[erlang-questions] Maths Problem -> 2452.45*100. = 245244.99999999997

[Michael Turner]

Gee, now might be a good time to bore everyone with a war story: a
startup I once worked for independently /pre/-invented the Pentium
Floating Point Bug, against my strenuous protests. How could this
happen, I wondered, as I was losing the battle for IEEE-standard
purity? Maybe, I thought, it's because we're a small company that
can't afford to hire experts to ratify my opinion?

We were using Weitek FPU adder and multiplier chips in our hardware,
and I implemented floating point division the way they said to do it -
the right way. Tried to, that is. I got overruled.

Not too many years later, for the Pentium design, Intel acquired
Weitek's IP. Then they introduced very similar mistakes in floating
point arithmetic, through no fault of Weitek. Ours, at least, could
have been patched by sending customers a new ROM to socket. Intel's
errors went onto a single chip.

What I learned about IEEE floating point arithmetic is that if you try
to clean up floating point "errors" one way, you just squeeze the
errors out some place else.

What I learned about people is that Size Doesn't Matter.

The proximate provocation in both cases: somebody didn't like how the
number printed. I first saw "errors" like when I was in high school,
programming in Dartmouth BASIC. I was told, "floating point is like
that - if it weren't that, it would be something worse." So I was
prepared. But lots of people now graduate with C.S. degrees who aren't
prepared.


Regards,
Michael Turner
Executive Director
Project Persephone
K-1 bldg 3F
7-2-6 Nishishinjuku
Shinjuku-ku Tokyo 160-0023
Tel: +81 (3) 6890-1140
Fax: +81 (3) 6890-1158
Mobile: +81 (90) 5203-8682
turner@REDACTED
http://www.projectpersephone.org/

"Love does not consist in gazing at each other, but in looking outward
together in the same direction." -- Antoine de Saint-Exupéry


On Wed, Nov 27, 2013 at 12:28 PM, Richard A. O'Keefe <ok@REDACTED> wrote:
>
> On 27/11/2013, at 3:01 PM, Sanath Prasanna wrote:
>
>> I accepted your comment. But generally we expected full value. I did same in another scripting language like php. It is given correct result.
>
> Wrong.  It *calculated* the same number as Erlang did,
> but then it *displayed* a different number, a rounded version.
>
> m% cat >foo.php
> <?php
> $a = 2452.45*100;
> echo $a; echo "\n";
> $a = $a - 245245.0;
> echo $a; echo "\n";
> exit;
> ?>
> <EOF>
> m% php foo.php
> 245245
> -2.9103830456734E-11
>
> We see from this that while the number *displays as*
> 245245, it is not *equal to* 245245.
>
> As I said, PHP computed *exactly* the same answer as Erlang.
> Who is this "we" who "expect full value"?
> This is floating-point arithmetic, not rational arithmetic.
>
> (Our 2nd year students are taught this stuff, but it doesn't
> really take.  In 3rd year they have an exercise they have to
> do where they start to really understand that floating point
> arithmetic is *not* arithmetic on the mathematical real
> numbers, but a different mathematical system which sort of
> approximates the algebra of the reals.  They learn that
> adding up an array of numbers from left to right generally
> gives you a different answer from adding up from right to
> left, for example.)
>
> Numbers in IEEE floating point arithmetic are precise,
> but are bounded in their precision, and arithmetic operations
> on them (including conversion between decimal and binary)
> have to round their answers to something that is representable.
> There's a rounding in the conversion of 2452.45 to binary,
> and then the multiplication incurs another rounding.
>
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