[squeak-dev] Exact #sqrtJuan Vuletich juan at jvuletich.orgFri Oct 14 11:07:13 UTC 2011
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 -------------- next part -------------- '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! -------------- next part -------------- A non-text attachment was scrubbed... Name: 1105-sqrtAndnThRoot-jmv.13.cs.zip Type: application/zip Size: 4364 bytes Desc: not available Url : http://lists.squeakfoundation.org/pipermail/squeak-dev/attachments/20111014/c37f86f4/1105-sqrtAndnThRoot-jmv.13.cs.zip
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