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<div class="moz-cite-prefix">On 13/06/2018 11:07, J. Gareth Moreton
wrote:<br>
</div>
<blockquote type="cite"
cite="mid:4982.1528844845@web-cluster.fastnet.co.uk">
Well, I would argue that when computing log(x) of any base, as x
tends to 0 from a positive number, log(x) becomes a progressively
more negative number to the point that, yes, it's undefined, but
that's only going by the definition of limits.<br>
<br>
<div>The issue here is that my proposed logarithm functions work
not on floating-point numbers, but integers, primarily for the
sake of speed and application. There is no way to store a NaN
or plus or minus infinity in an integral type, which means the
only other option is to raise an exception.</div>
</blockquote>
That's why I am willing to withdraw my objections, as stated, the
moment you remove log from the function name. What you want to use
is a function of your own design, with value mappings different from
what a logarithm does. My objection is not about your optimization
work, it is about potential mis-use of your work, and
mis-understandings by any future maintainer of your work.<br>
Wolf<br>
<blockquote type="cite"
cite="mid:4982.1528844845@web-cluster.fastnet.co.uk">
<div><br>
</div>
<div>For truly mathematical functions with a continuous domain,
then yes, it should return proper values. I suppose what I'm
proposing is not the true base-2 logarithm, but two functions
that do the following:<br>
<br>
</div>
<div>FLog2(x), x element of N (natural numbers), including zero<br>
<br>
0, x = 0<br>
floor(log2(x)); x ≠ 0<br>
</div>
<br>
CLog2(x), x element of N, including zero<br>
<br>
0, x = 0<br>
ceil(log2(x)); x ≠ 0<br>
<div><br>
</div>
<div>(Not easy to describe when you can't use proper mathematical
notation in an e-mail!)</div>
<div><br>
</div>
<div>In this instance, it's less about mathematical correctness
and more for the sake of convenience, because only a single
input (zero) would be undefined, and for the sake of bit lengths
and loop iterations, 0 is a more convenient value than throwing
an exception or returning something undefined or potentially
hazardous like $FFFFFFFF, which if passed blindly into a
for-loop, will cause 4.29 billion iterations..</div>
</blockquote>
I understand that, and if you want to use a Bit Scan Reverse
instruction, use it. But do not call it a logarithm, because that
has implications . . . Take a look at the <b>[fpc-pascal]
round(2.5)=2</b> thread. Why is nobody there suggesting to look
for Intel to sort out his /her rounding issues? That thread displays
the kind of blindness I am concerned about. The answers are
available, but hidden in massive documentation, as you yourself
noticed so recently.<br>
Wolf<br>
<blockquote type="cite"
cite="mid:4982.1528844845@web-cluster.fastnet.co.uk">
<div><br>
</div>
<div>Gareth<br>
</div>
<br>
<br>
<span style="font-weight: bold;">On Wed 13/06/18 00:45 , Wolf
<a class="moz-txt-link-abbreviated" href="mailto:wv99999@gmail.com">wv99999@gmail.com</a> sent:<br>
</span>
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<p>Hi</p>
<p>I object to one component of Gareth's proposal - to make
log2(0)=0. The problem lies not with what Gareth wants to
do with it, but what other people will use it for once it
becomes available. log2(0) is undefined (and undefinable,
as it is not a computable number), the appropriate choice
for log2(0) is to make it Not-A-Number (NaN). <br>
</p>
<p align="center">FLog2(0) = NaN = CLog2. <br>
</p>
<p>Such a choice would avoid the mess Gustavson got himself
into when he <a moz-do-not-send="true"
href="www.johngustafson.net/pdfs/BeatingFloatingPoint.pdf">mapped</a>
0 and 1/0 onto the same number - a mapping that has many
advantages for computing, but eventually <a
moz-do-not-send="true"
href="https://arxiv.org/pdf/1701.00722">destroys
computability</a>. To a lesser degree, this mess is
already present in the IEEE 754 standard for
floating-point arithmetic, and thus led to, to put it
mildly, <a moz-do-not-send="true"
href="www.itu.dk/%7Esestoft/bachelor/IEEE754_article.pdf">computing
difficulties</a>, difficulties that many programmers
gloss over - or simply ignore.<br>
</p>
I will have to say more about this when I am ready to file a
bug report on floating point exceptions, since Gareth's
proposal has deep links to how floating point numbers are
defined - and why they were defined such.<br>
<br>
Wolf<br>
<br>
<div class="moz-cite-prefix">On 13/06/2018 00:42, J. Gareth
Moreton wrote:<br>
</div>
<blockquote type="cite"
cite="mid:2794.1528807356@web-cluster.fastnet.co.uk">
<div> Hi everyone,</div>
<div><br>
</div>
<div>Sorry to flood the mailing list again with more ideas
and experiments.<br>
<br>
I would like to propose introducing a new pair of
in-built functions for the compiler.<br>
<br>
function FLog2(X: Cardinal): Cardinal;<br>
function CLog2(X: Cardinal): Cardinal;<br>
<br>
FLog2 returns the base-2 logarithm of X, rounded down,
while CLog2 returns the base-2 logarithm of X, rounded
up.<br>
<br>
To give examples where these functions could be useful,
FLog2(X) + 1 indicates the minimum number of bits
required to store X as an unsigned integer, while
CLog2(X) is equal to the maximum number of iterations
required for a binary search of an X-sized list.<br>
</div>
</blockquote>
Given the stated purpose, I could withdraw my objection if
any reference to logarithm was removed from the function and
its name. Then Gareth would be free to create his function
any way he likes and assign to it the properties he chooses.
The only requirement left then would be to state in the
comments around the function what it is supposed to achieve,
as a deterrence to mis-use and guidance to future
maintainers, who may not think the same as Gareth does.<br>
<blockquote type="cite"
cite="mid:2794.1528807356@web-cluster.fastnet.co.uk">
<div> Why should they be in-built though? With the binary
search example, the size of the list is sometimes known
at compile time, hence is a constant - therefore, its
logarithm is also a constant. By pre-calculating the
logarithm using the in-built function, it can be used to
aid optimization such as loop unrolling. It also makes
them easier to inline, where FLog2(X) on x86_64-win64
translates to a single line of assembly language: BSR
EAX, ECX (unless X is zero, in which case ZF is set and
EAX is undefined).<br>
<br>
If there had to be a slight nuance though, it's what
happens if X = 0, since log(0) = -oo</div>
</blockquote>
This statement is false. log(0) is not infinity. To obtain a
numerical value for log(0) by e.g. Taylor series expansion,
at one stage you have to divide by zero since the
differential<br>
<div align="center">(d ln x )/ d x = 1/x.<br>
</div>
And since 1/0 is not an element of the set of computable
numbers, log(0) is not either. The only valid assignment can
be log(0):=NaN, for any base.<br>
<blockquote type="cite"
cite="mid:2794.1528807356@web-cluster.fastnet.co.uk">
<div>, which cannot be stored as an integer and may cause
problems with the compiler. I would propose it return 0
as a special case, as this will fix most problems when
it comes to loops and storage, and also ensure that
FLog2 and CLog2 are "entire functions". To give an
optimised example of FLog(2) in x86-64 assembly
language:<br>
<br>
XOR EDX, EDX<br>
BSR EAX, ECX // ZF is set if ECX is zero<br>
CMOVZ EAX, EDX // Zero (EDX) written to result if ZF is
set.<br>
<br>
Some kind of deep optimization could be used if the
input is known to be non-zero and remove the
instructions either side of BSR.<br>
<br>
(Alternative names: FILog2 and CILog2, to indicate they
work on integers and to distinguish them from versions
that work with floating-point numbers)<br>
<br>
Gareth<br>
</div>
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