How do I count the number of taxonomy terms which have both a) an associated post and b) where that post also has an associated term from another taxonomy?

Specifically, I have...

• Custom post type "article" (9,000+ posts)
• Custom taxonomy "format" (11 terms)
• Custom taxonomy "company" (1,624 terms)

An article is taggable with both a "format" and a "company".

I want to get a count of the number of "companies" about which there is an "article" with the "format" term "viewpoint" or, more importantly, any of its 3 children. How can I do this?

Ordinarily, I think the question would be "How do I count the number of 'articles' tagged with any 'company'"? Except that I would like specifically those qualifying articles to be of the "format" taxonomy's "viewpoint" term or one of its children.

Am I thinking of this backward? Is this about asking: "Find all 'articles' tagged with both any 'company' and 'format' of 'viewpoint', and then return a count of those 'articles'"?

I think I should avoid a WP_Query as we are talking about a large number of items here, especially for "article". What is an efficient way?

I feel like get_object_term coupled with children may be useful here?

The following code does something different (counts the number of objects with "viewpoint" or a child alone. But maybe it's a start...

  // Get Viewpoints total count
  $term = get_term_by('slug', 'viewpoint', 'format');
  $id = $term->term_id;
  $count = 0;
  $taxonomy = 'format';
  $args = array('child_of' => $id);
  $tax_terms = get_terms($taxonomy,$args);
  foreach ($tax_terms as $tax_term) {
    if ($tax_term->parent == $id) {
          $count +=$tax_term->count;
    }
  }