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Message-ID: <1785564.ahg88INnCG@PointedEars.de>
From: Thomas 'PointedEars' Lahn <PointedEars@web.de>
Reply-To: Thomas 'PointedEars' Lahn <cljs@PointedEars.de>
Organization: PointedEars Software (PES)
Date: Sat, 04 Feb 2012 15:12:27 +0100
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Subject: Re: Fun With Numbers
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Tom de Neef wrote:

> "Gene Wirchenko" <genew@ocis.net>
>>     I need to do work with fixed-decimal quantities (mainly dollar
>> amounts but others as well).  I need to be able to do reliable
>> arithmetic with them.
>>
>>     0.1 + 0.1 + 0.1 equals 0.15 + 0.15 mathematically, but not with
>> floating point.  I need to have it equal.
>>
>>     So I am cheating.  I am storing fixed-decimal amounts internally
>> as integers.  When I need to output a value, I scale it, but any
>> arithmetic or comparison operations will be on the integers.
> 
> Just wondering: how do you calculate 3,62% of ($4.56 + $14.13) and equal
> it to 3,62% of $4.56 plus 3,62% of $14.13 ?
> Or go from euro to dollar, etc ?

You store the floating-point number n as an integer value (i.e., this.value 
% 1 = 0) and store the number of significant decimal digits of n 
(this.scale) along with it in an object.  Then

  n = this.value × 10^(−this.scale).

Of course, in order to do basic arithmetic with the object, you need special 
methods that account for the scale.  That is,

    a.b + c.de
  ~ ab:scale1 + cde:scale2
  ~ (ab × 10^(scale2 − scale1) + cde)):max(scale1, scale2)
  ~ (ab × 10^(scale2 − scale1) + cde) × 10^(−max(scale1, scale2)),

and

    a.b × c.de
  ~ ab:scale1 × cde:scale2
  ~ (ab × 10^(scale2 − scale1) × cde):(scale1 × scale2)
  ~ (ab × 10^(scale2 − scale1) × cde) × 10^(−scale1 × scale2)

where scale1 <= scale2 (here: scale1 = 1, scale2 = 2).

The idea is anything but new.  For example, Java has had this for many 
years: 
<http://docs.oracle.com/javase/1.5.0/docs/api/java/math/BigDecimal.html>

The precision of an implementation of this in an ECMAScript implementation 
is limited by the precision for integer values (number values `i' with i % 1 
== 0), as the unscaled value and the scale are still IEEE-754 floating-point 
values.  But the precision of computation is indeed better than with plain 
floating-point values, that is 1:1 + 1:1 + 1:1 ~ 3:1 ~ 0.3.


PointedEars
-- 
When all you know is jQuery, every problem looks $(olvable).