Inspired by some Conor Hoekstra YouTube videos, I tried doing some baby steps in APL and also convert my small lines to point-free style. But for this (percentage of rolls of 4, 5, or 6 in 1000 die-6 rolls) I can't wrap my head around how to eliminate the omega before the reshape.
{(+/3<?⍵⍴6)×100÷⍵}1000
Let's take it step by step:
{(+/3<?⍵⍴6)×100÷⍵}
First we need to express every part of the function that uses the argument, as a function of the argument. The multiplication combines the two main parts:
{+/3<?⍵⍴6}×{100÷⍵}
In the rightmost part, {100÷⍵}
, we need the argument. There are a couple of ways we can deal with this:
⊢
to represent it: 100÷⊢
100
, to the function ÷
yielding a monadic function: 100∘÷
Let's take the last approach:
{+/3<?⍵⍴6}×100∘÷
In the left part, {+/3<?⍵⍴6}
, we can do the same, but need to watch out for two things, and each can be dealt with in a few different ways:
6
, as the rightmost part of our function.
6⍨
⍴
and use an identity function: 6⍴⍨⊢
6
, to the function ⍴
yielding a monadic function: ⍴∘6
?
, in the middle.
?
atop ⍴
: ?⍤⍴
?
beside <
: <∘?
Let's take the last approach for each problem:
(+/3<∘?⍴∘6)×100∘÷
This is a fully tacit equivalent to the monadic function {(+/3<?⍵⍴6)×100÷⍵}
. However, there's one more trick we can use to eliminate the parenthesis. Since ×
is commutative, we can swap its arguments to put the more involved expression on the right:
100∘÷×(+/3<∘?⍴∘6)
However, now we have the problem of the monadic +/
in the middle. Observe that <
sees a vector on the right and a scalar on the left. In the case of F/s G v
for scalar functions F
and G
with scalar s
and vector v
the inner product s F.G v
is equivalent, so we can combine the summation with the comparison as follows:
100∘÷×3+.<∘?⍴∘6
Alternatively, we can observer that summation is equivalent to evaluation in base 1 because the place values in base 1 are (…,12, 11, 10) = (…, 1, 1, 1) so if we have the list (…, c, b, a) and evaluate it as a number in base 1, we get:
(… + c×12 + b×11 + a×10) =
(… + c×1 + b×1 + a×10) =
(… + c + b×1 + a×1) =
(… + c + b + a)
That is, the sum of our list. We can write this as:
100∘÷×1⊥3<∘?⍴∘6
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