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<p>I agree that choice of algorithm is much more important than
compiler optimisations. Indeed, a reliance on compiler
optimisations is often, in my experience, a sign of underlying
ignorance of relevant numerical algorithms. This is particularly
visible when, using an iterative numerical method, one does not
pay sufficient - or as is often the case absolutely no - attention
to getting good starting values. Another mortal sin is to forget
about strict error bounds. Chebyshev always insisted on them and
he was one of the founders of modern approximation theory.</p>
<p>And as regards summing numbers in ascending order of magnitude,
yes, it is better than summing them in no particular order, but
there still pitfalls for the unwary even when ascending order is
used. The best short text on this is, IMHO:</p>
<p><i>Rounding Errors in Algebraic Processes</i>, J. H. Wilkinson,
Dover Publications</p>
<p>though it's hard to get hold of and can cost an arm and a leg
even when you can find it.</p>
<p><br>
</p>
<p>regards,</p>
<p>Olwen</p>
<p><br>
</p>
<div class="moz-cite-prefix">On 10/05/2020 15:54, Peter Bishop
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:db5cf447-66b0-e251-bb44-04e1ddb524be@adelard.com">
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<div class="moz-cite-prefix">Exact results are pretty unlikely for
FP.<br>
But on the other hand, I think the *numerical algorithm* is
probably more important than any local HLL optimisations in
reducing rounding errors, <br>
</div>
<div class="moz-cite-prefix">e.g. when adding an array of FP
numbers, the best option is to sort them an ascending order
before sequentially adding them. <br>
<br>
</div>
<div class="moz-cite-prefix">Peter Bishop</div>
<div class="moz-cite-prefix"><br>
</div>
<div class="moz-cite-prefix">On 09/05/2020 18:21, Peter Bernard
Ladkin wrote:<br>
</div>
<blockquote type="cite"
cite="mid:024617e1-2a58-d457-8bee-c4e0858daab3@causalis.com">
<pre class="moz-quote-pre" wrap="">On 2020-05-09 18:41 , Derek M Jones wrote:
</pre>
<blockquote type="cite">
<pre class="moz-quote-pre" wrap="">Section 5.1.2.3 of the C Standard,
or sentence 189: <a class="moz-txt-link-freetext" href="https://c0x.coding-guidelines.com/5.1.2.3.html" moz-do-not-send="true">https://c0x.coding-guidelines.com/5.1.2.3.html</a>
"In the abstract machine, all expressions are evaluated as specified by the semantics."
</pre>
</blockquote>
<pre class="moz-quote-pre" wrap="">Which should mean that the expression "x^2 - y^2" returns the square of the value of x minus the
square of the value of y. What is called the exact value.
What a poor piece of writing.
PBL
Prof. Peter Bernard Ladkin, Bielefeld, Germany
Styelfy Bleibgsnd
Tel+msg +49 (0)521 880 7319 <a class="moz-txt-link-abbreviated" href="http://www.rvs-bi.de" moz-do-not-send="true">www.rvs-bi.de</a>
</pre>
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<pre class="moz-signature" cols="72">--
Peter Bishop
Chief Scientist
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