Table with programmatically specified number of iterators in Compile

Question

Compare two ways of computing a table of values of a multivariable Gaussian:

First@AbsoluteTiming[
  ta = Table[Exp[-Total[Array[r, 5]^2]],
  {r[1], 1., 2.},
  {r[2], 1., 3.},
  {r[3], 1., 3.},
  {r[4], 1., 23.},
  {r[5], 1., 421.}
  ];
]
(* 2.1069 *)

comp = Compile[{},
  Table[Exp[-(r[1]^2 + r[2]^2 + r[3]^2 + r[4]^2 + r[5]^2)],
  {r[1], 1., 2.},
  {r[2], 1., 3.},
  {r[3], 1., 3.},
  {r[4], 1., 23.},
  {r[5], 1., 421.}
  ], CompilationTarget->"C"
];
First@AbsoluteTiming[
  tb = comp[];
]
(* 0.002853 *)

I would like to get such a speedup in a case where I specify the number of iterators/variables programmatically, and where the iterator values are stored in a list itvals. Non-compiled is easy, e.g.:

n = 5; (* number of iterators *)
itvals = RandomReal[{0,1}, {n, 8}];
tc = Fold[Table[#1,{r[#2],itvals[[#2]]}]&, Exp[-Total[Array[r,n]^2]], Range[n]];

Or:

td = Table[Exp[-Total[Array[r,n]^2]],##] & @@ Table[{r[i], itvals[[i]]}, {i,n,1,-1}];

tc == td
(* True *)

But I can't get this to work inside Compile! If I try

c1 = Compile[{{n, _Integer}},
  Block[{itvals},
   itvals = RandomReal[{0, 1}, {n, 5}];
   Fold[Table[#1, {x[#2], itvals[[#2]]}] &, 1, Range[n]]
   ]
  ]

where I use the "function" 1 instead of the Gaussian for simplicity. CompilePrint[c1] shows that the whole Fold is wrapped in a MainEvaluate. I can't do it similarly to td either because Compile only supports @@ with the functions Plus, Times or List...

Anybody know a way to do it?

Answer 1

Verily, this is a headache, since Table wants its iterators as Sequence rather than nested list of lists. Here is a method I used quite recently to get sufficiently fast code for this.

c1c[n_] := 
 With[{itvals = RandomReal[{0, 1}, {n, 5}]}, 
  With[{iters = Apply[Sequence, Table[{x[j], itvals[[j]]}, {j, n}]]},
   c1[n] = cCompile[{},
      tTable[1, iters]] /. {cCompile -> Compile, tTable -> Table}]]

In[55]:= c1c[2][]

(* Out[55]= {{1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1, 1, 1}, {1, 1, 1,
   1, 1}, {1, 1, 1, 1, 1}} *)