## [squeak-dev] The Trunk: Kernel-nice.1220.mcz

commits at source.squeak.org commits at source.squeak.org
Sat Apr 27 12:36:17 UTC 2019

```Nicolas Cellier uploaded a new version of Kernel to project The Trunk:
http://source.squeak.org/trunk/Kernel-nice.1220.mcz

==================== Summary ====================

Name: Kernel-nice.1220
Author: nice
Time: 27 April 2019, 2:36:07.52994 pm
UUID: b2eb6e2e-1dfa-4d23-9421-ec0e5e8d4f86
Ancestors: Kernel-fn.1216

Accelerate some huge Integer arithmetic
(This is a squash of inbox commits)

These algorithms use divide and conquer strategy that recursively split a problem into simpler problems.

1) Multiplication and Squaring

- Implement Karatsuba and 3-way Toom-Cook algorithms for fast large integer multiplication (a Bodrato Zanoni variant)

- Implement the 2-way asymetrical Karatsuba, 3-way symetrical, and 4-way asymetrical Toom-Cook squaring variant of Chung-Hasan. Note that asymetrical squareToom4 over-performs the symetrical squaredToom3 even for medium size (800 bytes).

- provide a fastMultiply: that dispatch on appropriate variant based on heuristics. Currently this fast multiplication is used only in LargePositiveInteger>>squared, and other algorithme below.

- slightly modify raisedToInteger: to use that squared.
This result in a small penalty for small number exponentation, but large speed up for big integers.

2) Square-root

- implement the Karatsuba-like square root algorithm from Paul Zimmerman.

3) Division

- Implement the recursive fast division of Burnikel-Ziegler for large integers and connect it to digitDiv:neg: when operands are large enough.

- Use this recursive algorithm to accelerate digitDiv:neg:

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

4) Printing

- 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 4x for 50000 factorial printString.

Examples on a 64 bits image/VM (MBP 2015 i5 2.7GHz)

[10 raisedToInteger: 30] bench.
'3,770,000 per second. 265 nanoseconds per run.' - before
'3,500,000 per second. 286 nanoseconds per run.' - after
[10.0 raisedToInteger: 30] bench.
'10,900,000 per second. 91.6 nanoseconds per run.' - before
'9,540,000 per second. 105 nanoseconds per run.' - after

| x |
x := 100 factorial.
[x raisedToInteger: 30] bench.
'15,900 per second. 63 microseconds per run.' - before
'19,500 per second. 51.3 microseconds per run.' - after

| x |
x := 1000 factorial.
[x raisedToInteger: 30] bench.
'58.8 per second. 17 milliseconds per run.' - before
'179 per second. 5.6 milliseconds per run.' - after

| x |
x := 500 factorial - 100 factorial.
{
[x sqrtFloor] bench.
[x sqrtFloorNewtonRaphson] bench.
}
#(
'42,300 per second. 23.6 microseconds per run.'
'8,240 per second. 121 microseconds per run.')

| x |
x := 50000 factorial - 10000 factorial + 5000 factorial - 1000 factorial + 500 factorial - 100 factorial + 50 factorial - 10 factorial + 5 factorial - 1.
{
[x printString] bench.
}
#('1.29 per second. 778 milliseconds per run.') - before
#('5.12 per second. 195 milliseconds per run.') - after

=============== Diff against Kernel-fn.1216 ===============

+ ----- Method: Integer>>digitDiv21: (in category 'private') -----
+ digitDiv21: anInteger
+
+ 	^(self digitDiv: anInteger neg: false) collect: #normalize!

+ ----- 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!

+ ----- Method: Integer>>fastMultiply: (in category 'arithmetic') -----
+ fastMultiply: anInteger
+ 	^self * anInteger!

+ ----- Method: Integer>>karatsubaTimes: (in category 'arithmetic') -----
+ karatsubaTimes: anInteger
+ 	"Eventually use Karatsuba algorithm to perform the multiplication.
+ 	This is efficient only for large integers (see subclass).
+ 	By default, use regular multiplication."
+
+ 	^self * anInteger!

+ ----- 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 changed:
----- Method: Integer>>sqrtFloor (in category 'mathematical functions') -----
sqrtFloor
+ 	"Return the integer part of the square root of self
+ 	Assume self >= 0
+ 	The following invariants apply:
+ 	1) self sqrtFloor squared <= self
+ 	2) (self sqrtFloor + 1) squared > self"
- 	"Return the integer part of the square root of self"

+ 	^self sqrtFloorNewtonRaphson!
- 	| guess delta |
- 	guess := 1 bitShift: self highBit + 1 // 2.
- 	[
- 		delta := guess squared - self // (guess bitShift: 1).
- 		delta = 0 ] whileFalse: [
- 			guess := guess - delta ].
- 	^guess - 1!

+ ----- Method: Integer>>sqrtFloorNewtonRaphson (in category 'mathematical functions') -----
+ sqrtFloorNewtonRaphson
+ 	"Return the integer part of the square root of self.
+ 	Use a Newton-Raphson iteration since it converges quadratically
+ 	(twice more bits of precision at each iteration)"
+
+ 	| guess delta |
+ 	guess := 1 bitShift: self highBit + 1 // 2.
+ 	[
+ 		delta := guess squared - self // (guess bitShift: 1).
+ 		delta = 0 ] whileFalse: [
+ 			guess := guess - delta ].
+ 	^guess - 1!

+ ----- Method: Integer>>sqrtRem (in category 'mathematical functions') -----
+ sqrtRem
+ 	"Return an array with floor sqrt and sqrt remainder.
+ 	Assume self >= 0.
+ 	The following invariants apply:
+ 	1) self  sqrtRem first squared <= self
+ 	2) (self sqrtRem first + 1) squared > self
+ 	3) self sqrtRem first squared + self sqrtRem last = self"
+
+ 	| s |
+ 	s := self sqrtFloorNewtonRaphson.
+ 	^{ s. self - s squared } !

+ ----- Method: Integer>>toom3Times: (in category 'arithmetic') -----
+ toom3Times: anInteger
+ 	^self * anInteger!

+ ----- Method: LargePositiveInteger>>butLowestNDigits: (in category 'private') -----
+ butLowestNDigits: N
+ 	"make a new integer removing N least significant digits of self."
+
+ 	^self bitShiftMagnitude: -8 * N!

+ ----- Method: LargePositiveInteger>>copyDigitsFrom:to: (in category 'private') -----
+ copyDigitsFrom: start to: stop
+ 	"Make a new integer keeping only some digits of self."
+
+ 	| len slice |
+ 	start > 0 ifFalse: [^self error: 'start index should be at least 1'].
+ 	len := self digitLength.
+ 	(start > len or: [start > stop]) ifTrue: [^0].
+ 	stop >= len
+ 		ifTrue: [start = 1 ifTrue: [^self].
+ 				len := len - start + 1]
+ 		ifFalse: [len := stop - start + 1].
+ 	slice := self class new: len.
+ 	slice replaceFrom: 1 to: len with: self startingAt: start.
+ 	^slice normalize!

+ ----- 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!

+ ----- 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
+ 	!

+ ----- 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.
+ 	self negative ifTrue: [qr at: 2 put: qr second negated].
+ 	aBoolean ifTrue: [qr at: 1 put: qr first negated].
+ 	^qr!

+ ----- Method: LargePositiveInteger>>fastMultiply: (in category 'arithmetic') -----
+ fastMultiply: anInteger
+ 	"Eventually use Karatsuba or 3-way Toom-Cook algorithm to perform the multiplication"
+
+ 	| xLen |
+ 	"arrange to have the receiver be the shorter"
+ 	(xLen := self digitLength) > anInteger digitLength
+ 		ifTrue: [^ anInteger fastMultiply: self ].
+
+ 	"If too short to be worth, fallback to naive algorithm"
+ 	(xLen >= 600) ifFalse: [^self * anInteger].
+ 	(xLen >= 6000) ifFalse: [^self karatsubaTimes: anInteger].
+ 	^self toom3Times: anInteger!

+ ----- Method: LargePositiveInteger>>karatsubaTimes: (in category 'arithmetic') -----
+ karatsubaTimes: anInteger
+ 	"Eventually use Karatsuba algorithm to perform the multiplication
+ 	The Karatsuba algorithm divide number in two smaller parts.
+ 	Then it uses tricks to perform only 3 multiplications.
+ 	See https://en.wikipedia.org/wiki/Karatsuba_algorithm"
+
+ 	| half xHigh xLow yHigh yLow low high mid xLen yLen |
+ 	"arrange to have the receiver be the shorter"
+ 	(xLen := self digitLength) > (yLen := anInteger digitLength)
+ 		ifTrue: [^ anInteger karatsubaTimes: self ].
+
+ 	"If too short to be worth, fallback to naive algorithm"
+ 	(xLen >= 600) ifFalse: [^self * anInteger].
+
+ 	"Seek for integers of about the same length, else split the long one"
+ 	yLen >= (7 * xLen bitShift: -2)
+ 		ifTrue:
+ 			[^(0 to: yLen - 1 by: xLen + 1)  digitShiftSum: [:yShift |
+ 				self karatsubaTimes:
+ 					(anInteger copyDigitsFrom: yShift + 1 to: yShift + 1 + xLen)]].
+
+ 	"At this point, lengths are similar, divide each integer in two halves"
+ 	half := (yLen + 1 bitShift: -1)  bitClear: 2r11.
+ 	xLow := self lowestNDigits: half.
+ 	xHigh := self butLowestNDigits: half.
+ 	yLow := anInteger lowestNDigits: half.
+ 	yHigh := anInteger butLowestNDigits: half.
+
+ 	"Karatsuba trick: perform with 3 multiplications instead of 4"
+ 	low := xLow karatsubaTimes: yLow.
+ 	high := xHigh karatsubaTimes: yHigh.
+ 	mid := high + low + (xHigh - xLow karatsubaTimes: yLow - yHigh).
+
+ 	"Sum the parts of decomposition"
+ 	^low + (mid bitShiftMagnitude: 8*half) + (high bitShiftMagnitude: 16*half)
+
+ "
+ | a b |
+ a := 650 factorial-1.
+ b := 700 factorial-1.
+ {a digitLength. b digitLength}.
+ self assert: (a karatsubaTimes: b) - (a * b) = 0.
+ [Smalltalk garbageCollect.
+ [1000 timesRepeat: [a karatsubaTimes: b]] timeToRun] value /
+ [Smalltalk garbageCollect.
+ [1000 timesRepeat: [a * b]] timeToRun] value asFloat
+ "!

+ ----- Method: LargePositiveInteger>>lowestNDigits: (in category 'private') -----
+ lowestNDigits: N
+ 	"Make a new integer keeping only N least significant digits of self"
+
+ 	| low |
+ 	N >= self digitLength ifTrue: [^self].
+ 	low := self class new: N.
+ 	low replaceFrom: 1 to: N with: self startingAt: 1.
+ 	^low normalize!

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).

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!

+ ----- 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)!

+ ----- Method: LargePositiveInteger>>sqrtFloor (in category 'mathematical functions') -----
+ sqrtFloor
+ 	"Like super, but use a faster divide and conquer strategy if it's worth"
+
+ 	self digitLength >= 64 ifFalse: [^self sqrtFloorNewtonRaphson].
+ 	^self sqrtRem first!

+ ----- 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 := 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.
+ 	r negative
+ 		ifTrue:
+ 			[r := (s bitShift: 1) + r - 1.
+ 			s := s - 1].
+ 	sr at: 1 put: s; at: 2 put: r.
+ 	^sr
+ 	!

+ ----- 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!

+ ----- Method: LargePositiveInteger>>squaredKaratsuba (in category 'mathematical functions') -----
+ squaredKaratsuba
+ 	"Use a divide and conquer algorithm to perform the multiplication.
+ 	Split in two parts like Karatsuba, but economize 2 additions by using asymetrical product."
+
+ 	| half xHigh xLow low high mid |
+
+ 	"Divide digits in two halves"
+ 	half := self digitLength + 1 // 2 bitClear: 2r11.
+ 	xLow := self lowestNDigits: half.
+ 	xHigh := self butLowestNDigits: half.
+
+ 	"eventually use karatsuba"
+ 	low := xLow squared.
+ 	high := xHigh squared.
+ 	mid := xLow karatsubaTimes: xHigh.
+
+ 	"Sum the parts of decomposition"
+ 	^low + (mid bitShift: 8*half+1) + (high bitShift: 16*half)
+
+ "
+ | a |
+ a := 440 factorial-1.
+ a digitLength.
+ self assert: a * a - a squaredKaratsuba = 0.
+ [Smalltalk garbageCollect.
+ [2000 timesRepeat: [a squaredKaratsuba]] timeToRun] value /
+ [Smalltalk garbageCollect.
+ [2000 timesRepeat: [a * a]] timeToRun] value asFloat
+ "!

+ ----- Method: LargePositiveInteger>>squaredToom3 (in category 'mathematical functions') -----
+ squaredToom3
+ 	"Use a 3-way Toom-Cook divide and conquer algorithm to perform the multiplication"
+
+ 	| third x0 x1 x2 x20 z0 z1 z2 z3 z4 |
+ 	"divide in 3 parts"
+ 	third := self digitLength + 2 // 3 bitClear: 2r11.
+ 	x2 := self butLowestNDigits: third * 2.
+ 	x1 := self copyDigitsFrom: third + 1 to: third * 2.
+ 	x0 := self lowestNDigits: third.
+
+ 	"Toom-3 trick: 5 multiplications instead of 9"
+ 	z0 := x0 squared.
+ 	z4 := x2 squared.
+ 	x20 := x2 + x0.
+ 	z1 := (x20 + x1) squared.
+ 	x20 := x20 - x1.
+ 	z2 := x20 squared.
+ 	z3 := ((x20 + x2 bitShift: 1) - x0) squared.
+
+ 	"Sum the parts of decomposition"
+ 	z3 := z3 - z1 quo: 3.
+ 	z1 := z1 - z2 bitShift: -1.
+ 	z2 := z2 - z0.
+
+ 	z3 := (z2 - z3 bitShift: -1) + (z4 bitShift: 1).
+ 	z2 := z2 + z1 - z4.
+ 	z1 := z1 - z3.
+ 	^z0 + (z1 bitShift: 8*third) + (z2 bitShift: 16*third) + (z3 + (z4 bitShift: 8*third) bitShift: 24*third)
+
+ "
+ | a |
+ a := 1400 factorial-1.
+ a digitLength.
+ self assert: a * a - a squaredToom3 = 0.
+ [Smalltalk garbageCollect.
+ [1000 timesRepeat: [a squaredToom3]] timeToRun] value /
+ [Smalltalk garbageCollect.
+ [1000 timesRepeat: [a squaredKaratsuba]] timeToRun] value asFloat
+ "!

+ ----- 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
+ "!

+ ----- Method: LargePositiveInteger>>toom3Times: (in category 'arithmetic') -----
+ toom3Times: anInteger
+ 	"Eventually use a variant of Toom-Cooke for performing multiplication.
+ 	Toom-Cooke is a generalization of Karatsuba divide and conquer algorithm.
+ 	It divides operands in n parts instead of 2.
+ 	See https://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication
+ 	Here we divide each operand in 3 parts, thus the name Toom-3.
+ 	And use a Bodrato-Zanoni variant for the choice of interpolation points and matrix inversion
+ 	See What about Toom-Cook matrices optimality? - Marco Bodrato, Alberto Zanoni - Oct. 2006
+
+ 	| third x2 x1 x0 y2 y1 y0 xLen yLen y20 z4 z3 z2 z1 z0 x20 |
+ 	"arrange to have the receiver be the shorter"
+ 	(xLen := self digitLength) > (yLen := anInteger digitLength)
+ 		ifTrue: [^anInteger toom3Times: self ].
+
+ 	"If too short to be worth, fallback to Karatsuba algorithm"
+ 	(xLen > 6000) ifFalse: [^self karatsubaTimes: anInteger].
+
+ 	"Seek for well balanced integers"
+ 	yLen > (5 * xLen bitShift: -2)
+ 		ifTrue: [^(0 to: yLen - 1 by: xLen + 1)  digitShiftSum: [:yShift |
+ 				self toom3Times: (anInteger copyDigitsFrom: yShift + 1 to: yShift +1 + xLen)]].
+
+ 	"At this point, lengths are well balanced, divide in 3 parts"
+ 	third := yLen + 2 // 3 bitClear: 2r11.
+ 	x2 := self butLowestNDigits: third * 2.
+ 	x1 := self copyDigitsFrom: third + 1 to: third * 2.
+ 	x0 := self lowestNDigits: third.
+ 	y2 := anInteger butLowestNDigits: third * 2.
+ 	y1 := anInteger copyDigitsFrom: third + 1 to: third * 2.
+ 	y0 := anInteger lowestNDigits: third.
+
+ 	"Toom-3 trick: 5 multiplications instead of 9"
+ 	z0 := x0 toom3Times: y0.
+ 	z4 := x2 toom3Times: y2.
+ 	x20 := x2 + x0.
+ 	y20 := y2 + y0.
+ 	z1 := x20 + x1 toom3Times: y20 + y1.
+ 	x20 := x20 - x1.
+ 	y20 := y20 - y1.
+ 	z2 := x20 toom3Times: y20.
+ 	z3 := (x20 + x2 bitShift: 1) - x0 toom3Times: (y20 + y2 bitShift: 1) - y0.
+
+ 	"Sum the parts of decomposition"
+ 	z3 := z3 - z1 quo: 3.
+ 	z1 := z1 - z2 bitShift: -1.
+ 	z2 := z2 - z0.
+
+ 	z3 := (z2 - z3 bitShift: -1) + (z4 bitShift: 1).
+ 	z2 := z2 + z1 - z4.
+ 	z1 := z1 - z3.
+ 	^z0 + (z1 bitShift: 8*third) + (z2 bitShift: 16*third) + (z3 + (z4 bitShift: 8*third) bitShift: 24*third)
+
+ "
+ | a b |
+ a :=5000 factorial - 1.
+ b := 6000 factorial - 1.
+ a digitLength min: b digitLength.
+ a digitLength max: b digitLength.
+ (a toom3Times: b) = (a * b) ifFalse: [self error].
+ [Smalltalk garbageCollect.
+ [300 timesRepeat: [a toom3Times: b]] timeToRun] value /
+ [Smalltalk garbageCollect.
+ [300 timesRepeat: [a karatsubaTimes: b]] timeToRun] value asFloat
+ "!

Item was changed:
----- Method: Number>>raisedToInteger: (in category 'mathematical functions') -----
raisedToInteger: anInteger
"The 0 raisedToInteger: 0 is an special case. In some contexts must be 1 and in others must
be handled as an indeterminate form.
I take the first context because that's the way that was previously handled.
Maybe further discussion is required on this topic."

| bitProbe result |
anInteger negative ifTrue: [^(self raisedToInteger: anInteger negated) reciprocal].
bitProbe := 1 bitShift: anInteger highBit - 1.
result := self class one.
[
(anInteger bitAnd: bitProbe) > 0 ifTrue: [ result := result * self ].
(bitProbe := bitProbe bitShift: -1) > 0 ]
+ 		whileTrue: [ result := result squared ].
- 		whileTrue: [ result := result * result ].
^result!

```