[squeak-dev] Exact #sqrt

Nicolas Cellier nicolas.cellier.aka.nice at gmail.com
Thu Oct 13 23:37:52 UTC 2011


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>:
> 2011/10/13 Juan Vuletich <juan at jvuletich.org>:
>> Juan Vuletich wrote:
>>>
>>> Hi Folks,
>>>
>>> Yesterday I realized that there's no good reason for 8 sqrt to answer 2.0
>>> and not just 2. To fix this inelegant situation I wrote the attached. This
>>> way, for instance, 4 sqrt = (8 raisedTo: 1/3) answers true. I also
>>> introduced #nthRoot: , which can be of some use. A rather complete set of
>>> tests is included, to illustrate the desired behavior.
>>>
>>> ...
>>
>> This new version includes performance improvements and better checking of
>> valid domains. Enjoy!
>>
>> Cheers,
>> Juan Vuletich
>>
>> 'From Cuis 3.3 of 2 June 2011 [latest update: #1024] on 13 October 2011 at
>> 9:18:36 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: '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
>> 10/11/2011 22:27'!
>> 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.
>>        #( 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
>> 10/13/2011 09:07'!
>> testExactRaisedToErrorConditions
>>        "
>>        IntegerTest new testExactRaisedToErrorConditions
>>        "
>>
>>        self should: [ -2 raisedTo: 1/4 ] raise: ArithmeticError.
>>        self should: [ -2 raisedTo: 1.24 ] raise: ArithmeticError.! !
>>
>> !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: '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 10/11/2011 08:54'!
>> maxExactInteger
>>        "Answer the biggest integer such that it is exactly represented in a
>> float, and all smaller integers also are"
>>        ^1 bitShift: 53! !
>
> or         ^1 bitShift: self precision
>
>
>>
>>
>> !Fraction methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011
>> 08:36'!
>> nthRoot: aPositiveInteger
>>        "Answer the nth root of the receiver."
>>
>>        ^ (numerator nthRoot: aPositiveInteger) / (denominator nthRoot:
>> aPositiveInteger)! !
>>
>> !Fraction methodsFor: 'mathematical functions' stamp: 'jmv 10/11/2011
>> 10:51'!
>> sqrt
>>        ^ numerator sqrt / denominator sqrt! !
>>
>
> This can cause an overflow...
>
> ((1 << 1024 + 1) / (1 << 1024 + 3)) sqrt
>
> I'd expect 1.0 (inexact, but not that much).
>
>>
>> !Integer methodsFor: 'mathematical functions' stamp: 'jmv 10/13/2011 09:18'!
>> nthRoot: aPositiveInteger
>>        "Answer the nth root of the receiver."
>>
>>        | selfAsFloat floatResult guess raised higher lower delta |
>>        selfAsFloat _ self asFloat.
>>        floatResult _ selfAsFloat nthRoot: aPositiveInteger.
>>        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 10/12/2011 15:17'!
>> sqrt
>>        "Answer the square root of the receiver."
>>
>>        | selfAsFloat floatResult integerResult |
>>        selfAsFloat _ self asFloat.
>>        floatResult _ selfAsFloat sqrt.
>>        integerResult _ floatResult truncated.
>>
>>        "If got an exact answer, answer it. Otherwise answer float
>> approximate answer."
>>        integerResult squared = self
>>                ifTrue: [ ^ integerResult ].
>>
>>        "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: [
>>                        integerResult _ self sqrtFloor.
>>                        integerResult * integerResult = self ifTrue: [
>>                                ^integerResult ]].
>>
>>        "We need an approximate result"
>>        ^floatResult! !
>>
>>
>> !FloatTest reorganize!
>> ('testing - arithmetic' testDivide)
>> ('infinity behavior' testHugeIntegerCloseTo testInfinity1 testInfinity2
>> testInfinityCloseTo)
>> ('IEEE 754' test32bitGradualUnderflow test32bitRoundingMode testInfinity3
>> testNaN5 testZero2)
>> ('zero behavior' testIsZero testNegativeZeroAbs testNegativeZeroSign
>> testZero1 testZeroSignificandAsInteger)
>> ('NaN behavior' testNaN1 testNaN2 testNaN3 testNaN4 testNaNCompare
>> testNaNisLiteral testReciprocal)
>> ('testing - conversion' testAsTrueFraction testFloatRounded
>> testFloatTruncated testFractionAsFloat testFractionAsFloat2
>> testIntegerAsFloat testStringAsNumber)
>> ('testing compare' testCloseTo testComparison
>> testComparisonWhenPrimitiveFails)
>> ('printing' testStoreBase16)
>> ('testing' testArcTan testCopy testMaxExactInteger testSetOfFloat
>> testStoreOn)
>> ('characterization' testCharacterization)
>> ('tests - mathematical functions' testNthRoot testRaisedTo)
>> !
>>
>>
>>
>>
>>
>



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