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From Gilles <gil...@harfang.homelinux.org>
Subject Re: [numbers] Fraction() and Knuth 4.5.1 -- overflow, BigInteger, long, and rounding
Date Sat, 01 Dec 2018 01:23:31 GMT
On Fri, 30 Nov 2018 15:56:54 -0800, Eric Barnhill wrote:
> Here is what I propose for the Fraction doc text regarding this 
> issue:
>
>      * Implement add and subtract. This algorithm is similar to that
>      * described in Knuth 4.5.1. while making some concessions to
>      * performance. Note Knuth 4.5.1 Exercise 7, which observes that
>      * adding two fractions with 32-bit numerators and denominators
>      * requires 65 bits in extreme cases. Here calculations are 
> performed
>      * with 64-bit longs and the BigFraction class is recommended for
> numbers
>      * that may grow large enough to be in danger of overflow.

Does this mean that computations can "unpredictably" overflow
(or throw an exception)?
Is it acceptable, or should we enclose the problematic code in
a "try" block and redo the computation with "BigInteger" when
necessary?

What is the performance hit of using "BigFraction" rather than
"Fraction"?
Are there use-cases that would need the ultimate performance from
"Fraction" while not worry about overflow?

Regards,
Gilles

>
>
> On Fri, Nov 9, 2018 at 4:33 PM Eric Barnhill <ericbarnhill@gmail.com> 
> wrote:
>
>> Addendum to the above. In an exercise in the Knuth book Knuth does 
>> indeed
>> state that "If the inputs are n-bit binary numbers, 2N+1 bits may be
>> necessary to represent t." where t is a derived quantity that would 
>> take
>> some time to explain.
>>
>> So that means in extreme cases, the needed precision to represent a
>> fraction operation with 32 bits ints is 65 bits, one more than a 
>> long has.
>>
>> The present code solves this by using BigInteger briefly in the 
>> code,
>> which strikes me as an awfully big performance hit for what must 
>> surely be
>> very occasional and very  extreme cases.
>>
>> I think the most sensible strategy would be to restrict the 
>> precision of
>> Fraction to longs, with user guidance to use BigFraction if there is
>> concern of overflow.
>>
>> Eric
>>
>>
>>
>>
>>
>>
>>
>> On Thu, Nov 8, 2018 at 11:11 AM Gary Gregory 
>> <garydgregory@gmail.com>
>> wrote:
>>
>>> I'm all for the Javadoc made to reflect the reality of the code. It 
>>> is
>>> fine
>>> to have an additional section that points out Knuth and how we may 
>>> want to
>>> change things as a hint or request to contributors.
>>>
>>> Gary
>>>
>>> On Wed, Nov 7, 2018 at 10:52 AM Eric Barnhill 
>>> <ericbarnhill@gmail.com>
>>> wrote:
>>>
>>> > I read Kunth's "Art of Computer Programming 4.5.1" that is 
>>> referenced
>>> many
>>> > times in the doc as the guidance for the 
>>> commons-math/commons-numbers
>>> > Fraction class. It is an interesting read. Also, for all the 
>>> times it is
>>> > cited in the doc, it is interesting that Fraction doesn't really 
>>> use it
>>> as
>>> > implemented. Here is one example.
>>> >
>>> > Knuth is concerned about overflow in multiplication and division,
>>> because
>>> > numerator of f1 is multiplied by denominator of f2 and so forth, 
>>> so he
>>> > suggests a technique called "mediant rounding" that allows for
>>> intermediate
>>> > quantities in fraction multiplication to be rounded.
>>> >
>>> > It is a clever technique and probably works well, however the 
>>> current
>>> > Fraction class cites this chapter, then implements multiplication 
>>> with
>>> > BigInteger instead, ignoring this suggestion.
>>> >
>>> > First of all, the doc should be clear that the code is NOT 
>>> following
>>> 4.5.1,
>>> > while it gives the opposite impression. And that's ok but the use 
>>> of
>>> > BigInteger creates additional inconsistency: Multiply and divide 
>>> are
>>> > accomplished using ArithmeticUtils.addAndCheck and
>>> > ArithmeticUtils.mulAndCheck . These convert the relevant ints to 
>>> longs,
>>> > then perform the operation, then if the resulting long is greater 
>>> than
>>> the
>>> > range of an int, throw an OverflowException. So some parts of 
>>> Fraction
>>> > check for overflow using longs and others use BigInteger.
>>> >
>>> > It seems to me that BigInteger is overkill here for the vast 
>>> majority of
>>> > practical uses of Fraction in a way that could be damaging for
>>> performance.
>>> > And furthermore, we already have a BigFraction class to handle 
>>> cases
>>> that
>>> > require BigInteger.
>>> >
>>> > So, I propose rewriting the doc to say the opposite of what it 
>>> currently
>>> > says when appropriate, and get usages of BigInteger out of 
>>> Fraction, use
>>> > them only in BigFraction, and use the long-based ArithmeticUtils
>>> methods to
>>> > check for overflow and underflow in fraction addition and 
>>> subtraction.
>>> >
>>> > Eric
>>> >
>>>
>>


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