Segmented Sieve in PHP

<?php
declare(strict_types=1);
// Segmented Sieve in PHP
//
// PURPOSE
// A segmented sieve marks composites only inside a requested interval, reducing memory
// compared with sieving all the way to its upper endpoint.
//
// METHOD
// Sieve base primes up to square root of high. For each prime, mark its multiples in
// [low,high], starting no earlier than p squared. Count unmarked values at least two.
//
// COMPLEXITY
// O(sqrt(high) log log high + (high-low+1) log log high) time, O(sqrt(high)+high-low+1)
// space.
//
// CONTRACT
// 0 <= low <= high <= 1000000. Returns number of primes in the inclusive interval.
// 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: 6.
// 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 segmented_sieve(int $low, int $high): int {
    alg_require((($low >= 0) && ($low <= $high) && ($high <= 1000000)));
    $limit = 1;
    while (($limit <= intdiv($high, $limit))) {
        $limit = alg_add($limit, 1);
    }
    $base = array_fill(0, alg_size(alg_add($limit, 1)), 0);
    $marked = array_fill(0, alg_size(alg_add(alg_sub($high, $low), 1)), 0);
    $p = 2;
    $multiple = 0;
    $count = 0;
    $i = 0;
    while (($p < $limit)) {
        if (($base[$p] == 0)) {
            $multiple = alg_mul($p, $p);
            while (($multiple < $limit)) {
                $base[$multiple] = 1;
                $multiple = alg_add($multiple, $p);
            }
            $multiple = max(alg_mul($p, $p), alg_mul(intdiv(alg_sub(alg_add($low, $p), 1), $p), $p));
            while (($multiple <= $high)) {
                $marked[alg_sub($multiple, $low)] = 1;
                $multiple = alg_add($multiple, $p);
            }
        }
        $p = alg_add($p, 1);
    }
    $i = max(2, $low);
    while (($i <= $high)) {
        if (($marked[alg_sub($i, $low)] == 0)) {
            $count = alg_add($count, 1);
        }
        $i = alg_add($i, 1);
    }
    return $count;
}

$low = 10;
$high = 30;
$result = segmented_sieve($low, $high);
echo $result, PHP_EOL;