🌺 💹 ↗️ The cornerstones of the destruction of slow code in Wolfram Language: speeding up code tens, hundreds and thousands of times 🏴󠁧󠁢󠁥󠁮󠁧󠁿 👩🏽‍🔧 🤠

You can download the file with the code and data in the original post on my blog

The picture for the webinar and the post was taken for a reason: in a sense, the symbolic core of Wolfram Language can be compared with Thanos - if his power were directed in the right direction, he could become the most powerful and useful “good man”. It is the same with the Wolfram symbolic core - its monstrous power must be used correctly, and if this is not done, it can become a real “evil” that slows down everything very much. Novice developers do not know many of the most important paradigms, ideas and principles of the Wolfram Language, write code that is actually wildly inefficient and are disappointed after that, although there is no Wolfram Language fault. This article is intended to correct this article.

I had a chance to work with Wolfram Language starting from (already quite distant) 2005 (then there was still version Mathematica 5.2, now it is the 12th). Over the past 15 years, a lot has happened: thousands of new built-in functions and areas in which they work have been added ( machine learning , precise geometry , working with audio , working on the web , cloud capabilities , deep unit support, integration with Wolfram databases | on Alpha , geographical calculation , support for working with the CUDA , the Python , parallelization of operations, and much more), new services - cloud by Wolfram cloud , a well-known system of computer-value by Wolfram | on Alpha , functions of the repository , the repository of neural networks and etc. .

Over the years, I used Wolfram Language very much (yes, earlier, in fact, it was called Mathematica , before Stephen Wolfram decided not to single out the language and make it common for a whole galaxy of his products) and in different areas: I managed to make a sales model for a pharmaceutical company; create parsers and automatic movie album generators for the movie rental industry (when they were just in their infancy); apply actively in science (work in chemistry and solid state physics and destruction theory); to develop prototypes of teaching systems for the platforms of domestic school and university education and leading companies and publishers in this market in Russia.

Records of some of my speeches at MBLT DEV and Wolfram Conferences

, , , Wolfram Language : JavaScript, Python, Java, PHP . — Wolfram , , , .

, , Wolfram Language — (., , "Performance tuning in Mathematica?" — ) — Wolfram Language — , , ( ) .

, , : , C - Wolfram Language, CUDA, Java C (, Wolfram-).

, Wolfram Language . , , , 15 ( , — , Factorial

):

Module[{st, stInput, data, systemFunction}, 
st=Style[#, FontFamily->"Arial", Gray, FontSize->14]&;

SetAttributes[stInput, HoldAll];

stInput[code_]:=(Clear[f];

code;

{Style[HoldForm@code, "Input", FontSize->14, Background->None], code;

RepeatedTiming[f[9];

, 2][[1]]});

data=Flatten/@{
{st@" Wolfram", stInput[f=Factorial]}, 
{"", stInput[f[n_]:=n!]}, 
{st[" "], stInput[f[n_]:=n f[n-1];

f[1]=1]}, 
{st[""], stInput[f[n_]:=Module[{t=1}, Do[t=t*i, {i, n}];

t]]}, 
{"", stInput[f[n_]:=Module[{t=1, i}, For[i=1, i<=n, i++, t*=i];

t]]}, 
{st["  ()"], stInput[f[n_]:=Apply[Times, Range[n]]]}, 
{"", stInput[f[n_]:=Fold[Times, 1, Range[n]]]}, 
{st[""], stInput[f[n_]:=If[n==1, 1, n*f[n-1]]]}, 
{st[""], stInput[f=If[#1==1, 1, #1*#0[#1-1]]&]}, 
{"", stInput[f[n_]:=Fold[#2[#1]&, 1, Array[Function[t, #t]&, n]]]}, 
{st[""], stInput[f[n_]:=Length[Permutations[Range[n]]]]}, 
{st["  ()"], stInput[f[n_]:=First[{1, n}//.{a_, b_/;

b>0}:>{b*a, b-1}]]}, 
{st[" "], stInput[f[n_]:=StringLength[Fold[StringJoin[Table[#1, {#2}]]&, "A", Range[n]]]]}, 
{st[""], stInput[f[n_]:=Gamma[n+1]]}, 
{"", stInput[f[n_]:=Product[i, {i, n}]]}
};

systemFunction=data[[1, 3]];

Framed[#, Background->White, FrameMargins->5, FrameStyle->None]&@Grid[{Style[st[#], Bold]&/@{"", "", " \n,  "}}~Join~({#[[1]], #[[2]], st[Round[#[[3]]/systemFunction, 1/10]//N]}&/@data), 
Alignment->{{Center, Left, Left}, Center}, 
Dividers->{None, {None, {LightGray}, None}}, Background->{None, {LightOrange, None}}
]]

, , , , Python. Wolfram Language, C, .

Wolfram Language.

Wolfram Language (Mathematica , ) — , , , , , Matlab (Sympy, Maxima, Maple .), , . , , , , , , :

Wolfram Language, .

1.

1.1. , ,

Wolfram ( Java), ( !) , . , , .

ClearAll[data];

data=RandomReal[1, {10^6, 2, 2}];

compareTiming[{Map[Flatten, data], Flatten[data, {{1}, {2, 3}}]}]

1.2. (`Map`

/@, `Apply`

@@ )

(-) #-&, — , , , ( Function

) , ( , , ).

ClearAll[data];

data=Range[1, 10^6];

compareTiming[{f[x_]:=x^2;

Table[f[i], {i, data}], Map[#^2&, data]}]

1.3.

Transpose

, Flatten

, Partition

, Part

. . — , .

ClearAll[data];

data=RandomReal[1, {10^5, 2}];

compareTiming[{{#[[1]], #[[2]]^2}&/@data, Transpose[{#1, #2^2}&@@Transpose[data]]}]

1.4.

. , ( . .). «» .

ClearAll[fibonacci, f];

compareTiming[{fibonacci={1, 1};

n=3;

While[n<=20000, AppendTo[fibonacci, fibonacci[[n-1]]+fibonacci[[n-2]]];

n++];

fibonacci, f[1]=1;

f[2]=1;

f[n_]:=f[n]=f[n-1]+f[n-2];

f/@Range[20000]}]

2. , ,

, , 6/47 . , , , .

2.1. ,

, , .

compareTiming[{data=Table[{x, Sin[x]/Gamma[x]}, {x, 0, 10, 1/100}];

ListPlot[data], data=Table[{x, Sin[x]/Gamma[x]}, {x, 0, 10, 1/100.}];

ListPlot[data]}]

— 100 100. — , — . 4 ! , , .

Table[{x, Sin[x]/Gamma[x]}, {x, 0, 10, 1/2}]
Table[{x, Sin[x]/Gamma[x]}, {x, 0, 10, 1/2.}]

, Wolfram Language : , ( ), , , , 100 ( Wolfram Language ), , . , .

2.2.

, . ,

a x^{3} + b x + c = 0

$ax^3+bx+c=0$ a, b, c. :

Solve[a x^3+b x+c==0, x]/.{a->1.23, b->3.45, c->2.6}

, 15 :

compareTiming[{Solve[a x^3+b x+c==0, x]/.{a->1.23, b->3.45, c->2.6}, Solve[a x^3+b x+c==0/.{a->1.23, b->3.45, c->2.6}, x]}]

, Wolfram Language :

Solve[a x^3+b x+c==0, x]

Solve[a x^3+b x+c==0, x]/.{a->1.23, b->3.45, c->2.6}

a x^3+b x+c==0/.{a->1.23, b->3.45, c->2.6}

Solve[a x^3+b x+c==0/.{a->1.23, b->3.45, c->2.6}, x]

2.3.

— (. ). N

, Rationalize

, Round

, Floor

, Ceiling

, SetPrecision

, SetAccuracy

.

3. , , CUDA

3.1.

(Listability) — , . Wolfram Language List

— . , .

, , Map

.

ClearAll[data];

data=RandomReal[{-Pi, Pi}, 10^7];

compareTiming[{Sin/@data, Sin[data]}]

3.2. `Compile`

, . C- ( , Visual C++ Build Tools 2017 — , , . ). (RuntimeAttributes) (Parallelization).

ClearAll[data];

data=Range[-50, 50, 0.001];

fNotCompiled=Function[{x}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum]];

, - Wolfram Language (JIT-) ( JIT-)

fJITCompiled=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum]];

JIT- :

fJITCompiledParallelized=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum], Parallelization->True];

JIT- :

fJITCompiledListable=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum], RuntimeAttributes->{Listable}];

JIT- :

fJITCompiledListableParallelized=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum], Parallelization->True, RuntimeAttributes->{Listable}];

, C ( ):

fCompiled=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum], CompilationTarget->"C"];

fCompiledParallelized=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum], CompilationTarget->"C", Parallelization->True];

fCompiledListable=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum], CompilationTarget->"C", RuntimeAttributes->{Listable}];

fCompiledListableParallelized=Compile[{{x}}, Block[{sum=1.0, inc=1.0}, Do[inc=inc*x/i;

sum=sum+inc, {i, 100}];

sum], CompilationTarget->"C", Parallelization->True, RuntimeAttributes->{Listable}];

( !) . , C 50 !

compareTiming[
{fNotCompiled/@data, 
fJITCompiled/@data, 
fJITCompiledListable@data, 
fJITCompiledParallelized/@data, 
fJITCompiledListableParallelized@data, 
fCompiled/@data, 
fCompiledListable@data, 
fCompiledParallelized/@data, 
fCompiledListableParallelized@data}]

Compile , «», . MathGroup.

3.3. Compile

, (UnitStep

, Clip

, Sign

, Abs

. .) , Compile

, « (-)», If

, ( , ) Compile. - Wolfram Language, C .

f1Compiled=Compile[{{x, _Real}}, SeedRandom[1];

Block[{data}, data=RandomReal[{-x, x}, 10^5];

Table[If[data[[i]]>0, 1, 0], {i, 1, Length[data]}]], CompilationTarget->"C", Parallelization->True];

f2Compiled=Compile[{{x, _Real}}, SeedRandom[1];

Block[{data}, data=RandomReal[{-x, x}, 10^5];

UnitStep@data], CompilationTarget->"C", Parallelization->True];

compareTiming[{f1Compiled@10, f2Compiled@10}]

3.4.

— , . , : Parallelize, ParallelMap, ParallelTable.

compareTiming[{Table[Solve[Sin[n x+n^2]==0, x], {n, 1, 100}], ParallelTable[Solve[Sin[n x+n^2]==0, x], {n, 1, 100}]}]

3.5. CUDA, OpenCL — ()

, CUDA OpenCL. , , . .

4. , ({...} List) Wolfram Language,

4.1. , «», .

, . , 100-1000 , , XMLObject

, -, , .

4.2. Append, Prepend, AppendTo PrependTo

Append

, Prepend

, AppendTo

and PrependTo

, . . — , , , .

compareTiming[{list={};

n=1;

While[n<=10000, list=Append[list, n++]];

list, Range[1, 10000]}]

4.3.

« » (, {1,{2,{3,{}}}}) «» ( {1,2,3,4}) .

: A = {new element, A}.

A — , .

compareTiming[{list1={};

Table[If[PrimeQ[i], list1=list1~Join~{i}], {i, 1, 100000}];

list1, list2={};

Table[If[PrimeQ[i], list2={list2, i}], {i, 1, 100000}];

Flatten@list2}]

4.4.

, (Pattern matching) , (__ BlankSequence

___ BlankNullSequence

) , (Sequence

) . , , {fst_,rest___}:>{f[fst],g[rest]} . , , . , (. ).

5. . .

, , (Patterns) Wolfram Language :

5.1.

, , , , .

, , , ( ).

:

list //. {left___, x_, middle__, y_, right__} /; x > y :> {left, y, middle, x, right};

SequenceCases

, SequencePosition

, SequenceCount

, SequenceReplace

, , 10.1 Wolfram.

5.2.

, .

ClearAll[data];

data=RandomInteger[{0, 100}, {10^6, 2}];

compareTiming[{data/.{p:({x_/;

x>90&&PrimeQ[x], y_/;

y>34})/;

x^2+y^2>1000:>p, t:{_, _}:>Nothing}, ((data/.{_, y_/;

y<=34}:>Nothing)/.{x_/;

(x<=90||Not[PrimeQ[x]]), _}:>Nothing)/.({x_, y_}/;

x^2+y^2<=1000)->Nothing, 
Cases[Cases[Cases[data, {x_/;

x>90&&PrimeQ[x], _}], {_, y_/;

y>34}], {x_, y_}/;

x^2+y^2>1000]}]

5.3.

. , ( , ..). , ( / ;) (?), , , .

ClearAll[data];

data=RandomSample[{##}]&@@@Transpose[{RandomInteger[{0, 100}, 10^5], RandomReal[{0, 100}, 10^5]}];

compareTiming[{Cases[data, {_Integer, x_/;

Sin[x]<FractionalPart[x]}], Module[{preData, seconds}, preData=Cases[data, {_Integer, _}];

seconds=Developer`ToPackedArray[preData[[;;, 2]]];

Pick[preData, UnitStep[FractionalPart[seconds]-Sin[seconds]], 1]]}]

6.

Wolfram Language, , . , , .

, ( ), — [[...]] (Part

).

6.1. ,

, .

compareTiming[{NestWhile[Drop[#, 1]&, Range[1000], (Length[#]>10)&], Drop[Range[1000], 990]}]

6.2. Part

Part ( ). , (Part ).

:

list={1, 2, 3, 4, 5};

list[[{1, 3}]]=list[[{4, 1}]];

list

— :

ClearAll[list1, list2];

list1=Range[1000000];

list2=list1;

newArrangement=RandomSample[list1];

compareTiming[{list1=Table[list1[[i]], {i, newArrangement}];

list1, 
list2=list2[[newArrangement]];

list2}]

6.3. Extract

Extract

, .

ClearAll[list1, list2];

list=Range[10000000];

elems=RandomInteger[{1, 10000000}, 10000];

compareTiming[{Table[list[[elems[[i]]]], {i, 1, Length@elems}], 
Extract[list, List/@elems]}]

7.

7.1.

, Developer`ToPackedArray. Developer`PackedArrayQ.

ClearAll[data];

data=RandomInteger[{0, 1}, 10^6];

dataUnPacked=data~Join~{1.};

dataPacked=Developer`ToPackedArray[N[data]~Join~{1.}];

compareTiming[{Total[dataUnPacked], Total[dataPacked]}]

7.2.

, , , , . SparseArray

.

sparseArray=SparseArray[RandomInteger[{1, 100}, {100, 2}]->RandomReal[{0, 10}, 100], {100, 100}];

notSparseArray=Normal@sparseArray;

compareTiming[{Det@notSparseArray, Det@sparseArray}]

7.3. -

10 Wolfram Language , <|...|> (Association

).

, , - (« », -). Wolfram Language .

assoc=<|a->1, b->2, c->3, d->4, e->5|>;

assoc[d]+assoc[a]

Dataset — , Wolfram Language.

Dispatch

, - .

8. ,

8.1. ()

Wolfram Language .

: f[x_]:=functionBody f[x_]:=f[x]=functionBody — , . 1 , , , Save

DumpSave

.

ClearAll[f1, f2];

f1[x_]:=(Pause[0.1];

x^2);

f2[x_]:=f2[x]=(Pause[0.1];

x^2);

compareTiming[{f1/@{1, 1, 1, 1, 1}, f2/@{1, 1, 1, 1, 1}}]

8.2. (closures)

Wolfram Language , () , , , .

ClearAll[f1, f2]

f1[x_]:=(y/.Solve[y^2+x y+x==0, y]);

Module[{solution=FullSimplify@Solve[y^2+x y+x==0, y]}, 
f2[xValue_]:=(y/.solution)/.x->xValue];

compareTiming[{Table[f1[x], {x, 0, 100}], Table[f2[x], {x, 0, 100}]}]

8.3.

Wolfram Language := (SetDelayed

), :> (RuleDelayed

). «» , , = (Set

) -> (Rule

). , .

ClearAll[f1, f2]

f1[x_]:=(y/.FullSimplify[Solve[y^2+x y+x==0, y]]);

f2[x_]=(y/.FullSimplify[Solve[y^2+x y+x==0, y]]);

compareTiming[{Table[f1[x], {x, 0, 100}], Table[f2[x], {x, 0, 100}]}]

9.

9.1. Reap + Sow

Reap Sow , «» , . .

9.2.

Wolfram Language , , Mathematica. . . FullForm

.

FullForm[{{1, 2}, {3, {4}}, {{5}}}]

FullForm[Hold[x5+y2+1/z+4/2]]

9.3.

, , ( ). .

9.4. $HistoryLength

Wolfram Language , . $HistoryLength

.

Wolfram Language

— , . , Wolfram Language, : .

, , Wolfram Mathematica, Wolfram Cloud, Wolfram One . , , « ». Wolfram Language Wolfram Mathematica, , , . : , , .

: , — Wolfram Language . , , .

, , Wolfram Language :

, ( (, , ) ). , — 15, .

, Wolfram Language , . ( 12.0) 5436 188 . .

data=SortBy[#,-Length[#]&]&@GroupBy[Flatten[Thread/@EntityValue[WolframLanguageData[],{name,functionality areas}],1],Last,Sort[#[[;;,1]]]&];

Framed[Labeled[Style["  Wolfram Language   ,   \n",30,FontFamily->"Intro Regular"],#],Background->GrayLevel[0.95],FrameMargins->5,FrameStyle->None]&@WordCloud[KeyValueMap[{StringRiffle[StringCases[#1,RegularExpression["[A-Z][a-z]+"]]/."Symbols"->""," "]<>" ("<>ToString[Length@#2]<>")",N@Length@#2}&,data],ImageSize->{1200,800},Background->GrayLevel[0.95],WordOrientation->{{-Pi/4,0,Pi/4}},MaxItems->All,WordSpacings->3,FontFamily->"Intro Regular"]

Export[FileNameJoin[{NotebookDirectory[], "fileNames.nb"}], Notebook@Flatten@KeyValueMap[{Cell[#1<>" ("<>ToString[Length[#2]]<>")", "Title"], Cell[BoxData@ToBoxes@Row[Map[Hyperlink[#, "http://reference.wolfram.com/language/ref/"<>#<>".html"]&, #2], ",  "], "Text"]}&, data]];

Wolfram Language

Messages And Printing (18)

Assert, MessageList, MessageName, Messages, Off, On, PrintTemporary, Quiet, Shallow, Short, Skeleton, $AssertFunction, $Language, $MessageGroups, $MessageList, $MessagePrePrint, $Messages, $NewMessage

Binary Data (17)

BaseDecode, BaseEncode, BinaryDeserialize, BinaryRead, BinaryReadList, BinarySerialize, BinaryWrite, ByteArray, ByteArrayFormat, ByteArrayQ, ByteArrayToString, ByteOrdering, ExportByteArray, ImportByteArray, ReadByteArray, StringToByteArray, $ByteOrdering

Debug (17)

Echo, EchoFunction, MatchLocalNames, Stack, StackBegin, StackComplete, StackInhibit, Trace, TraceAbove, TraceBackward, TraceDepth, TraceForward, TraceOff, TraceOn, TraceOriginal, TracePrint, TraceScan

Discrete Fourier (17)

Cepstrogram, CepstrogramArray, CepstrumArray, FourierDCT, FourierDST, FourierParameters, InverseFourier, InverseShortTimeFourier, InverseSpectrogram, ListFourierSequenceTransform, ListZTransform, Periodogram, PeriodogramArray, ShortTimeFourier, ShortTimeFourierData, Spectrogram, SpectrogramArray

Histogram (17)

BinCounts, BinLists, DateHistogram, DateReduction, DensityHistogram, Histogram, Histogram3D, HistogramDistribution, HistogramList, HistogramTransform, HistogramTransformInterpolation, ImageHistogram, PairedHistogram, PairedSmoothHistogram, SmoothDensityHistogram, SmoothHistogram, SmoothHistogram3D

Numeric (17)

DifferenceQuotient, DiscreteVariables, EvaluationMonitor, Gradient, MaxIterations, MaxRecursion, MaxStepFraction, MaxSteps, MaxStepSize, Method, NormFunction, PerformanceGoal, PrecisionGoal, StartingStepSize, StepMonitor, Tolerance, WorkingPrecision

Permutation (17)

Cycles, FindPermutation, InversePermutation, PermutationCycles, PermutationCyclesQ, PermutationLength, PermutationList, PermutationListQ, PermutationMax, PermutationMin, PermutationOrder, PermutationPower, PermutationProduct, PermutationReplace, PermutationSupport, Permute, RandomPermutation

Symbol Information (17)

Constant, Flat, Locked, NHoldAll, NHoldFirst, NHoldRest, NumericFunction, OneIdentity, OptionValue, Orderless, Protected, ReadProtected, SequenceHold, SetSystemOptions, Stub, SystemOptions, Temporary

System Model (17)

ConnectSystemModelComponents, CreateDataSystemModel, CreateSystemModel, FindSystemModelEquilibrium, SetSystemModel, SystemModel, SystemModeler, SystemModelExamples, SystemModelLinearize, SystemModelParametricSimulate, SystemModelPlot, SystemModelProgressReporting, SystemModelReliability, SystemModels, SystemModelSimulate, SystemModelSimulateSensitivity, SystemModelSimulationData

Cryptography (16)

Decrypt, DecryptFile, DerivedKey, DigitalSignature, Encrypt, EncryptedObject, EncryptFile, GenerateAsymmetricKeyPair, GenerateDerivedKey, GenerateDigitalSignature, GenerateSymmetricKey, PrivateKey, PublicKey, SymmetricKey, VerifyDerivedKey, VerifyDigitalSignature

Structural (16)

GroupBy, Groupings, Heads, Insert, Nothing, Order, Ordering, OrderingBy, Position, PositionIndex, ReverseSort, ReverseSortBy, Riffle, SortedBy, Split, SplitBy

Time Memory (16)

AbsoluteTiming, MaxMemoryUsed, MemoryAvailable, MemoryConstrained, MemoryConstraint, MemoryInUse, Pause, RepeatedTiming, SessionTime, Share, TimeConstrained, TimeConstraint, TimeGoal, TimeUsed, $TimedOut, $TimeUnit

Code Action (15)

BlockMap, FixedPoint, FixedPointList, FoldPair, FoldPairList, MapAll, MapIndexed, NestList, NestWhile, NestWhileList, Operate, SequenceFold, SequenceFoldList, SubsetMap, Through

Continuous Fourier (15)

FourierCoefficient, FourierCosCoefficient, FourierCosSeries, FourierCosTransform, FourierSeries, FourierSinCoefficient, FourierSinSeries, FourierSinTransform, FourierTransform, FourierTrigSeries, InverseFourierCosTransform, InverseFourierSinTransform, InverseFourierTransform, InverseLaplaceTransform, LaplaceTransform

Domain (15)

Element, NegativeIntegers, NegativeRationals, NegativeReals, NonNegativeIntegers, NonNegativeRationals, NonNegativeReals, NonPositiveIntegers, NonPositiveRationals, NonPositiveReals, NotElement, PositiveIntegers, PositiveRationals, PositiveReals, Rationals

External Process (15)

KillProcess, ProcessConnection, ProcessDirectory, ProcessEnvironment, Processes, ProcessInformation, ProcessObject, ProcessStatus, RemoteAuthorizationCaching, RemoteConnect, RemoteConnectionObject, RemoteRun, RemoteRunProcess, RunProcess, StartProcess

Integer (15)

DigitCount, FromCharacterCode, FromDigits, FromRomanNumeral, IntegerDigits, IntegerExponent, IntegerLength, IntegerName, IntegerPart, IntegerReverse, Integers, IntegerString, QuotientRemainder, RealDigits, RomanNumeral

Interpreter (15)

DelimitedSequence, ExcludedLines, GeoLocation, HeaderLines, ImportOptions, Interpreter, MissingDataRules, RectangularRepeatingElement, RepeatingElement, Restricted, SemanticImport, SemanticImportString, SemanticInterpretation, SquareRepeatingElement, $InterpreterTypes

Socket (15)

NetworkPacketCapture, NetworkPacketRecording, NetworkPacketTrace, SocketConnect, SocketListen, SocketListener, SocketObject, SocketOpen, SocketReadMessage, SocketReadyQ, Sockets, SocketWaitAll, SocketWaitNext, $DefaultNetworkInterface, $NetworkInterfaces

Vector Calculus (15)

ArcCurvature, ArcLength, CoordinateChartData, CoordinateTransform, CoordinateTransformData, Curl, Div, FrenetSerretSystem, FromPolarCoordinates, FromSphericalCoordinates, Grad, Laplacian, ToPolarCoordinates, ToSphericalCoordinates, TransformedField

Blockchain (14)

BlockchainAddressData, BlockchainBase, BlockchainBlockData, BlockchainContractValue, BlockchainData, BlockchainGet, BlockchainKeyEncode, BlockchainPut, BlockchainTokenData, BlockchainTransaction, BlockchainTransactionData, BlockchainTransactionSign, BlockchainTransactionSubmit, $BlockchainBase

Dynamic (14)

AbsoluteCurrentValue, Deinitialization, Dynamic, DynamicEvaluationTimeout, DynamicModule, DynamicModuleValues, DynamicSetting, DynamicWrapper, FinishDynamic, MouseAnnotation, MouseAppearance, Mouseover, NotebookDynamicExpression, $DynamicEvaluation

Library Link (14)

ConnectLibraryCallbackFunction, CreateManagedLibraryExpression, FindLibrary, LibraryDataType, LibraryFunction, LibraryFunctionError, LibraryFunctionInformation, LibraryFunctionLoad, LibraryFunctionUnload, LibraryLoad, LibraryUnload, ManagedLibraryExpressionID, ManagedLibraryExpressionQ, $LibraryPath

Sound (14)

Beep, EmitSound, ListPlay, Play, PlayRange, SampleDepth, SampledSoundFunction, SampledSoundList, SampleRate, Sound, SoundNote, SoundVolume, Speak, $SoundDisplayFunction

Dialog (13)

AuthenticationDialog, ChoiceDialog, CreateDialog, Dialog, DialogInput, DialogNotebook, DialogProlog, DialogReturn, DialogSymbols, MessageDialog, Modal, SystemDialogInput, TraceDialog

Environment (13)

ClearSystemCache, Exit, In, InString, Quit, Update, $HistoryLength, $IterationLimit, $Post, $Pre, $PrePrint, $PreRead, $RecursionLimit

Signal Processing (13)

BandpassFilter, BandstopFilter, BesselFilterModel, BiquadraticFilterModel, ButterworthFilterModel, Chebyshev1FilterModel, Chebyshev2FilterModel, EllipticFilterModel, EquirippleFilterKernel, FrequencySamplingFilterKernel, KalmanFilter, LeastSquaresFilterKernel, ResamplingAlgorithmData

Astronomy (12)

AltitudeMethod, CelestialSystem, EclipseType, LunarEclipse, MoonPhase, MoonPosition, SiderealTime, SolarEclipse, SunPosition, Sunrise, Sunset, TimeDirection

Geo Graphics Primitive (12)

DayHemisphere, DayNightTerminator, GeoBoundsRegion, GeoCircle, GeoDisk, GeoHemisphere, GeoHemisphereBoundary, GeoMarker, GeoPath, GeoVisibleRegion, GeoVisibleRegionBoundary, NightHemisphere

Root (12)

Cubics, FindRoot, Modulus, NRoots, Quartics, Root, RootOfUnityQ, RootReduce, Roots, RootSum, ToRadicals, $MaxRootDegree

Symbol Assignment (12)

AddTo, AppendTo, Decrement, DivideBy, Increment, PreDecrement, PreIncrement, PrependTo, Protect, SubtractFrom, TimesBy, Unprotect

Algebraic (11)

AlgebraicIntegerQ, AlgebraicNumber, AlgebraicNumberDenominator, AlgebraicNumberNorm, AlgebraicNumberPolynomial, AlgebraicNumberTrace, Algebraics, AlgebraicUnitQ, IsolatingInterval, MinimalPolynomial, RootApproximant

Alignment (11)

Above, Alignment, AlignmentPoint, Baseline, BaselinePosition, Below, Bottom, Center, Left, Right, Top

Asymptotic (11)

AsymptoticDSolveValue, AsymptoticEqual, AsymptoticEquivalent, AsymptoticGreater, AsymptoticGreaterEqual, AsymptoticIntegrate, AsymptoticLess, AsymptoticLessEqual, AsymptoticRSolveValue, AsymptoticSolve, AsymptoticSum

Button (11)

Button, ButtonBar, ButtonData, ButtonFunction, ButtonMinHeight, ButtonNotebook, ButtonSource, CancelButton, ChoiceButtons, DefaultButton, PasteButton

Code Interruption (11)

Abort, AbortProtect, Catch, CheckAbort, Failure, FailureQ, Interrupt, Success, Throw, $Aborted, $Failed

Discrete Calculus (11)

DifferenceDelta, DiscreteLimit, DiscreteMaxLimit, DiscreteMinLimit, NProduct, NSum, Product, Regularization, Sum, SumConvergence, VerifyConvergence

Finance (11)

Annuity, AnnuityDue, Cashflow, CurrencyConvert, EffectiveInterest, FinancialBond, FinancialDerivative, FinancialIndicator, InflationAdjust, InflationMethod, TimeValue

Geometric Scene (11)

AngleBisector, CircleThrough, FindGeometricConjectures, GeometricAssertion, Midpoint, PerpendicularBisector, RandomInstance, TriangleCenter, TriangleConstruct, TriangleMeasurement, UnconstrainedParameters

Pattern (11)

Alternatives, Except, Longest, Optional, OptionsPattern, OrderlessPatternSequence, PatternSequence, PatternTest, Repeated, RepeatedNull, Shortest

Resource (11)

DefineResourceFunction, PublisherID, ResourceData, ResourceFunction, ResourceObject, ResourceRegister, ResourceRemove, ResourceSearch, ResourceSubmit, ResourceUpdate, $PublisherID

Service (11)

EmbeddedService, SaveConnection, ServiceConnect, ServiceDisconnect, ServiceExecute, ServiceObject, ServiceRequest, ServiceSubmit, $EmbeddableServices, $ServiceCreditsAvailable, $Services

Annotation (10)

Annotation, AnnotationDelete, AnnotationRules, DefaultTooltipStyle, Monitor, ProgressIndicator, StatusArea, Tooltip, TooltipDelay, TooltipStyle

Bit (10)

BitAnd, BitClear, BitGet, BitLength, BitNot, BitOr, BitSet, BitShiftLeft, BitShiftRight, BitXor

Code Evaluation (10)

Activate, Defer, EvaluationData, HoldAllComplete, HoldComplete, IgnoringInactive, Inactivate, Inactive, ReleaseHold, Unevaluated

External Session (10)

ExternalEvaluate, ExternalFunction, ExternalObject, ExternalSessionObject, ExternalSessions, ExternalValue, FindExternalEvaluators, RegisterExternalEvaluator, StartExternalSession, UnregisterExternalEvaluator

Initialization (10)

EvaluationEnvironment, InitializationObjects, InitializationValue, Initialize, LocalSubmit, ValuePreprocessingFunction, $Initialization, $InitializationContexts, $NoValue, $PreInitialization

Legend (10)

BarLegend, LegendFunction, LegendLabel, LegendLayout, LegendMargins, LegendMarkers, LegendMarkerSize, LineLegend, PointLegend, SwatchLegend

Number Field (10)

Extension, NumberFieldClassNumber, NumberFieldDiscriminant, NumberFieldFundamentalUnits, NumberFieldIntegralBasis, NumberFieldNormRepresentatives, NumberFieldRegulator, NumberFieldRootsOfUnity, NumberFieldSignature, ToNumberField

Optimization (10)

ConicOptimization, LinearFractionalOptimization, LinearOptimization, QuadraticOptimization, SecondOrderConeOptimization, SemidefiniteOptimization, VectorGreater, VectorGreaterEqual, VectorLess, VectorLessEqual

Task (10)

SessionSubmit, TaskAbort, TaskExecute, TaskObject, TaskRemove, TaskResume, Tasks, TaskSuspend, TaskWait, $CurrentTask

Ask (9)

Ask, AskAppend, AskConfirm, AskDisplay, AskedQ, AskedValue, AskFunction, AskState, AskTemplateDisplay

Code Flow (9)

Break, Continue, Goto, Label, LengthWhile, Once, Reap, Scan, Sow

Expression Testing (9)

Between, DuplicateFreeQ, EqualTo, GreaterEqualThan, GreaterThan, LessEqualThan, LessThan, Unequal, UnequalTo

Font (9)

FontColor, FontFamily, FontSize, FontSlant, FontSubstitutions, FontTracking, FontVariations, FontWeight, $FontFamilies

Fractal (9)

CantorStaircase, EscapeRadius, JuliaSetBoettcher, JuliaSetIterationCount, JuliaSetPoints, MandelbrotSetBoettcher, MandelbrotSetDistance, MandelbrotSetIterationCount, MandelbrotSetMemberQ

Grammar (9)

AllowLooseGrammar, AnyOrder, FixedOrder, GrammarApply, GrammarRules, GrammarToken, IgnoreDiacritics, NoWhitespace, OptionalElement

Persistent Object (9)

ExpirationDate, MergingFunction, PersistenceLocation, PersistenceTime, PersistentObject, PersistentObjects, PersistentValue, $PersistenceBase, $PersistencePath

Dataset (8)

Dataset, DeleteMissing, FailureAction, Missing, MissingBehavior, MissingQ, PartBehavior, Query

Expression Size (8)

ByteCount, Count, CountDistinct, CountDistinctBy, Counts, CountsBy, Entropy, LeafCount

Formatting (8)

DisableFormatting, Framed, Labeled, Legended, Multicolumn, Pane, Panel, TextGrid

Graph Distribution (8)

BarabasiAlbertGraphDistribution, BernoulliGraphDistribution, DegreeGraphDistribution, GraphPropertyDistribution, PriceGraphDistribution, SpatialGraphDistribution, UniformGraphDistribution, WattsStrogatzGraphDistribution

Page (8)

GroupPageBreakWithin, PageBreakAbove, PageBreakBelow, PageBreakWithin, PageFooters, PageHeaders, PageWidth, ShowPageBreaks

Web Session (8)

StartWebSession, WebElementObject, WebExecute, WebImage, WebSessionObject, WebSessions, WebWindowObject, $CurrentWebSession

Databin (7)

CopyDatabin, CreateDatabin, Databin, DatabinAdd, DatabinRemove, Databins, DatabinUpload

Interpolation (7)

FunctionInterpolation, InterpolatingFunction, InterpolatingPolynomial, Interpolation, InterpolationOrder, InterpolationPoints, ListInterpolation

Number (7)

Negative, NonNegative, NonPositive, NumericalOrder, NumericalSort, Positive, PossibleZeroQ

Quantity Variable (7)

ExcludedPhysicalQuantities, QuantityVariable, QuantityVariableCanonicalUnit, QuantityVariableDimensions, QuantityVariableIdentifier, QuantityVariablePhysicalQuantity, RequiredPhysicalQuantities

Series (7)

ComposeSeries, InverseSeries, O, PadeApproximant, Series, SeriesCoefficient, SeriesData

Symbol Value (7)

DownValues, FullDefinition, Information, OwnValues, SyntaxInformation, UpValues, ValueQ

Viewer (7)

FlipView, GalleryView, OpenerView, Pagination, PopupView, SlideView, TabView

Wolfram Alpha (7)

Asynchronous, ExcludePods, IncludePods, InputAssumptions, PodStates, PodWidth, WolframAlpha

Controller (6)

ControllerInformation, ControllerLinking, ControllerManipulate, ControllerMethod, ControllerPath, ControllerState

Local Object (6)

LocalObject, LocalObjects, LocalSymbol, $DefaultLocalBase, $LocalBase, $LocalSymbolBase

Numeric Constant (6)

Catalan, EulerGamma, Glaisher, GoldenAngle, GoldenRatio, Khinchin

Slider (6)

ColorSlider, IntervalSlider, MinIntervalSize, Slider, Slider2D, VerticalSlider

Test (6)

TestID, TestReport, TestReportObject, TestResultObject, VerificationTest, $TestFileName

Text String (6)

BooleanStrings, ElidedForms, ListFormat, MissingString, TextString, TimeFormat

Travel (6)

TravelDirections, TravelDirectionsData, TravelDistance, TravelDistanceList, TravelMethod, TravelTime

Continued Fraction (5)

ContinuedFraction, ContinuedFractionK, Convergents, FromContinuedFraction, QuadraticIrrationalQ

Infinity (5)

ComplexInfinity, DirectedInfinity, Indeterminate, Overflow, Underflow

Menu (5)

DefaultMenuStyle, MenuCommandKey, MenuSortingValue, MenuStyle, MenuView

Package (5)

BeginPackage, DeclarePackage, EndPackage, Needs, $Packages

Raw Expression (5)

Hash, NumericArray, NumericArrayQ, NumericArrayType, RawData

Set (5)

AnySubset, DisjointQ, IntersectingQ, SameTest, SubsetQ

Symbol Context (5)

Begin, Context, End, $Context, $ContextPath

Uncertainty (5)

Around, AroundReplace, ComputeUncertainty, MeanAround, VectorAround

Function (4)

ComposeList, Construct, Curry, InverseFunction

Real (4)

MantissaExponent, Rationalize, RealExponent, Reals

Rule (4)

Dispatch, ReplaceList, ReplaceRepeated, TwoWayRule

Scheduled Task (4)

ContinuousTask, IncludeGeneratorTasks, RestartInterval, ScheduledTask

Angle (3)

DMSList, DMSString, FromDMS

Asynchronous Task (3)

HandlerFunctions, HandlerFunctionsKeys, NotificationFunction

Complex (3)

AbsArg, Complexes, ReIm

Front End Execution (3)

FrontEndExecute, FrontEndToken, FrontEndTokenExecute

Interval (3)

IntervalIntersection, IntervalMemberQ, IntervalUnion

Special (3)

After, Before, Full

(3)

Remove, SymbolName, Unique

Autocomplete (2)

Autocomplete, AutocompletionFunction

(2)

AlternateImage, CDFDeploy

Global Options (2)

FindSettings, Language

Palette (2)

CreatePalette, PaletteNotebook

Rational (2)

MixedFractionParts, NumeratorDenominator

Module (1)

BlockRandom

Option (1)

AbsoluteOptions

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ClearAll[timing];

SetAttributes[timing, HoldAll];

timing[expr_]:=(ClearSystemCache[];

AbsoluteTiming[expr]);

ClearAll[compareTiming];

SetAttributes[compareTiming, HoldAll];

compareTiming[expr_List, OptionsPattern[{"OrderQ"->True}]]:=Module[{t, v, c1, c2, holdExpr, colorF, order, rescale}, 
colorF=Blend[{ColorSwatchGraphicsFontCapHeightRGBColor[19255, 167255, 107255]DeployedRGBColorValueSelectorClosingActionsSelectionDepartureParentChangedEvaluatorQuitPreemptive, ColorSwatchGraphicsFontCapHeightRGBColor[1, 101255, 47255]DeployedRGBColorValueSelectorClosingActionsSelectionDepartureParentChangedEvaluatorQuitPreemptive}, #]&;

holdExpr=ReleaseHold@Map[HoldForm, HoldForm@expr, {2}];

{t, v}=Transpose[timing[ReleaseHold[#]]&/@holdExpr];

order=If[OptionValue["OrderQ"], Reverse[Ordering[t]], Range[1, Length[expr]]];

t=t[[order]];

rescale=Evaluate[Rescale[#, MinMax[t], {0, 1}]]&;

v=v[[order]];

holdExpr=holdExpr[[order]];

If[Length[DeleteDuplicates[v]]>1, Echo["    "]];

Framed[Grid[{{Style[#, Bold]&@"      Wolfram Language"}, 
{Grid[{{"", ",  "}}~Join~Table[{Style[holdExpr[[ind]], "Input", Background->Transparent], t[[ind]]}, {ind, 1, Length[expr]}], Alignment->{{Left, Center}, {Center, Center}}, Dividers->White, Background->{None, {LightGray}~Join~(colorF@*rescale/@t)}, ItemSize->{{30, 8}, Automatic}]}, 
If[Length[expr]==2, {Row[{"   ", Style[Round[Max[t]/Min[t], 1/100]//N, Bold], " "}]}, {Row[{"   ", Style[Round[Max[t]/Min[t], 1/100]//N, Bold], " "}]}], 
{Graphics[
Table[{Opacity[0.6], colorF@*rescale@t[[ind]], Rectangle[{0, 1-ind-0.2(ind-1)}, {t[[ind]], 2-ind-0.2(ind-1)}]}, {ind, 1, Length[expr]}], 
ImageSize->{500, Automatic}, PlotRange->{{0, Max[t]}, {1.2(1-Length[expr]), 1}}, AspectRatio->1/(9-Length[expr]/2), 
Axes->False, Ticks->{Automatic, None}, FrameTicks->{All, None}, 
GridLines->{Automatic, None}, Background->White, PlotRangePadding->0, Frame->True, FrameLabel->{{None, None}, {",  ", None}}]}
}, 
ItemStyle->Directive[FontSize -> 14, FontFamily->"Open Sans Light"]], Background->White, FrameStyle->None]];

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The cornerstones of the destruction of slow code in Wolfram Language: speeding up code tens, hundreds and thousands of times

1.

1.1. , ,

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1.4.

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2.2.

2.3.

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3.1.

3.2. Compile (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({}); (adsbygoogle = window.adsbygoogle || []).push({});

3.3. Compile

3.4.

3.5. CUDA, OpenCL — ()

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6.2. Part

6.3. Extract

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8.2. (closures)

8.3.

9.

9.1. Reap + Sow

9.2.

9.3.

9.4. $HistoryLength

Wolfram Language

Math Functions (279)

Basic (243)

Image (183)

Graph Theory (179)

Statistical Distribution (152)

Graphics (126)

Region (125)

String (115)

Plotting (110)

Statistics (110)

Paclet (103)

Cloud (97)

Net (90)

Control System (88)

Operator (86)

Front End (85)

Cell (78)

File System (78)

Comparison Operator (73)

Color (71)

Audio (69)

Linear Algebra (62)

Machine Learning (60)

Notebook (60)

Date (59)

Clustering (52)

Graphics Primitive (51)

Form (48)

Stream (44)

Statistical Test (42)

Statistical Process (41)

Chart (39)

Polynomial (39)

Arrow Operator (38)

Machine (38)

(36)

Image Filter (35)

Parallel (35)

3.2. `Compile`