Coin Change Combination Count in C

#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
#include <limits.h>
#include <errno.h>
// Coin Change Combination Count in C
//
// 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
// C99: checked calloc/free workspace, const input pointers, explicit size_t lengths. Errors
// return INT_MIN with errno=EINVAL, ERANGE, or ENOMEM. Check errno before using a result.
// Input/output arrays must not alias. No variable-length stack arrays.
//
// EXAMPLE
// Expected return: 4.
// Run: gcc -std=c99 -O2 example.c -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 int alg_checked(int64_t v, int *error) {
    if (v < INT_MIN || v > INT_MAX) { *error = 1; return 0; }
    return (int)v;
}
static int alg_add(int a, int b, int *e) { return alg_checked((int64_t)a+b,e); }
static int alg_sub(int a, int b, int *e) { return alg_checked((int64_t)a-b,e); }
static int alg_mul(int a, int b, int *e) { return alg_checked((int64_t)a*b,e); }

int coin_combinations(const int *coins, size_t coins_length, int amount) {
    int alg_error = 0;
    int alg_result = INT_MIN;
    errno = 0;
    int *dp = NULL;
    if (coins_length > 1000000 || (coins_length && !coins)) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    int m = (int)coins_length;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((m >= 0) && (m <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((int)coins_length == m))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    if (!(((amount >= 0) && (amount <= 1000000)))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    for (int alg_i = 0; alg_i < (int)coins_length; alg_i++) {
        int alg_value = coins[alg_i];
        if (!((alg_value > 0))) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    {
        int alg_length = alg_add(amount, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (alg_length < 0 || alg_length > 4000000) { errno = EINVAL; alg_result = INT_MIN; goto alg_cleanup; }
        dp = calloc((size_t)(alg_length ? alg_length : 1), sizeof(*dp));
        if (!dp) { errno = ENOMEM; alg_result = INT_MIN; goto alg_cleanup; }
    }
    dp[0] = 1;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int j = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    int x = 0;
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    while (1) {
        int alg_condition = (j < m);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        if (!alg_condition) break;
        x = coins[j];
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        while (1) {
            int alg_condition = (x <= amount);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            if (!alg_condition) break;
            dp[x] = alg_add(dp[x], dp[alg_sub(x, coins[j], &alg_error)], &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
            x = alg_add(x, 1, &alg_error);
            if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
        }
        j = alg_add(j, 1, &alg_error);
        if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    }
    alg_result = dp[amount];
    if (alg_error) { errno = ERANGE; alg_result = INT_MIN; goto alg_cleanup; }
    goto alg_cleanup;
alg_cleanup:
    free(dp);
    return alg_result;
}

int main(void) {
    int coins[] = {1, 2, 5};
    int amount = 5;
    int result = coin_combinations(coins, 3, amount);
    printf("%d\n", result);
    if (errno) { perror("algorithm"); return 1; }
    return 0;
}