seems fine
I am glad to hear that you did not find any errors in my analysis.
However, I also need to ask you to answer a question that was implicit
in my earlier note. I said:

"I assume you are familiar with the relevant mathematics and how it has
been used to derive the algorithm. If so then I would like you to review
this rewrite and ensure that there are nor mathematical errors in it. I
would also like you to check that the explanatory comments for of the
individual steps in the algorithm do not contain any errors.

If you are not familiar with the mathematics then please let me know. I
need to know whether this has been reviewed by someone competent to do so."

As you didn't respond to this I will have to ask you explicitly this
time. Do you have a background in mathematics and numerical analysis
that means you understand how the original algorithm has been arrived
at? equally, how your algorithm may legitimately vary from that original?

I'll break this down into several steps:

Do you understand the (elementary) theory that explains how the various
polynomial expansions I described in my comments converge to the
original log and exp functions?

Do you understand the theory that explains how partial polynomial sums
(Remez polynomials) can be used used to approximate these polynomial
expansions within specified ranges?

Do you know how the coefficients of these Remez polynomial can be
derived to any necessary accuracy?

Do you understand how the computation of the values of those Remez
polynomials must proceed in order to guarantee accuracy in the computed
result in the presence of rounding errors?

Can you provide a mathematical proof that the variations you have
introduced into the computational process (specifially the move from
Horner form to Estrin form) will not introduce rounding errors?

I certainly cannot lay claim to a /thorough/ understanding of most, if
not all, those topics. If you also cannot then I think we need to bring
in someone who does. In particular, it is the last point that matters
most of all here as this is where you have /chosen/ to make your
algorithm diverge from the code you inherited.

As regards the rest of the background maths, we do at least know that
the other aspects of the algorithm -- in its original manifestation --
have been checked by numerical experts. Hence, if we ensure that your
algorithm implements /equivalent/ steps then it ought to inherit the
same guarantees of correctness. So, the only task as far as most of the
code is concerned is to iron out any errors you might inadvertently have
introduced. I have several nits to pick in that regard that which I will
be posting shortly.

regards,


Andrew Dinn
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Senior Principal Software Engineer
Red Hat UK Ltd
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