Sieve of Eratosthenes in C++

#include <stdio.h>
#include <vector>
#include <algorithm>
#include <stdexcept>
#include <cstdint>
#include <limits>
// Sieve of Eratosthenes in C++
//
// PURPOSE
// The sieve identifies all primes up to a limit by marking multiples of smaller primes. This
// example returns their count.
//
// METHOD
// Keep a composite marker array. For each unmarked p up to the square root, mark multiples
// starting at p squared; smaller multiples were already handled. Count the remaining
// unmarked values from two onward.
//
// COMPLEXITY
// O(n log log n) time and O(n) auxiliary space.
//
// CONTRACT
// n >= 0. Returns the number of primes <= n. The marker array requires memory proportional
// to n.
// 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
// C++17: const vector references for inputs, mutable references for outputs, RAII workspace,
// standard lower_bound for LIS. Invalid inputs throw invalid_argument; overflow throws
// overflow_error; allocation failure propagates bad_alloc.
//
// EXAMPLE
// Expected return: 4.
// Run: g++ -std=c++17 -O2 example.cpp -o example && ./example
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

static int alg_min(int a, int b) { return a < b ? a : b; }
static int alg_max(int a, int b) { return a > b ? a : b; }
static void alg_require(bool valid) { if (!valid) throw std::invalid_argument("invalid algorithm input"); }
static int alg_checked(int64_t v) { if (v < INT32_MIN || v > INT32_MAX) throw std::overflow_error("32-bit result overflow"); return (int)v; }
static int alg_add(int a,int b) { return alg_checked((int64_t)a+b); }
static int alg_sub(int a,int b) { return alg_checked((int64_t)a-b); }
static int alg_mul(int a,int b) { return alg_checked((int64_t)a*b); }
static size_t alg_size(int n) { alg_require(n>=0 && n<=4000000); return (size_t)n; }

int sieve_count(int n) {
    alg_require(((n >= 0) && (n <= 1000000)));
    std::vector<int> composite(alg_size(alg_add(n, 1)), 0);
    int p = 2;
    int multiple = 0;
    int count = 0;
    while ((p <= (n / p))) {
        if ((composite[p] == 0)) {
            multiple = alg_mul(p, p);
            while ((multiple <= n)) {
                composite[multiple] = 1;
                multiple = alg_add(multiple, p);
            }
        }
        p = alg_add(p, 1);
    }
    p = 2;
    while ((p <= n)) {
        if ((composite[p] == 0)) {
            count = alg_add(count, 1);
        }
        p = alg_add(p, 1);
    }
    return count;
}

int main(void) {
    int n = 10;
    int result = sieve_count(n);
    printf("%d\n", result);
    return 0;
}