[squeak-dev] The Inbox: Kernel-nice.1218.mcz
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commits at source.squeak.org
Sat Apr 27 08:41:36 UTC 2019
Nicolas Cellier uploaded a new version of Kernel to project The Inbox:
http://source.squeak.org/inbox/Kernel-nice.1218.mcz
==================== Summary ====================
Name: Kernel-nice.1218
Author: nice
Time: 27 April 2019, 10:41:24.794539 am
UUID: 21f74fbe-a0cd-4b6f-86e9-be13465d57fe
Ancestors: Kernel-nice.1217
Implement the recursive fast division of Burnikel-Ziegler for large integers and connect it to digitDiv:neg: when operands are large enough.
This is not the fastest known division which is a composition of Barrett and Newton-Raphson inversion - but is easy to implement and should have similar performances for at least a few thousand bytes long integers - see for example http://bioinfo.ict.ac.cn/~dbu/AlgorithmCourses/Lectures/Lec5-Fast-Division-Hasselstrom2003.pdf
Use digitDiv:neg: in large integer printString so as to obtain the quotient (head) and remainder (tail) in a single operation. Together with divide and conquer division, this results in a factor of about 3x for 50000 factorial printString.
Implement the 4-way Toom-Cook squaring variant of Chung-Hasan. This over-performs the symetrical squaredToom3 even for medium size (800 bytes).
=============== Diff against Kernel-nice.1217 ===============
Item was added:
+ ----- Method: Integer>>digitDiv21: (in category 'private') -----
+ digitDiv21: anInteger
+
+ ^(self digitDiv: anInteger neg: false) collect: #normalize!
Item was added:
+ ----- Method: Integer>>digitDiv32: (in category 'private') -----
+ digitDiv32: anInteger
+
+ ^(self digitDiv: anInteger neg: false) collect: #normalize!
Item was changed:
----- Method: Integer>>digitDiv:neg: (in category 'private') -----
+ digitDiv: anInteger neg: aBoolean
+ ^self primDigitDiv: anInteger neg: aBoolean!
- digitDiv: arg neg: ng
- "Answer with an array of (quotient, remainder)."
- | quo rem ql d div dh dnh dl qhi qlo j l hi lo r3 a t divDigitLength remDigitLength |
- <primitive: 'primDigitDivNegative' module:'LargeIntegers'>
- arg = 0 ifTrue: [^ (ZeroDivide dividend: self) signal].
- "TFEI added this line"
- l := self digitLength - arg digitLength + 1.
- l <= 0 ifTrue: [^ Array with: 0 with: self].
- "shortcut against #highBit"
- d := 8 - arg lastDigit highBitOfByte.
- div := arg digitLshift: d.
- divDigitLength := div digitLength + 1.
- div := div growto: divDigitLength.
- "shifts so high order word is >=128"
- rem := self digitLshift: d.
- rem digitLength = self digitLength ifTrue: [rem := rem growto: self digitLength + 1].
- remDigitLength := rem digitLength.
- "makes a copy and shifts"
- quo := Integer new: l neg: ng.
- dl := divDigitLength - 1.
- "Last actual byte of data"
- ql := l.
- dh := div digitAt: dl.
- dnh := dl = 1
- ifTrue: [0]
- ifFalse: [div digitAt: dl - 1].
- 1 to: ql do:
- [:k |
- "maintain quo*arg+rem=self"
- "Estimate rem/div by dividing the leading to bytes of rem by dh."
- "The estimate is q = qhi*16+qlo, where qhi and qlo are nibbles."
- j := remDigitLength + 1 - k.
- "r1 := rem digitAt: j."
- (rem digitAt: j)
- = dh
- ifTrue: [qhi := qlo := 15
- "i.e. q=255"]
- ifFalse:
- ["Compute q = (r1,r2)//dh, t = (r1,r2)\\dh.
- Note that r1,r2 are bytes, not nibbles.
- Be careful not to generate intermediate results exceeding 13
- bits."
- "r2 := (rem digitAt: j - 1)."
- t := ((rem digitAt: j)
- bitShift: 4)
- + ((rem digitAt: j - 1)
- bitShift: -4).
- qhi := t // dh.
- t := (t \\ dh bitShift: 4)
- + ((rem digitAt: j - 1)
- bitAnd: 15).
- qlo := t // dh.
- t := t \\ dh.
- "Next compute (hi,lo) := q*dnh"
- hi := qhi * dnh.
- lo := qlo * dnh + ((hi bitAnd: 15)
- bitShift: 4).
- hi := (hi bitShift: -4)
- + (lo bitShift: -8).
- lo := lo bitAnd: 255.
- "Correct overestimate of q.
- Max of 2 iterations through loop -- see Knuth vol. 2"
- r3 := j < 3
- ifTrue: [0]
- ifFalse: [rem digitAt: j - 2].
- [(t < hi
- or: [t = hi and: [r3 < lo]])
- and:
- ["i.e. (t,r3) < (hi,lo)"
- qlo := qlo - 1.
- lo := lo - dnh.
- lo < 0
- ifTrue:
- [hi := hi - 1.
- lo := lo + 256].
- hi >= dh]]
- whileTrue: [hi := hi - dh].
- qlo < 0
- ifTrue:
- [qhi := qhi - 1.
- qlo := qlo + 16]].
- "Subtract q*div from rem"
- l := j - dl.
- a := 0.
- 1 to: divDigitLength do:
- [:i |
- hi := (div digitAt: i)
- * qhi.
- lo := a + (rem digitAt: l) - ((hi bitAnd: 15)
- bitShift: 4) - ((div digitAt: i)
- * qlo).
- rem digitAt: l put: lo - (lo // 256 * 256).
- "sign-tolerant form of (lo bitAnd: 255)"
- a := lo // 256 - (hi bitShift: -4).
- l := l + 1].
- a < 0
- ifTrue:
- ["Add div back into rem, decrease q by 1"
- qlo := qlo - 1.
- l := j - dl.
- a := 0.
- 1 to: divDigitLength do:
- [:i |
- a := (a bitShift: -8)
- + (rem digitAt: l) + (div digitAt: i).
- rem digitAt: l put: (a bitAnd: 255).
- l := l + 1]].
- quo digitAt: ql + 1 - k put: (qhi bitShift: 4)
- + qlo].
- rem := rem
- digitRshift: d
- bytes: 0
- lookfirst: dl.
- ^ Array with: quo with: rem!
Item was added:
+ ----- Method: Integer>>primDigitDiv:neg: (in category 'private') -----
+ primDigitDiv: arg neg: ng
+ "Answer with an array of (quotient, remainder)."
+ | quo rem ql d div dh dnh dl qhi qlo j l hi lo r3 a t divDigitLength remDigitLength |
+ <primitive: 'primDigitDivNegative' module:'LargeIntegers'>
+ arg = 0 ifTrue: [^ (ZeroDivide dividend: self) signal].
+ "TFEI added this line"
+ l := self digitLength - arg digitLength + 1.
+ l <= 0 ifTrue: [^ Array with: 0 with: self].
+ "shortcut against #highBit"
+ d := 8 - arg lastDigit highBitOfByte.
+ div := arg digitLshift: d.
+ divDigitLength := div digitLength + 1.
+ div := div growto: divDigitLength.
+ "shifts so high order word is >=128"
+ rem := self digitLshift: d.
+ rem digitLength = self digitLength ifTrue: [rem := rem growto: self digitLength + 1].
+ remDigitLength := rem digitLength.
+ "makes a copy and shifts"
+ quo := Integer new: l neg: ng.
+ dl := divDigitLength - 1.
+ "Last actual byte of data"
+ ql := l.
+ dh := div digitAt: dl.
+ dnh := dl = 1
+ ifTrue: [0]
+ ifFalse: [div digitAt: dl - 1].
+ 1 to: ql do:
+ [:k |
+ "maintain quo*arg+rem=self"
+ "Estimate rem/div by dividing the leading to bytes of rem by dh."
+ "The estimate is q = qhi*16+qlo, where qhi and qlo are nibbles."
+ j := remDigitLength + 1 - k.
+ "r1 := rem digitAt: j."
+ (rem digitAt: j)
+ = dh
+ ifTrue: [qhi := qlo := 15
+ "i.e. q=255"]
+ ifFalse:
+ ["Compute q = (r1,r2)//dh, t = (r1,r2)\\dh.
+ Note that r1,r2 are bytes, not nibbles.
+ Be careful not to generate intermediate results exceeding 13
+ bits."
+ "r2 := (rem digitAt: j - 1)."
+ t := ((rem digitAt: j)
+ bitShift: 4)
+ + ((rem digitAt: j - 1)
+ bitShift: -4).
+ qhi := t // dh.
+ t := (t \\ dh bitShift: 4)
+ + ((rem digitAt: j - 1)
+ bitAnd: 15).
+ qlo := t // dh.
+ t := t \\ dh.
+ "Next compute (hi,lo) := q*dnh"
+ hi := qhi * dnh.
+ lo := qlo * dnh + ((hi bitAnd: 15)
+ bitShift: 4).
+ hi := (hi bitShift: -4)
+ + (lo bitShift: -8).
+ lo := lo bitAnd: 255.
+ "Correct overestimate of q.
+ Max of 2 iterations through loop -- see Knuth vol. 2"
+ r3 := j < 3
+ ifTrue: [0]
+ ifFalse: [rem digitAt: j - 2].
+ [(t < hi
+ or: [t = hi and: [r3 < lo]])
+ and:
+ ["i.e. (t,r3) < (hi,lo)"
+ qlo := qlo - 1.
+ lo := lo - dnh.
+ lo < 0
+ ifTrue:
+ [hi := hi - 1.
+ lo := lo + 256].
+ hi >= dh]]
+ whileTrue: [hi := hi - dh].
+ qlo < 0
+ ifTrue:
+ [qhi := qhi - 1.
+ qlo := qlo + 16]].
+ "Subtract q*div from rem"
+ l := j - dl.
+ a := 0.
+ 1 to: divDigitLength do:
+ [:i |
+ hi := (div digitAt: i)
+ * qhi.
+ lo := a + (rem digitAt: l) - ((hi bitAnd: 15)
+ bitShift: 4) - ((div digitAt: i)
+ * qlo).
+ rem digitAt: l put: lo - (lo // 256 * 256).
+ "sign-tolerant form of (lo bitAnd: 255)"
+ a := lo // 256 - (hi bitShift: -4).
+ l := l + 1].
+ a < 0
+ ifTrue:
+ ["Add div back into rem, decrease q by 1"
+ qlo := qlo - 1.
+ l := j - dl.
+ a := 0.
+ 1 to: divDigitLength do:
+ [:i |
+ a := (a bitShift: -8)
+ + (rem digitAt: l) + (div digitAt: i).
+ rem digitAt: l put: (a bitAnd: 255).
+ l := l + 1]].
+ quo digitAt: ql + 1 - k put: (qhi bitShift: 4)
+ + qlo].
+ rem := rem
+ digitRshift: d
+ bytes: 0
+ lookfirst: dl.
+ ^ Array with: quo with: rem!
Item was added:
+ ----- Method: LargePositiveInteger>>digitDiv21: (in category 'private') -----
+ digitDiv21: anInteger
+ "This is part of the recursive division algorithm from Burnikel - Ziegler
+ Divide a two limbs receiver by 1 limb dividend
+ Each limb is decomposed in two halves of p bytes (8*p bits)
+ so as to continue the recursion"
+
+ | p qr1 qr2 |
+ p := anInteger digitLength + 1 bitShift: -1.
+ p <= 256 ifTrue: [^(self primDigitDiv: anInteger neg: false) collect: #normalize].
+ qr1 := (self butLowestNDigits: p) digitDiv32: anInteger.
+ qr2 := (self lowestNDigits: p) + (qr1 last bitShift: 8*p) digitDiv32: anInteger.
+ qr2 at: 1 put: (qr2 at: 1) + ((qr1 at: 1) bitShift: 8*p).
+ ^qr2!
Item was added:
+ ----- Method: LargePositiveInteger>>digitDiv32: (in category 'private') -----
+ digitDiv32: anInteger
+ "This is part of the recursive division algorithm from Burnikel - Ziegler
+ Divide 3 limb (a2,a1,a0) by 2 limb (b1,b0).
+ Each limb is made of p bytes (8*p bits).
+ This step transforms the division problem into multiplication
+ It must use the fastMultiply: to be worth the overhead costs."
+
+ | a2 b1 d p q qr r |
+ p := anInteger digitLength + 1 bitShift: -1.
+ (a2 := self butLowestNDigits: 2*p)
+ < (b1 := anInteger butLowestNDigits: p)
+ ifTrue:
+ [qr := (self butLowestNDigits: p) digitDiv21: b1.
+ q := qr first.
+ r := qr last]
+ ifFalse:
+ [q := (1 bitShift: 8*p) - 1.
+ r := (self butLowestNDigits: p) - (b1 bitShift: 8*p) + b1].
+ d := q fastMultiply: (anInteger lowestNDigits: p).
+ r := (self lowestNDigits: p) + (r bitShift: 8*p) - d.
+ [r < 0]
+ whileTrue:
+ [q := q - 1.
+ r := r + anInteger].
+ ^Array with: q with: r
+ !
Item was added:
+ ----- Method: LargePositiveInteger>>digitDiv:neg: (in category 'private') -----
+ digitDiv: anInteger neg: aBoolean
+ "If length is worth, engage a recursive divide and conquer strategy"
+ | qr |
+ (anInteger digitLength <= 256
+ or: [self digitLength <= anInteger digitLength])
+ ifTrue: [^ self primDigitDiv: anInteger neg: aBoolean].
+ qr := self abs recursiveDigitDiv: anInteger abs.
+ ^ aBoolean
+ ifTrue: [qr collect: #negated]
+ ifFalse: [qr]!
Item was changed:
----- Method: LargePositiveInteger>>printOn:base: (in category 'printing') -----
printOn: aStream base: b
"Append a representation of this number in base b on aStream.
In order to reduce cost of LargePositiveInteger ops, split the number in approximately two equal parts in number of digits."
+ | halfDigits halfPower head tail nDigitsUnderestimate qr |
- | halfDigits halfPower head tail nDigitsUnderestimate |
"Don't engage any arithmetic if not normalized"
(self digitLength = 0 or: [(self digitAt: self digitLength) = 0]) ifTrue: [^self normalize printOn: aStream base: b].
nDigitsUnderestimate := b = 10
ifTrue: [((self highBit - 1) * 1233 >> 12) + 1. "This is because (2 log)/(10 log)*4096 is slightly greater than 1233"]
ifFalse: [self highBit quo: b highBit].
"splitting digits with a whole power of two is more efficient"
halfDigits := 1 bitShift: nDigitsUnderestimate highBit - 2.
halfDigits <= 1
ifTrue: ["Hmmm, this could happen only in case of a huge base b... Let lower level fail"
^self printOn: aStream base: b nDigits: (self numberOfDigitsInBase: b)].
"Separate in two halves, head and tail"
halfPower := b raisedToInteger: halfDigits.
+ qr := self digitDiv: halfPower neg: self negative.
+ head := qr first normalize.
+ tail := qr last normalize.
- head := self quo: halfPower.
- tail := self - (head * halfPower).
"print head"
head printOn: aStream base: b.
"print tail without the overhead to count the digits"
tail printOn: aStream base: b nDigits: halfDigits!
Item was changed:
----- Method: LargePositiveInteger>>printOn:base:nDigits: (in category 'printing') -----
printOn: aStream base: b nDigits: n
"Append a representation of this number in base b on aStream using n digits.
In order to reduce cost of LargePositiveInteger ops, split the number of digts approximatily in two
Should be invoked with: 0 <= self < (b raisedToInteger: n)"
+ | halfPower half head tail qr |
- | halfPower half head tail |
n <= 1 ifTrue: [
n <= 0 ifTrue: [self error: 'Number of digits n should be > 0'].
"Note: this is to stop an infinite loop if one ever attempts to print with a huge base
This can happen because choice was to not hardcode any limit for base b
We let Character>>#digitValue: fail"
^aStream nextPut: (Character digitValue: self) ].
halfPower := n bitShift: -1.
half := b raisedToInteger: halfPower.
+ qr := self digitDiv: half neg: self negative.
+ head := qr first normalize.
+ tail := qr last normalize.
- head := self quo: half.
- tail := self - (head * half).
head printOn: aStream base: b nDigits: n - halfPower.
tail printOn: aStream base: b nDigits: halfPower!
Item was added:
+ ----- Method: LargePositiveInteger>>recursiveDigitDiv: (in category 'private') -----
+ recursiveDigitDiv: anInteger
+ "This is the recursive division algorithm from Burnikel - Ziegler
+ See Fast Recursive Division - Christoph Burnikel, Joachim Ziegler
+ Research Report MPI-I-98-1-022, MPI Saarbrucken, Oct 1998
+ https://pure.mpg.de/rest/items/item_1819444_4/component/file_2599480/content"
+
+ | s m t a b z qr q i |
+ "round digits up to next power of 2"
+ s := anInteger digitLength.
+ m := 1 bitShift: (s - 1) highBit.
+ "shift so that leading bit of leading byte be 1, and digitLength power of two"
+ s := m * 8 - anInteger highBit.
+ a := self bitShift: s.
+ b := anInteger bitShift: s.
+
+ "Decompose a into t limbs - each limb have m bytes
+ choose t such that leading bit of leading limb of a be 0"
+ t := (a highBit + 1 / (m * 8)) ceiling.
+ z := a butLowestNDigits: t - 2 * m.
+ i := t - 2.
+ q := 0.
+ "and do a division of two limb by 1 limb b for each pair of limb of a"
+ [qr := z digitDiv21: b.
+ q := (q bitShift: 8*m) + qr first. "Note: this naive recomposition of q cost O(t^2) - it is possible in O(t log(t))"
+ (i := i - 1) >= 0] whileTrue:
+ [z := (qr last bitShift: 8*m) + (a copyDigitsFrom: i * m + 1 to: i + 1 * m)].
+ ^Array with: q with: (qr last bitShift: s negated)!
Item was changed:
----- Method: LargePositiveInteger>>sqrtRem (in category 'mathematical functions') -----
sqrtRem
"Like super, but use a divide and conquer method to perform this operation.
See Paul Zimmermann. Karatsuba Square Root. [Research Report] RR-3805, INRIA. 1999, pp.8. <inria-00072854>
https://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf"
+ | n qr q s r sr high mid low |
- | n qr s r sr high mid low |
n := self digitLength bitShift: -2.
n >= 16 ifFalse: [^super sqrtRem].
high := self butLowestNDigits: n * 2.
mid := self copyDigitsFrom: n + 1 to: n * 2.
low := self lowestNDigits: n.
sr := high sqrtRem.
qr := (sr last bitShift: 8 * n) + mid digitDiv: (sr first bitShift: 1) neg: false.
+ q := qr first normalize.
+ s := (sr first bitShift: 8 * n) + q.
+ r := (qr last normalize bitShift: 8 * n) + low - q squared.
- s := (sr first bitShift: 8 * n) + qr first.
- r := (qr last bitShift: 8 * n) + low - qr first squared.
r negative
ifTrue:
[r := (s bitShift: 1) + r - 1.
s := s - 1].
sr at: 1 put: s; at: 2 put: r.
^sr
!
Item was changed:
----- Method: LargePositiveInteger>>squared (in category 'mathematical functions') -----
squared
"Eventually use a divide and conquer algorithm to perform the multiplication"
(self digitLength >= 400) ifFalse: [^self * self].
+ (self digitLength >= 800) ifFalse: [^self squaredKaratsuba].
+ ^self squaredToom4!
- (self digitLength >= 1600) ifFalse: [^self squaredKaratsuba].
- ^self squaredToom3!
Item was added:
+ ----- Method: LargePositiveInteger>>squaredToom4 (in category 'mathematical functions') -----
+ squaredToom4
+ "Use a 4-way Toom-Cook divide and conquer algorithm to perform the multiplication.
+ See Asymmetric Squaring Formulae Jaewook Chung and M. Anwar Hasan
+ https://www.lirmm.fr/arith18/papers/Chung-Squaring.pdf"
+
+ | p a0 a1 a2 a3 a02 a13 s0 s1 s2 s3 s4 s5 s6 t2 t3 |
+ "divide in 4 parts"
+ p := (self digitLength + 3 bitShift: -2) bitClear: 2r11.
+ a3 := self butLowestNDigits: p * 3.
+ a2 := self copyDigitsFrom: p * 2 + 1 to: p * 3.
+ a1 := self copyDigitsFrom: p + 1 to: p * 2.
+ a0 := self lowestNDigits: p.
+
+ "Toom-4 trick: 7 multiplications instead of 16"
+ a02 := a0 - a2.
+ a13 := a1 - a3.
+ s0 := a0 squared.
+ s1 := (a0 fastMultiply: a1) bitShift: 1.
+ s2 := (a02 + a13) fastMultiply: (a02 - a13).
+ s3 := ((a0 + a1) + (a2 + a3)) squared.
+ s4 := (a02 fastMultiply: a13) bitShift: 1.
+ s5 := (a3 fastMultiply: a2) bitShift: 1.
+ s6 := a3 squared.
+
+ "Interpolation"
+ t2 := s1 + s5.
+ t3 := (s2 + s3 + s4 bitShift: -1) - t2.
+ s3 := t2 - s4.
+ s4 := t3 - s0.
+ s2 := t3 - s2 - s6.
+
+ "Sum the parts of decomposition"
+ ^s0 + (s1 bitShift: 8*p) + (s2 + (s3 bitShift: 8*p) bitShift: 16*p)
+ +(s4 + (s5 bitShift: 8*p) + (s6 bitShift: 16*p) bitShift: 32*p)
+
+ "
+ | a |
+ a := 770 factorial-1.
+ a digitLength.
+ [a * a - a squaredToom4 = 0] assert.
+ [Smalltalk garbageCollect.
+ [1000 timesRepeat: [a squaredToom4]] timeToRun] value /
+ [Smalltalk garbageCollect.
+ [1000 timesRepeat: [a squaredKaratsuba]] timeToRun] value asFloat
+ "!
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