The Non-Zero Digital Product Challenge

Haskell, 27 bytes

foldr((*).max 1.read.pure)1

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Ungolfed with UniHaskell and -XUnicodeSyntax

import UniHaskell

f ∷ String → Int
f = product ∘ map (max 1 ∘ read ∘ pure)

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Explanation

I'll start with what I initially had:

product.map(max 1.read.pure)

This is a point-free expression that evaluates to a function taking a string (or a list of characters) s ("301") as an argument. It maps max 1.read.pure over s, essentially taking each character i, injecting it into a list (which makes it a string) (["3", "0", "1"]), then reading it, which evaluates the string ([3, 0, 1]) and finally taking the greater of i and 1 ([3, 1, 1]). Then it takes the product of the resulting list of integers (3).

I then golfed it by a byte with:

foldr((*).max 1.read.pure)1

This works because product is equivalent to foldr (*) 1. Instead of mapping and folding, I combined the two by folding with (*).max 1.read.pure which takes each non-zero digit and multiplies it with the accumulator.

Python 2, 34 bytes

f=lambda n:n<1or(n%10or 1)*f(n/10)

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Jelly, 4 bytes

Do1P

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How it works

Do1P - Main link. Argument: n (integer)  e.g. 1230456
D    - Digits                                 [1, 2, 3, 0, 4, 5, 6]
 o1  - Replace 0 with 1                       [1, 2, 3, 1, 4, 5, 6]
   P - Product                                720