[squeakdev] The Inbox: Kernelnice.903.mcz
Nicolas Cellier
nicolas.cellier.aka.nice at gmail.com
Sat Feb 14 01:29:12 UTC 2015
20150214 2:20 GMT+01:00 <commits at source.squeak.org>:
> Nicolas Cellier uploaded a new version of Kernel to project The Inbox:
> http://source.squeak.org/inbox/Kernelnice.903.mcz
>
> ==================== Summary ====================
>
> Name: Kernelnice.903
> Author: nice
> Time: 14 February 2015, 2:19:44.184 am
> UUID: d4f06790d47b4cb0b9ff9e0ff7dea9fc
> Ancestors: Kernelnice.902
>
> Introduce two new alternatives to integer division: #ratio: and #residue:
> are like #quo: and #rem: except that they round the quotient to nearest
> integer (tie to even) instead of truncating (note that // and \\ floor the
> quotient...).
>
> The second thing that they do differently is that they compute the exact
> ratio and exact residue when given a pair of Float.
>
> The third thing that they do differently is that they first coerce a pair
> of numbers to the highest generality before attempting any operation.
>
> =============== Diff against Kernelnice.902 ===============
>
>
Hi folks,
I've already bent Squeak/Pharo handling of Float a bit, and I'd like to
increase further Smalltalk dialects compliance to language independent
arithmetic standard (ISO/IEC 10967).
The subject of the day is state of the art quotient and remainder
operations for Floats.
I noticed that the libm function fmod which computes the remainder of
division of two floats (quotient being the truncated fraction) is and
should be exact.
Our equivalent Float version of #rem: is not, and I had the feeling that it
ain't good.
So I wrote a pair of methods #fquo: and #fmod: that evaluates the
(truncated) quotient (quo:) and remainder (rem:) of 2 floats exactly.
Then I plugged these functions to implementors of quo: and rem:.
They are not only exact, they also support extreme Floats without overflow,
like
(Float fmax quo: Float fmin*5).
(Float fmax rem: Float fmin*5).
We will let // and \\ apart for the moment  see below for the reason.
Once plugged, some results might unfortunately look surprising like:
(1 to: 100) count: [:i  ((i/100.0) fquo: 0.01) ~= i].
> 58
(1 to: 100) count: [:i  ((i/100.0) fmod: 0.01) ~= 0].
> 93
The seven exact multiple are of course (0 to: 6) collect:[:i  0.01
timesTwoPower: i]
The others just tell the awful truth about floating point...
While with the old implementation was a little less surprising:
(1 to: 100) count: [:i  ((i/100.0) / 0.01) truncated ~= i].
> 6
(1 to: 100) count: [:i  (((i/100.0) / 0.01) truncated * 0.01  (i/100.0))
~= 0].
> 13
Then I realized that language independent arithmetic standard and also
older floating point standard (IEEE 754), do rather define quotient and
remainder (ratio and residue) in term of rounded rather than truncated
quotient.
So I implemented fratio: and fresidue: as prescribed by the standard (with
an unbiased roundedTieToEven).
And these two operations better fit our expectations, at least for the
ratio:
(1 to: 100) count: [:i  ((i/100.0) fratio: 0.01) ~= i].
> 0
The residues are exact allways small:
(1 to: 100) count: [:i 
 a b fa fb ea eb 
a := 1/100. fa := a asFloat. ea := a  fa asFraction. "exact, float,
and error"
b := i*a. fb := b asFloat.. eb := b  fb asFraction.
[eb <= (0.5*fb ulp) and: [ea <= (0.5*fa ulp)]] assert.
(fb fresidue: fa) abs > (ea*ieb) abs].
> 0
The naive implementation works well too for the ratio:
(1 to: 100) count: [:i  ((i/100.0) / 0.01) rounded ~= i].
> 0
but of course with inexact residue:
(1 to: 100) count: [:i 
 a b fa fb ea eb 
a := 1/100. fa := a asFloat. ea := a  fa asFraction. "exact, float,
and error"
b := i*a. fb := b asFloat.. eb := b  fb asFraction.
[eb <= (0.5*fb ulp) and: [ea <= (0.5*fa ulp)]] assert.
(fb  ((fb/fa) rounded*fb)) abs > (ea*ieb) abs].
> 99
Rounded quotient is exactly what we want for Float. The rounding problems
still exist, but they are rejected near exact tie.
But that means implementing yet another quo/rem message (#ratio: #residue:
using terms of the standard) while we already have two pairs...
Do you think that such extension ratio/residue is valuable?
Do you think that modification of quo:/rem: is bearable?
The alternative would be to provide an external package (eventually with
overrides) to turn Squeak/Pharo behavior into standard, but I don't like
the idea that the kernel does not respect the standards by default...

About // and \\:
The problem with \\ is that it can't always be exact when expressed as
Float.
(b \\ a) can be near a when b abs < a abs, a*b < 0.
So when b is of very small magnitude relative to a, remainder requires many
bits.
For example Float fmin negated asFraction \\ 1.0 asFraction would exactly
be (1.0 asFraction  Float fmin asFraction) and thus require a significand
of 1074 bits.
We don't have ArbitraryPrecisionFloat in trunk, and Fraction is overkill...
We can't either let the exact remainder be rounded to nearest Float because
it will be equal to the divider (it should be less than in magnitude).
So, there is no easy solution for this case in Float...
But note that current naive implementation suffers exactly from same
problem and would answer 1.0 (the divider).

I've also asked peers for a good reason to implement an exact #rem: but
knowledgeable peers are unreachable yet ;)
http://cs.stackexchange.com/questions/24362/whydoesfloatingpointmodulusexactnessmatters
So maybe it's time to push this stuff
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