compile - List of compilable functions

Is there somewhere a list on the functions that Compile can compile, or the cases in which a particular function can be compiled that I haven't found? I'd be glad even with a list of some of them which surprisingly aren't compilable, and how to do without them.

I am not happy every time I have to rewrite or redesign code because it seems to make external calls for functions I didn't expect. I'd like to know how you handle all that, what you keep in mind.

Answer

Yes, but this only exists in version 8 onwards and is undocumented:

Compile`CompilerFunctions[] // Sort

giving, for reference:

{Abs, AddTo, And, Append, AppendTo, Apply, ArcCos, ArcCosh, ArcCot, ArcCoth, ArcCsc,
 ArcCsch, ArcSec, ArcSech, ArcSin, ArcSinh, ArcTan, ArcTanh, Arg, Array, ArrayDepth,

 Internal`Bag, Internal`BagPart, BitAnd, BitNot, BitOr, BitXor, Block, BlockRandom, Boole,
 Break, Cases, Catch, Ceiling, Chop, Internal`CompileError, System`Private`CompileSymbol,
 Complement, ComposeList, CompoundExpression, Conjugate, ConjugateTranspose, Continue,
 Cos, Cosh, Cot, Coth, Count, Csc, Csch, Decrement, Delete, DeleteCases, Dimensions,
 Divide, DivideBy, Do, Dot, Drop, Equal, Erf, Erfc, EvenQ, Exp, Fibonacci, First,
 FixedPoint, FixedPointList, Flatten, NDSolve`FEM`FlattenAll, Floor, Fold, FoldList, For,
 FractionalPart, FreeQ, Compile`GetElement, Goto, Greater, GreaterEqual, Gudermannian,
 Haversine, If, Im, Implies, Increment, Inequality, Compile`InnerDo, Insert,
 IntegerDigits, IntegerPart, Intersection, InverseGudermannian, InverseHaversine,
 Compile`IteratorCount, Join, Label, Last, Length, Less, LessEqual, List, Log, Log10,

 Log2, LucasL, Map, MapAll, MapAt, MapIndexed, MapThread, NDSolve`FEM`MapThreadDot,
 MatrixQ, Max, MemberQ, Min, Minus, Mod, Compile`Mod1, Module, Most, N, Negative, Nest,
 NestList, NonNegative, Not, OddQ, Or, OrderedQ, Out, Outer, Part, Partition, Piecewise,
 Plus, Position, Positive, Power, PreDecrement, PreIncrement, Prepend, PrependTo, Product,
 Quotient, Random, RandomChoice, RandomComplex, RandomInteger, RandomReal, RandomSample,
 RandomVariate, Range, Re, ReplacePart, Rest, Return, Reverse, RotateLeft, RotateRight,
 Round, RuleCondition, SameQ, Scan, Sec, Sech, SeedRandom, Select, Set, SetDelayed,
 Compile`SetIterate, Sign, Sin, Sinc, Sinh, Sort, Sqrt, Internal`Square, Internal`StuffBag,
 Subtract, SubtractFrom, Sum, Switch, Table, Take, Tan, Tanh, TensorRank, Throw, Times,
 TimesBy, Tr, Transpose, Unequal, Union, Unitize, UnitStep, UnsameQ, VectorQ, Which,

 While, With, Xor}

As of Mathematica 10.0.2, there are also the following functions:

{Gamma, Indexed, LogGamma, LogisticSigmoid, Internal`ReciprocalSqrt}

As of Mathematica 11, there are also the following functions:

{Internal`Expm1, Internal`Log1p, Ramp}

As of Mathematica 11.2, there are also the following functions:

{RealAbs, RealSign}

About Tr:

Please note that Tr appears in this list, but cannot actually be compiled without a call to MainEvaluate[]. It is unclear if this is deliberate or a bug.

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Edit: additional functions

I have just discovered the symbol Internal`CompileValues, which provides various definitions and function calls needed to compile further functions not in the list above. Using the following code,

Internal`CompileValues[]; (* to trigger auto-load *)
ClearAttributes[Internal`CompileValues, ReadProtected];
syms = DownValues[Internal`CompileValues] /. 
    HoldPattern[Verbatim[HoldPattern][Internal`CompileValues[sym_]] :> _] :>

        sym;
Complement[syms, Compile`CompilerFunctions[]]

we get some more compilable functions as follows:

{Accumulate, ConstantArray, Cross, Depth, Det, DiagonalMatrix,
Differences, NDSolve`FEM`FEMDot, NDSolve`FEM`FEMHold,
NDSolve`FEM`FEMInverse, NDSolve`FEM`FEMPart, NDSolve`FEM`FEMTDot,
NDSolve`FEM`FEMTotalTimes, NDSolve`FEM`FEMZeroMatrix, FromDigits,
Identity, IdentityMatrix, Inverse, LinearSolve, Mean, Median, Nand,
NestWhile, NestWhileList, Nor, Norm, Ordering, PadLeft, PadRight,

Permutations, Ratios, Signature, SquareWave, StandardDeviation,
Tally, Total, TrueQ, Variance}

Looking at the definition of Internal`CompileValues[sym] for sym in the list above will provide some additional information about how these functions are compiled. This can range from type information (for e.g. Inverse), through to an implementation in terms of lower-level functions (e.g. NestWhileList). One can presumably also make one's own implementations of non-compilable functions using this mechanism, giving Compile the ability to compile a wider range of functions than it usually would be able to.

As of Mathematica 10.3, there are also the following functions:

{DeleteDuplicates, Region`Mesh`SmallMatrixRank,
 Region`Mesh`SmallQRSolve, Region`Mesh`SmallSingularValues,
 Region`Mesh`SmallSingularValueSystem, Region`Mesh`SmallSVDSolve,
 NDSolve`SwitchingVariable}

As of Mathematica 11, there are also the following functions:

{NearestFunction, RegionDistanceFunction, RegionMemberFunction, RegionNearestFunction}

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Edit 2: the meaning of the second list

In response to a recent question, I want to be clear that the presence of a function in the second list given above does not necessarily mean it can be compiled into a form free of MainEvaluate calls. If a top-level function is already highly optimized (as e.g. LinearSolve is), the purpose of Internal`CompileValues[func] may be solely to provide type information on the return value, assuming that this can be inferred from the types of the arguments or some other salient information. This mechanism allows more complex functions that call these highly-optimized top-level functions to be compiled more completely since there is no longer any question of what the return type may be and so further unnecessary MainEvaluate calls may be avoided. It does not imply that the use of MainEvaluate is unnecessary to call the function itself.