Skip to content

Through

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Through[p[f1, f2, ...][x1, x2, ...]]
    distributes the trailing argument list across the inner functions,
    giving p[f1[x1, x2, ...], f2[x1, x2, ...], ...].
Through[expr, h]
    distributes only when the outer head equals h.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Through[{f, g, h}[x]]
Out[1]= {f[x], g[x], h[x]}

In[2]:= Through[(f + g)[x, y]]
Out[2]= f[x, y] + g[x, y]

Implementation notes

Algorithm. builtin_through implements Through[p[f1,f2,...][x,y,...]] -> p[f1[x,y,...], f2[x,y,...], ...]: it threads the argument tuple of the outer call through each operator sitting in the (compound) head. For the one-argument form it requires the head itself to be a function p[f1,...]; it builds each fi[args...], evaluates it, and reassembles under p. The optional second argument h restricts the rewrite to heads equal to h: transform_head recurses through the head structure, applying the threading only at sub-heads whose head matches h (via expr_eq) and copying everything else.

Whenever a transformation actually happened the rebuilt expression is passed through evaluate() so the inner fi[...] calls take effect; if nothing matched the original expression is returned unchanged (transformed flag). Atoms are returned by expr_copy.

Data structures. Expr-tree manipulation only; each inner call is built with expr_new_function over copied argument arrays.

  • Protected.

Attributes: none registered.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= Through[(f + g)[x]]
Out[1]= f[x] + g[x]

In[1]:= Through[{Sin, Cos, Tan}[Pi/4]]
Out[1]= {1/Sqrt[2], 1/Sqrt[2], 1}

In[1]:= f[x_] := x^2; g[x_] := x + 1; Through[(f + g)[3]]
Out[1]= 13

Notes

Through[p[f1, f2, ...][x]] distributes the trailing arguments across each inner operator, producing p[f1[x], f2[x], ...]. With p = Plus it turns a sum of functions into a function of a point — so once f and g are defined, Through[(f + g)[3]] evaluates f[3] + g[3]. Applying it to a list of heads ({Sin, Cos, Tan}) evaluates all of them at the same argument at once. Through[expr, h] distributes only when the outer head equals h.