Coin Change Combination Count in PHP

<?php
declare(strict_types=1);
// Coin Change Combination Count in PHP
//
// PURPOSE
// This algorithm counts unordered combinations of reusable denominations. Putting the coin
// loop outside the amount loop prevents counting different orders as different combinations.
//
// METHOD
// There is one way to make zero: choose no coins. Process each denomination once and update
// amounts upward so that the same denomination can be reused.
//
// COMPLEXITY
// O(m * amount) time and O(amount) auxiliary space; pseudo-polynomial.
//
// CONTRACT
// Denominations are distinct positive integers, amount >= 0. Counts and array sizes must
// fit. Duplicate denominations would overcount.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// PHP 8.1+ on a 64-bit runtime: packed arrays, strict scalar types, read-only inputs by
// value using copy-on-write, references only for outputs, and checked 32-bit arithmetic.
// Invalid input and overflow throw exceptions. PHP arrays have higher overhead than native
// packed buffers in C/Java.
//
// EXAMPLE
// Expected return: 4.
// Run: php example.php
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(bool $valid): void { if (!$valid) throw new InvalidArgumentException("invalid algorithm input"); }
function alg_checked(int|float $v): int { if ($v < -2147483648 || $v > 2147483647 || floor($v) != $v) throw new OverflowException("32-bit result overflow"); return (int)$v; }
function alg_add(int $a,int $b): int { return alg_checked($a+$b); }
function alg_sub(int $a,int $b): int { return alg_checked($a-$b); }
function alg_mul(int $a,int $b): int { return alg_checked($a*$b); }
function alg_size(int $n): int { alg_require($n>=0 && $n<=4000000); return $n; }

function coin_combinations(array $coins, int $amount): int {
    alg_require(array_is_list($coins));
    $m = count($coins);
    alg_require((($m >= 0) && ($m <= 1000000)));
    alg_require((count($coins) == $m));
    alg_require((($amount >= 0) && ($amount <= 1000000)));
    for ($alg_i = 0; $alg_i < count($coins); $alg_i++) {
        $alg_value = $coins[$alg_i];
        alg_require(($alg_value > 0));
    }
    $dp = array_fill(0, alg_size(alg_add($amount, 1)), 0);
    $dp[0] = 1;
    $j = 0;
    $x = 0;
    while (($j < $m)) {
        $x = $coins[$j];
        while (($x <= $amount)) {
            $dp[$x] = alg_add($dp[$x], $dp[alg_sub($x, $coins[$j])]);
            $x = alg_add($x, 1);
        }
        $j = alg_add($j, 1);
    }
    return $dp[$amount];
}

$coins = [1, 2, 5];
$amount = 5;
$result = coin_combinations($coins, $amount);
echo $result, PHP_EOL;