[squeak-dev] Exact #sqrt

Nicolas Cellier nicolas.cellier.aka.nice at gmail.com
Mon Oct 17 20:46:21 UTC 2011


Hi Juan,
Finally, why didn't you just tried some sort of Newton ?

nthRootTruncated: aPositiveInteger
	"Answer the integer part of the nth root of the receiver."
	| guess guessToTheNthMinusOne delta |
	self = 0 ifTrue: [^0].
	self negative
		ifTrue:
			[aPositiveInteger even ifTrue: [ ArithmeticError signal: 'Negative
numbers don''t have even roots.' ].
			^(self negated nthRootFloor: aPositiveInteger) negated].
	guess := 1 bitShift: self highBitOfMagnitude + aPositiveInteger - 1
// aPositiveInteger.
	[
		guessToTheNthMinusOne := guess raisedToInteger: aPositiveInteger - 1.
		delta := (guess * guessToTheNthMinusOne - self) //
(guessToTheNthMinusOne * aPositiveInteger).
		delta = 0 ] whileFalse:
			[ guess := guess - delta ].
	( (guess := guess - 1) raisedToInteger: aPositiveInteger) > self  ifTrue:
			[ guess := guess - 1 ].
	^guess

It's not exactly cheap (repeated raisedTo:) but it should work without Fraction.
Note that selector is named Truncated not Floor, because it better
reflects the operation on negative receivers.

Nicolas

2011/10/14 Juan Vuletich <juan at jvuletich.org>:
> Nicolas Cellier wrote:
>>
>> You might also want another way to quickly discriminate non square
>> large integers
>>
>> http://en.wikipedia.org/wiki/Square_number
>>
>> if (n & 7 == 1) or (n & 31 == 4) or (n & 127 == 16) or (n & 191 == 0) then
>>    return n is probably square
>> else
>>    return n is definitely not square
>>
>> LargePositiveInteger>>canBeASquare
>>    | lsb |
>>    lsb := self digitAt: 1.
>>    ^((lsb bitAnd: 7) = 1
>>        or: [(lsb bitAnd: 31) = 4
>>        or: [(lsb bitAnd: 127) = 16
>>        or: [(lsb bitAnd: 191) = 0]]])
>>
>> Nicolas
>>
>> 2011/10/13 Nicolas Cellier <nicolas.cellier.aka.nice at gmail.com>:
>>
>>>
>>> This can cause an overflow...
>>>
>>> ((1 << 1024 + 1) / (1 << 1024 + 3)) sqrt
>>>
>>> I'd expect 1.0 (inexact, but not that much).
>>>
>
> Thanks, Nicolas. This new version addresses all the issues you raised.
>
> Cheers,
> Juan Vuletich
>
> 'From Cuis 3.3 of 2 June 2011 [latest update: #1024] on 14 October 2011 at
> 7:57:15 am'!
>
> !FloatTest methodsFor: 'testing' stamp: 'jmv 10/11/2011 08:55'!
> testMaxExactInteger
>        "
>        FloatTest new testMaxExactInteger
>        "
>
>        self assert: Float maxExactInteger asFloat truncated = Float
> maxExactInteger.
>        0 to: 10000 do: [ :j |
>                self assert: (Float maxExactInteger-j) asFloat truncated =
> (Float maxExactInteger-j) ].
>        self deny: (Float maxExactInteger+1) asFloat truncated = (Float
> maxExactInteger+1)
>        ! !
>
> !FloatTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 09:06'!
> testNthRoot
>        "
>        FloatTest new testNthRoot
>        "
>        self should: [ -1.23 nthRoot: 4 ] raise: ArithmeticError! !
>
> !FloatTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 09:06'!
> testRaisedTo
>        "
>        FloatTest new testRaisedTo
>        "
>        self should: [ -1.23 raisedTo: 1/4 ] raise: ArithmeticError! !
>
>
> !FractionTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/11/2011 22:27'!
> testExactRaisedTo
>        "
>        FractionTest new testExactRaisedTo
>        "
>        | f |
>        self assert: (4/9 raisedTo: 1/2) classAndValueEquals: 2/3.
>        self assert: (9/4 raisedTo: 1/2) classAndValueEquals: 3/2.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) pairsDo: [ :a :b |
>                f _ a / b.
>                self assert: (f squared raisedTo: 1/2) classAndValueEquals:
> f.
>                self assert: (f negated squared raisedTo: 1/2)
> classAndValueEquals: f.
>                f _ b / a.
>                self assert: (f squared raisedTo: 1/2) classAndValueEquals:
> f.
>                self assert: (f negated squared raisedTo: 1/2)
> classAndValueEquals: f ].
>
>        self assert: (8/27 raisedTo: 1/3) classAndValueEquals: 2/3.
>        self assert: (27/8 raisedTo: 1/3) classAndValueEquals: 3/2.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) pairsDo: [ :a :b |
>                f _ a / b.
>                self assert: ((f raisedTo: 3) raisedTo: 1/3)
> classAndValueEquals: f.
>                self assert: ((f negated raisedTo: 3) raisedTo: 1/3)
> classAndValueEquals: f negated.
>                f _ b / a.
>                self assert: ((f raisedTo: 3) raisedTo: 1/3)
> classAndValueEquals: f.
>                self assert: ((f negated raisedTo: 3) raisedTo: 1/3)
> classAndValueEquals: f negated ].
>
>        self assert: (4/9 raisedTo: 3/2) classAndValueEquals: 8/27.
>        self assert: (8/27 raisedTo: 2/3) classAndValueEquals: 4/9.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) pairsDo: [ :a :b |
>                f _ a / b.
>                self assert: ((f raisedTo: 3) raisedTo: 2/3)
> classAndValueEquals: f*f.
>                self assert: ((f raisedTo: 2) raisedTo: 3/2)
> classAndValueEquals: f*f*f.
>                self assert: ((f negated raisedTo: 3) raisedTo: 2/3)
> classAndValueEquals: f*f.
>                self assert: ((f negated raisedTo: 2) raisedTo: 3/2)
> classAndValueEquals: f*f*f.
>                f _ b / a.
>                self assert: ((f raisedTo: 3) raisedTo: 2/3)
> classAndValueEquals: f*f.
>                self assert: ((f raisedTo: 2) raisedTo: 3/2)
> classAndValueEquals: f*f*f.
>                self assert: ((f negated raisedTo: 3) raisedTo: 2/3)
> classAndValueEquals: f*f.
>                self assert: ((f negated raisedTo: 2) raisedTo: 3/2)
> classAndValueEquals: f*f*f ].
>
>        self assert: (32/243 raisedTo: 3/5) classAndValueEquals: 8/27.
>        self assert: (8/27 raisedTo: 5/3) classAndValueEquals: 32/243.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) pairsDo: [ :a :b |
>                f _ a / b.
>                self assert: ((f raisedTo: 5) raisedTo: 3/5)
> classAndValueEquals: f*f*f.
>                self assert: ((f raisedTo: 3) raisedTo: 5/3)
> classAndValueEquals: f*f*f*f*f.
>                self assert: ((f negated raisedTo: 5) raisedTo: 3/5)
> classAndValueEquals: (f*f*f) negated.
>                self assert: ((f negated raisedTo: 3) raisedTo: 5/3)
> classAndValueEquals: (f*f*f*f*f) negated.
>
>                self assert: ((f raisedTo: -5) raisedTo: 3/5)
> classAndValueEquals: 1/(f*f*f).
>                self assert: ((f raisedTo: -3) raisedTo: 5/3)
> classAndValueEquals: 1/(f*f*f*f*f).
>                self assert: ((f negated raisedTo: -5) raisedTo: 3/5)
> classAndValueEquals: -1/(f*f*f).
>                self assert: ((f negated raisedTo: -3) raisedTo: 5/3)
> classAndValueEquals: -1/(f*f*f*f*f).
>                self assert: ((f raisedTo: 5) raisedTo: -3/5)
> classAndValueEquals: 1/(f*f*f).
>                self assert: ((f raisedTo: 3) raisedTo: -5/3)
> classAndValueEquals: 1/(f*f*f*f*f).
>                self assert: ((f negated raisedTo: 5) raisedTo: -3/5)
> classAndValueEquals: -1/(f*f*f).
>                self assert: ((f negated raisedTo: 3) raisedTo: -5/3)
> classAndValueEquals: -1/(f*f*f*f*f).
>
>                "No exact result => Float result"
>                self assert: ((f raisedTo: 3) +1 raisedTo: 5/3) isFloat.
>                self assert: ((f negated raisedTo: 3) -1 raisedTo: 5/3)
> isFloat.
>
>                f _ b / a.
>                self assert: ((f raisedTo: 5) raisedTo: 3/5)
> classAndValueEquals: f*f*f.
>                self assert: ((f raisedTo: 3) raisedTo: 5/3)
> classAndValueEquals: f*f*f*f*f.
>                self assert: ((f negated raisedTo: 5) raisedTo: 3/5)
> classAndValueEquals: (f*f*f) negated.
>                self assert: ((f negated raisedTo: 3) raisedTo: 5/3)
> classAndValueEquals: (f*f*f*f*f) negated.
>
>                "No exact result => Float result"
>                self assert: ((f raisedTo: 3) +1 raisedTo: 5/3) isFloat.
>                self assert: ((f negated raisedTo: 3) -1 raisedTo: 5/3)
> isFloat ].! !
>
> !FractionTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/11/2011 22:12'!
> testExactSqrt
>        "
>        FractionTest new testExactSqrt
>        "
>        | f |
>        self assert: (4/9) sqrt classAndValueEquals: 2/3.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) pairsDo: [ :i :j |
>                f _ i / j.
>                self assert: f squared sqrt classAndValueEquals: f.
>                f _ j / i.
>                self assert: f squared sqrt classAndValueEquals: f ]! !
>
> !FractionTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 21:38'!
> testInexactRaisedTo
>        "
>        FractionTest new testInexactRaisedTo
>        "
>        self assert: (((1 << 1024 + 1) / (1 << 1024 + 3)) raisedTo: 1/3) =
> 1.0.
>        self assert: (((1 << 1024 + 1) / (1 << 1024 + 3)) negated raisedTo:
> 1/3) = -1.0! !
>
> !FractionTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 21:27'!
> testInexactSqrt
>        "
>        FractionTest new testInexactSqrt
>        "
>        self assert: ((1 << 1024 + 1) / (1 << 1024 + 3)) sqrt = 1.0! !
>
> !FractionTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 21:49'!
> testRaisedToErrorConditions
>        "
>        FractionTest new testRaisedToErrorConditions
>        "
>        self should: [ (-1/16) raisedTo: 1/4 ] raise: ArithmeticError.
>        self should: [ ((1 << 1024 + 1) / (1 << 1024 + 3)) negated raisedTo:
> 1/4 ] raise: ArithmeticError! !
>
> !FractionTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 21:48'!
> testSqrtErrorConditions
>        "
>        FractionTest new testSqrtErrorConditions
>        "
>        self should: [ (-1/4) sqrt ] raise: DomainError.
>        self should: [ ((1 << 1024 + 1) / (1 << 1024 + 3)) negated sqrt ]
> raise: DomainError! !
>
> !FractionTest methodsFor: 'private' stamp: 'jmv 10/11/2011 22:12'!
> assert: a classAndValueEquals: b
>        self assert: a class = b class.
>        self assert: a = b! !
>
>
> !IntegerTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 21:46'!
> testExactRaisedTo
>        "
>        IntegerTest new testExactRaisedTo
>        "
>        self assert: (4 raisedTo: 1/2) classAndValueEquals: 2.
>        self assert: (9 raisedTo: 1/2) classAndValueEquals: 3.
>        self assert: (9 raisedTo: -1/2) classAndValueEquals: 1/3.
>        self assert: (-1 raisedTo: 1/3) classAndValueEquals: -1.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) do: [ :i |
>                self assert: (i squared raisedTo: 1/2) classAndValueEquals:
> i.
>                self assert: (i negated squared raisedTo: 1/2)
> classAndValueEquals: i ].
>
>        self assert: (8 raisedTo: 1/3) classAndValueEquals: 2.
>        self assert: (27 raisedTo: 1/3) classAndValueEquals: 3.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) do: [ :i |
>                self assert: ((i raisedTo: 3) raisedTo: 1/3)
> classAndValueEquals: i.
>                self assert: ((i negated raisedTo: 3) raisedTo: 1/3)
> classAndValueEquals: i negated ].
>
>        self assert: (4 raisedTo: 3/2) classAndValueEquals: 8.
>        self assert: (8 raisedTo: 2/3) classAndValueEquals: 4.
>        self assert: (8 raisedTo: -2/3) classAndValueEquals: 1/4.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) do: [ :i |
>                self assert: ((i raisedTo: 3) raisedTo: 2/3)
> classAndValueEquals: i*i.
>                self assert: ((i raisedTo: 2) raisedTo: 3/2)
> classAndValueEquals: i*i*i.
>                self assert: ((i negated raisedTo: 3) raisedTo: 2/3)
> classAndValueEquals: i*i.
>                self assert: ((i negated raisedTo: 2) raisedTo: 3/2)
> classAndValueEquals: i*i*i ].
>
>        self assert: (32 raisedTo: 3/5) classAndValueEquals: 8.
>        self assert: (8 raisedTo: 5/3) classAndValueEquals: 32.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) do: [ :i |
>                self assert: ((i raisedTo: 5) raisedTo: 3/5)
> classAndValueEquals: i*i*i.
>                self assert: ((i raisedTo: 3) raisedTo: 5/3)
> classAndValueEquals: i*i*i*i*i.
>                self assert: ((i negated raisedTo: 5) raisedTo: 3/5)
> classAndValueEquals: (i*i*i) negated.
>                self assert: ((i negated raisedTo: 3) raisedTo: 5/3)
> classAndValueEquals: (i*i*i*i*i) negated.
>
>                self assert: ((i raisedTo: -5) raisedTo: 3/5)
> classAndValueEquals: 1/(i*i*i).
>                self assert: ((i raisedTo: -3) raisedTo: 5/3)
> classAndValueEquals: 1/(i*i*i*i*i).
>                self assert: ((i negated raisedTo: -5) raisedTo: 3/5)
> classAndValueEquals: -1/(i*i*i).
>                self assert: ((i negated raisedTo: -3) raisedTo: 5/3)
> classAndValueEquals: -1/(i*i*i*i*i).
>
>                self assert: ((i raisedTo: 5) raisedTo: -3/5)
> classAndValueEquals: 1/(i*i*i).
>                self assert: ((i raisedTo: 3) raisedTo: -5/3)
> classAndValueEquals: 1/(i*i*i*i*i).
>                self assert: ((i negated raisedTo: 5) raisedTo: -3/5)
> classAndValueEquals: -1/(i*i*i).
>                self assert: ((i negated raisedTo: 3) raisedTo: -5/3)
> classAndValueEquals: -1/(i*i*i*i*i).
>
>                "No exact result => Float result"
>                self assert: ((i raisedTo: 3) +1 raisedTo: 5/3) isFloat.
>                self assert: ((i negated raisedTo: 3) -1 raisedTo: 5/3)
> isFloat ].! !
>
> !IntegerTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/11/2011 22:09'!
> testExactSqrt
>        "
>        IntegerTest new testExactSqrt
>        "
>        self assert: 4 sqrt classAndValueEquals: 2.
>        self assert: 9 sqrt classAndValueEquals: 3.
>        self assert: Float maxExactInteger squared sqrt classAndValueEquals:
> Float maxExactInteger.
>        self assert: (Float maxExactInteger+1) squared sqrt
> classAndValueEquals: Float maxExactInteger+1.
>        #( 1 5 29 135 1234 567890 123123123 456456456456
> 98765432109876543210987654321
> 987123987123987123987123987123987123987123987123) do: [ :i |
>                self assert: i squared sqrt classAndValueEquals: i ]! !
>
> !IntegerTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 09:09'!
> testNthRootErrorConditions
>        "
>        IntegerTest new testExactRaisedToErrorConditions
>        "
>
>        self should: [ -2 nthRoot: 1/4 ] raise: ArithmeticError.
>        self should: [ -2 nthRoot: 1.24 ] raise: ArithmeticError.! !
>
> !IntegerTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 21:45'!
> testRaisedToErrorConditions
>        "
>        IntegerTest new testRaisedToErrorConditions
>        "
>
>        self should: [ -2 raisedTo: 1/4 ] raise: ArithmeticError.
>        self should: [ -2 raisedTo: 1.24 ] raise: ArithmeticError.! !
>
> !IntegerTest methodsFor: 'tests - mathematical functions' stamp: 'jmv
> 10/13/2011 21:46'!
> testSqrtErrorConditions
>        "
>        IntegerTest new testSqrtErrorConditions
>        "
>
>        self should: [ -1 sqrt ] raise: ArithmeticError! !
>
> !IntegerTest methodsFor: 'private' stamp: 'jmv 10/11/2011 08:14'!
> assert: a classAndValueEquals: b
>        self assert: a class = b class.
>        self assert: a = b! !
>
>
> !Number methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011 08:36'!
> nthRoot: aPositiveInteger
>        "Answer the nth root of the receiver."
>
>        self subclassResponsibility! !
>
> !Number methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011 08:50'!
> raisedTo: aNumber
>        "Answer the receiver raised to aNumber."
>
>        aNumber isInteger ifTrue: [
>                "Do the special case of integer power"
>                ^ self raisedToInteger: aNumber].
>        aNumber isFraction ifTrue: [
>                "Special case for fraction power"
>                ^ (self nthRoot: aNumber denominator) raisedToInteger:
> aNumber numerator ].
>        self < 0 ifTrue: [
>                ^ ArithmeticError signal: 'Negative numbers can''t be raised
> to float powers.' ].
>        0 = aNumber ifTrue: [^ self class one]. "Special case of exponent=0"
>        1 = aNumber ifTrue: [^ self].   "Special case of exponent=1"
>        0 = self ifTrue: [                              "Special case of self
> = 0"
>                aNumber < 0
>                        ifTrue: [^ (ZeroDivide dividend: self) signal]
>                        ifFalse: [^ self]].
>        ^ (aNumber * self ln) exp               "Otherwise use logarithms"! !
>
> !Number methodsFor: 'mathematical functions' stamp: 'jmv 10/11/2011 08:34'!
> sqrt
>        "Answer the square root of the receiver."
>
>        self subclassResponsibility! !
>
>
> !Float methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011 09:03'!
> nthRoot: aPositiveInteger
>        "Answer the nth root of the receiver."
>        aPositiveInteger = 2 ifTrue: [
>                ^self sqrt ].
>
>        (aPositiveInteger isInteger not or: [ aPositiveInteger negative ])
>                ifTrue: [^ ArithmeticError signal: 'nth root only defined for
> positive Integer n.'].
>
>        ^self negative
>                ifTrue: [
>                        aPositiveInteger odd
>                                ifTrue: [ (self negated raisedTo: 1.0 /
> aPositiveInteger) negated ]
>                                ifFalse: [ ArithmeticError signal: 'Negative
> numbers don''t have even roots.' ]]
>                ifFalse: [ self raisedTo: 1.0 / aPositiveInteger ]! !
>
>
> !Float class methodsFor: 'constants' stamp: 'jmv 10/13/2011 19:57'!
> maxExactInteger
>        "Answer the biggest integer such that it is exactly represented in a
> float, and all smaller integers also are"
>        ^1 bitShift: self precision! !
>
>
> !Fraction methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011
> 21:38'!
> nthRoot: aPositiveInteger
>        "Answer the nth root of the receiver."
>
>        | d n |
>        n _numerator nthRoot: aPositiveInteger.
>        d _ denominator nthRoot: aPositiveInteger.
>        "The #sqrt method in integer will only answer a Float if there's no
> exact square root.
>        So, we need a float anyway."
>        (n isInfinite or: [ d isInfinite ]) ifTrue: [
>                ^self asFloat nthRoot: aPositiveInteger ].
>        ^n / d! !
>
> !Fraction methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011
> 20:16'!
> sqrt
>        | d n |
>        n _numerator sqrt.
>        d _ denominator sqrt.
>        "The #sqrt method in integer will only answer a Float if there's no
> exact square root.
>        So, we need a float anyway."
>        (n isInfinite or: [ d isInfinite ]) ifTrue: [
>                ^self asFloat sqrt ].
>        ^n / d! !
>
>
> !Integer methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011 20:19'!
> nthRoot: aPositiveInteger
>        "Answer the nth root of the receiver."
>
>        | selfAsFloat floatResult guess raised higher lower delta |
>        selfAsFloat _ self asFloat.
>        floatResult _ selfAsFloat nthRoot: aPositiveInteger.
>
>        "If we can't do Float arithmetic, currently we can't look for an
> exact answer"
>        floatResult isInfinite ifTrue: [
>                ^floatResult ].
>
>        guess _ floatResult rounded.
>
>        "If got an exact answer, answer it."
>        raised _ guess raisedToInteger: aPositiveInteger.
>        raised = self
>                ifTrue: [ ^ guess ].
>
>        "In this case, maybe it failed because we are such a big integer that
> the Float
>        method gets inexact, even if we are a whole square number.
>        Note (jmv): The algorithms I found for computing the nthRoot would
> havily use
>        very large fractions. I wrote this one, that doesn't create
> fractions."
>        selfAsFloat abs >= (Float maxExactInteger asFloat raisedToInteger:
> aPositiveInteger)
>                ifTrue: [
>                        raised > self
>                                ifTrue: [
>                                        higher _ guess.
>                                        delta _  floatResult predecessor -
> floatResult.
>                                        [
>                                                floatResult _ floatResult +
> delta.
>                                                lower _ floatResult rounded.
>                                                (lower raisedToInteger:
> aPositiveInteger) > self ] whileTrue: [
>                                                        delta _ delta * 2.
>                                                        higher _ lower ] ]
>                                ifFalse: [
>                                        lower _ guess.
>                                        delta _  floatResult successor -
> floatResult.
>                                        [
>                                                floatResult _ floatResult +
> delta.
>                                                higher _ floatResult rounded.
>                                                (higher raisedToInteger:
> aPositiveInteger) < self ] whileTrue: [
>                                                        delta _ delta * 2.
>                                                        lower _ higher ]].
>                        [ higher - lower > 1 ] whileTrue: [
>                                guess _ lower + higher // 2.
>                                raised _ guess raisedToInteger:
> aPositiveInteger.
>                                raised = self
>                                        ifTrue: [
>                                                ^ guess ].
>                                raised > self
>                                        ifTrue: [ higher _ guess ]
>                                        ifFalse: [ lower _ guess ]]].
>
>        "We need an approximate result"
>        ^floatResult! !
>
> !Integer methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011 20:17'!
> sqrt
>        "Answer the square root of the receiver."
>
>        | selfAsFloat floatResult guess |
>        selfAsFloat _ self asFloat.
>        floatResult _ selfAsFloat sqrt.
>
>        floatResult isInfinite ifFalse: [
>                guess _ floatResult truncated.
>
>                "If got an exact answer, answer it. Otherwise answer float
> approximate answer."
>                guess squared = self
>                        ifTrue: [ ^ guess ]].
>
>        "In this case, maybe it failed because we are such a big integer that
> the Float method becomes
>        inexact, even if we are a whole square number. So, try the slower but
> more general method"
>        selfAsFloat >= Float maxExactInteger asFloat squared
>                ifTrue: [
>                        guess _ self sqrtFloor.
>                        guess * guess = self ifTrue: [
>                                ^guess ]].
>
>        "We need an approximate result"
>        ^floatResult! !
>
>
> !LargePositiveInteger methodsFor: 'mathematical functions' stamp: 'jmv
> 10/14/2011 07:56'!
> mightBeASquare
>        "In base 16, a square number can end only with 0,1,4 or 9 and
>        - in case 0, only 0,1,4,9 can precede it,
>        - in case 4, only even numbers can precede it.
>        See http://en.wikipedia.org/wiki/Square_number
>        So, in hex, the last byte must be one of:
>                00
>                10
>                40
>                90
>                x1
>                e4
>                x9
>        where x is any hex digit and e is any even digit
>        "
>    | lsb |
>    lsb := self digitAt: 1.
>    ^((lsb bitAnd: 7) = 1                                       "any|1 or
> any|9"
>        or: [(lsb bitAnd: 31) = 4                       "even|4"
>        or: [(lsb bitAnd: 127) = 16                     "10 or 90"
>        or: [(lsb bitAnd: 191) = 0]]])          "00 or 40"! !
>
> !LargePositiveInteger methodsFor: 'mathematical functions' stamp: 'jmv
> 10/13/2011 22:11'!
> sqrt
>        "If we know for sure no exact solution exists, then just answer the
> cheap float approximation without wasting time."
>        self mightBeASquare ifFalse: [
>                ^self asFloat sqrt ].
>
>        "If some exact solution might exist,  call potentially expensive
> super"
>        ^super sqrt! !
>
>
> !LargeNegativeInteger methodsFor: 'mathematical functions' stamp: 'jmv
> 10/13/2011 21:40'!
> sqrt
>        "Answer the square root of the receiver."
>        ^ DomainError signal: 'sqrt undefined for number less than zero.'! !
>
>
> !SmallInteger methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011
> 21:41'!
> sqrt
>        self negative ifTrue: [
>                ^ DomainError signal: 'sqrt undefined for number less than
> zero.' ].
>        ^super sqrt! !
>
>
> !SmallInteger reorganize!
> ('arithmetic' * + - / // \\ gcd: quo:)
> ('bit manipulation' bitAnd: bitOr: bitShift: bitXor: byteReversed
> hashMultiply highBit highBitOfMagnitude lowBit)
> ('testing' even isLarge odd)
> ('comparing' < <= = > >= hash identityHash ~=)
> ('copying' clone shallowCopy)
> ('converting' as31BitSmallInt asFloat)
> ('printing' decimalDigitLength destinationBuffer: numberOfDigitsInBase:
> printOn:base: printOn:base:length:padded: printOn:base:nDigits: printString
> printStringBase: printStringBase:nDigits: threeDigitName)
> ('system primitives' digitAt: digitAt:put: digitLength instVarAt:
> nextInstance nextObject)
> ('private' fromString:radix: highBitOfPositiveReceiver)
> ('mathematical functions' sqrt)
> !
>
>
> !LargeNegativeInteger reorganize!
> ('arithmetic' abs negated)
> ('bit manipulation' bitAt: highBit)
> ('converting' asFloat normalize)
> ('testing' negative positive sign strictlyPositive)
> ('printing' printOn:base:)
> ('mathematical functions' sqrt)
> !
>
>
> !LargePositiveInteger reorganize!
> ('arithmetic' * + - / // \\ \\\ abs negated quo:)
> ('bit manipulation' bitAt: bitReverse: hashMultiply highBit
> highBitOfMagnitude)
> ('testing' isLarge isPrime negative positive sign strictlyPositive)
> ('comparing' < <= > >= hash)
> ('converting' as31BitSmallInt asFloat normalize withAtLeastNDigits:)
> ('system primitives' digitAt: digitAt:put: digitLength
> replaceFrom:to:with:startingAt:)
> ('printing' printOn:base: printOn:base:nDigits: printStringBase:)
> ('mathematical functions' mightBeASquare sqrt)
> !
>
> IntegerTest removeSelector: #testExactRaisedToErrorConditions!
>
>
>
>



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