# Project Euler 824
# Chess Sliders on a cylindrical board.
# L(10^9, 10^15) mod (10^7+19)^2.
# Pure Flow port of the native C solver.
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function malloc(n: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const P: i64 = 10000019
const MOD: i64 = 100000380000361
let mut invp: ptr<i32> = null
let mut fac: ptr<i64> = null
let mut H: ptr<i32> = null
let mut w: i64 = 0
function mulmod(a: i64, b: i64, m: i64) -> i64 {
let r: i128 = ((a as i128) * (b as i128)) % (m as i128)
return r as i64
}
function inv_mod_p2(a0: i64) -> i64 {
let a: i64 = a0 % MOD
let r: i64 = a % P
let x: i64 = invp[r] as i64
let ax: i64 = mulmod(a, x, MOD)
let term: i64 = (2 + MOD - ax) % MOD
return mulmod(x, term, MOD)
}
function Fpow(u: i64) -> i64 {
let um: i64 = u % P
let a1: i64 = um * P
let a2: i64 = mulmod(a1, w, MOD)
let mut tmp: i64 = (1 + MOD - a2) % MOD
if u % 2 == 1 { tmp = (MOD - tmp) % MOD }
return tmp
}
function unit_factorial(n0: i64) -> i64 {
let mut n: i64 = n0
let mut res: i64 = 1
while n != 0 {
let u: i64 = n / P
let v: i64 = n % P
res = mulmod(res, fac[v], MOD)
let corr: i64 = (u % P) * (H[v] as i64) % P
res = mulmod(res, 1 + corr * P, MOD)
res = mulmod(res, Fpow(u), MOD)
n = u
}
return res
}
function vp_fact(n: i64) -> i64 {
let q: i64 = n / P
return q + q / P
}
function binom_mod_p2(n: i64, k: i64) -> i64 {
if k > n { return 0 }
let nk: i64 = n - k
let e: i64 = vp_fact(n) - vp_fact(k) - vp_fact(nk)
if e >= 2 { return 0 }
let un: i64 = unit_factorial(n)
let uk: i64 = unit_factorial(k)
let unk: i64 = unit_factorial(nk)
let mut val: i64 = mulmod(un, inv_mod_p2(uk), MOD)
val = mulmod(val, inv_mod_p2(unk), MOD)
if e == 1 { val = mulmod(val, P, MOD) }
return val
}
function coeff_alpha_power(M: i64, d: i64) -> i64 {
if d == 0 { return 1 }
let B: i64 = binom_mod_p2(M - d - 1, d - 1)
let val: i64 = mulmod(M % MOD, inv_mod_p2(d % MOD), MOD)
return mulmod(val, B, MOD)
}
function main() -> i32 {
invp = calloc(P, 4) as ptr<i32>
H = calloc(P, 4) as ptr<i32>
fac = malloc(P * 8) as ptr<i64>
if invp == null || fac == null || H == null {
return 1
}
# Inverses mod p
invp[1] = 1
let mut i: i64 = 2
while i < P {
invp[i] = ((P - (P / i) * (invp[P % i] as i64) % P) % P) as i32
i = i + 1
}
# Factorial mod p^2 for 0..p-1
let mut f: i64 = 1
fac[0] = 1
let mut i2: i64 = 1
while i2 < P {
f = mulmod(f, i2, MOD)
fac[i2] = f
i2 = i2 + 1
}
# Harmonic numbers H[v] = sum_{i<=v} 1/i mod p
let mut h: i32 = 0
let mut i3: i64 = 1
while i3 < P {
h = ((h as i64 + invp[i3] as i64) % P) as i32
H[i3] = h
i3 = i3 + 1
}
# Wilson quotient: (p-1)! = -1 + p*w (mod p^2)
let F: i64 = fac[P - 1]
w = ((F + 1) / P) % P
# Main computation
let N: i64 = 1000000000
let K: i64 = 1000000000000000
let t_max: i64 = K / N
let mut comb: i64 = 1
let mut ans: i64 = 0
let mut d: i64 = K
let mut M: i64 = N * N
let mut t: i64 = 0
while t <= t_max {
if t != 0 {
comb = mulmod(comb, (N - t + 1) % MOD, MOD)
comb = mulmod(comb, inv_mod_p2(t), MOD)
}
ans = (ans + mulmod(comb, coeff_alpha_power(M, d), MOD)) % MOD
d = d - N
M = M - 2 * N
t = t + 1
}
free(invp as ptr<void>)
free(fac as ptr<void>)
free(H as ptr<void>)
printf("%lld\n", ans)
return 0
}
Generated C
#include <stdint.h>
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
/* Flow runtime helpers */
typedef struct flow_temp_node { struct flow_temp_node* next; } flow_temp_node;
static flow_temp_node* flow_temp_head = NULL;
static int flow_temp_atexit_set = 0;
__attribute__((unused)) static void flow_temp_free_all(void) {
while (flow_temp_head) {
flow_temp_node* n = flow_temp_head;
flow_temp_head = n->next;
free(n);
}
}
__attribute__((unused)) static void* flow_temp_alloc(size_t nbytes) {
flow_temp_node* node = (flow_temp_node*)malloc(sizeof(flow_temp_node) + nbytes);
if (!node) return NULL;
node->next = flow_temp_head;
flow_temp_head = node;
if (!flow_temp_atexit_set) {
flow_temp_atexit_set = 1;
atexit(flow_temp_free_all);
}
return (void*)(node + 1);
}
#ifndef FLOW_DIAG
#define FLOW_DIAG(msg) fprintf(stderr, "%s", (msg))
#endif
#ifndef FLOW_LOG
#define FLOW_LOG(fmt, ...) printf(fmt, __VA_ARGS__)
#endif
#ifndef FLOW_LOG_EMPTY
#define FLOW_LOG_EMPTY(fmt) printf(fmt)
#endif
static char* flow_strcat(const char* a, const char* b) {
size_t la = strlen(a ? a : ""), lb = strlen(b ? b : "");
char* r = (char*)flow_temp_alloc(la + lb + 1);
if (!r) return NULL;
if (la) memcpy(r, a, la);
if (lb) memcpy(r + la, b, lb);
r[la + lb] = '\0';
return r;
}
#define __flow_in_arr(arr, val) __extension__ ({ \
int _found = 0; \
size_t _n = sizeof(arr)/sizeof((arr)[0]); \
for (size_t _i = 0; _i < _n; _i++) { \
if ((arr)[_i] == (val)) { _found = 1; break; } \
} _found; })
/* Unified fault handler (MISRA #279) — override with -DFLOW_FAULT_HANDLER=fn */
#ifndef FLOW_FAULT_HANDLER
__attribute__((unused)) static inline void flow_fault_handler(const char* msg) {
fprintf(stderr, "flow: %s\n", msg ? msg : "fault");
abort();
#if defined(__GNUC__) || defined(__clang__)
__builtin_unreachable();
#endif
}
#else
#define flow_fault_handler FLOW_FAULT_HANDLER
#endif
#define flow_div_by_zero_handler() flow_fault_handler("division by zero")
#define flow_shift_ub_handler() flow_fault_handler("invalid shift (amount out of range or left-shift of negative)")
#ifndef FLOW_CHECKED_DIV
#define FLOW_CHECKED_DIV(L, R) (((R) != 0) ? ((L) / (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_MOD
#define FLOW_CHECKED_MOD(L, R) (((R) != 0) ? ((L) % (R)) : (flow_div_by_zero_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHL
#define FLOW_CHECKED_SHL(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull)) && ((L) >= 0)) ? ((L) << (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#ifndef FLOW_CHECKED_SHR
#define FLOW_CHECKED_SHR(L, R) ((((R) >= 0) && ((unsigned long long)(R) < (sizeof(L) * 8ull))) ? ((L) >> (R)) : (flow_shift_ub_handler(), (L) * 0))
#endif
#include <math.h>
void* _ui_state = NULL;
static inline float i32_to_f32(int32_t v) { return (float)v; }
/* Host stub for @gpu kernels (device codegen replaces this). */
static inline int32_t gpu_thread_id(void) { return 0; }
int64_t mulmod_i64_i64_i64(int64_t a, int64_t b, int64_t m);
int64_t inv_mod_p2_i64(int64_t a0);
int64_t Fpow_i64(int64_t u);
int64_t unit_factorial_i64(int64_t n0);
int64_t vp_fact_i64(int64_t n);
int64_t binom_mod_p2_i64_i64(int64_t n, int64_t k);
int64_t coeff_alpha_power_i64_i64(int64_t M, int64_t d);
int32_t main(void);
static const int64_t P = 10000019;
static const int64_t MOD = 100000380000361;
/* Module statics */
static int32_t* invp = NULL;
static int64_t* fac = NULL;
static int32_t* H = NULL;
static int64_t w = 0;
int64_t mulmod_i64_i64_i64(int64_t a, int64_t b, int64_t m) {
__int128 r = FLOW_CHECKED_MOD(((((__int128)(a)) * ((__int128)(b)))), (((__int128)(m))));
return ((int64_t)(r));
}
int64_t inv_mod_p2_i64(int64_t a0) {
int64_t a = FLOW_CHECKED_MOD((a0), (MOD));
int64_t r = FLOW_CHECKED_MOD((a), (P));
int64_t x = ((int64_t)(invp[r]));
int64_t ax = mulmod_i64_i64_i64(a, x, MOD);
int64_t term = FLOW_CHECKED_MOD((((2 + MOD) - ax)), (MOD));
return mulmod_i64_i64_i64(x, term, MOD);
}
int64_t Fpow_i64(int64_t u) {
int64_t um = FLOW_CHECKED_MOD((u), (P));
int64_t a1 = (um * P);
int64_t a2 = mulmod_i64_i64_i64(a1, w, MOD);
int64_t tmp = FLOW_CHECKED_MOD((((1 + MOD) - a2)), (MOD));
if (FLOW_CHECKED_MOD((u), (2)) == 1) {
tmp = FLOW_CHECKED_MOD(((MOD - tmp)), (MOD));
}
return tmp;
}
int64_t unit_factorial_i64(int64_t n0) {
int64_t n = n0;
int64_t res = 1;
while (n != 0) {
int64_t u = FLOW_CHECKED_DIV((n), (P));
int64_t v = FLOW_CHECKED_MOD((n), (P));
res = mulmod_i64_i64_i64(res, fac[v], MOD);
int64_t corr = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((u), (P)) * ((int64_t)(H[v])))), (P));
res = mulmod_i64_i64_i64(res, (1 + (corr * P)), MOD);
res = mulmod_i64_i64_i64(res, Fpow_i64(u), MOD);
n = u;
}
return res;
}
int64_t vp_fact_i64(int64_t n) {
int64_t q = FLOW_CHECKED_DIV((n), (P));
return (q + FLOW_CHECKED_DIV((q), (P)));
}
int64_t binom_mod_p2_i64_i64(int64_t n, int64_t k) {
if (k > n) {
return 0;
}
int64_t nk = (n - k);
int64_t e = ((vp_fact_i64(n) - vp_fact_i64(k)) - vp_fact_i64(nk));
if (e >= 2) {
return 0;
}
int64_t un = unit_factorial_i64(n);
int64_t uk = unit_factorial_i64(k);
int64_t unk = unit_factorial_i64(nk);
int64_t val = mulmod_i64_i64_i64(un, inv_mod_p2_i64(uk), MOD);
val = mulmod_i64_i64_i64(val, inv_mod_p2_i64(unk), MOD);
if (e == 1) {
val = mulmod_i64_i64_i64(val, P, MOD);
}
return val;
}
int64_t coeff_alpha_power_i64_i64(int64_t M, int64_t d) {
if (d == 0) {
return 1;
}
int64_t B = binom_mod_p2_i64_i64(((M - d) - 1), (d - 1));
int64_t val = mulmod_i64_i64_i64(FLOW_CHECKED_MOD((M), (MOD)), inv_mod_p2_i64(FLOW_CHECKED_MOD((d), (MOD))), MOD);
return mulmod_i64_i64_i64(val, B, MOD);
}
int32_t main(void) {
invp = ((int32_t*)(calloc(P, 4)));
H = ((int32_t*)(calloc(P, 4)));
fac = ((int64_t*)(malloc((P * 8))));
if (((invp == NULL || fac == NULL) || H == NULL)) {
return 1;
}
invp[1] = 1;
int64_t i = 2;
while (i < P) {
invp[i] = ((int32_t)(FLOW_CHECKED_MOD(((P - FLOW_CHECKED_MOD(((FLOW_CHECKED_DIV((P), (i)) * ((int64_t)(invp[FLOW_CHECKED_MOD((P), (i))])))), (P)))), (P))));
i = (i + 1);
}
int64_t f = 1;
fac[0] = 1;
int64_t i2 = 1;
while (i2 < P) {
f = mulmod_i64_i64_i64(f, i2, MOD);
fac[i2] = f;
i2 = (i2 + 1);
}
int32_t h = 0;
int64_t i3 = 1;
while (i3 < P) {
h = ((int32_t)(FLOW_CHECKED_MOD(((((int64_t)(h)) + ((int64_t)(invp[i3])))), (P))));
H[i3] = h;
i3 = (i3 + 1);
}
int64_t F = fac[(P - 1)];
w = FLOW_CHECKED_MOD((FLOW_CHECKED_DIV(((F + 1)), (P))), (P));
int64_t N = 1000000000;
int64_t K = 1000000000000000;
int64_t t_max = FLOW_CHECKED_DIV((K), (N));
int64_t comb = 1;
int64_t ans = 0;
int64_t d = K;
int64_t M = (N * N);
int64_t t = 0;
while (t <= t_max) {
if (t != 0) {
comb = mulmod_i64_i64_i64(comb, FLOW_CHECKED_MOD((((N - t) + 1)), (MOD)), MOD);
comb = mulmod_i64_i64_i64(comb, inv_mod_p2_i64(t), MOD);
}
ans = FLOW_CHECKED_MOD(((ans + mulmod_i64_i64_i64(comb, coeff_alpha_power_i64_i64(M, d), MOD))), (MOD));
d = (d - N);
M = (M - (2 * N));
t = (t + 1);
}
free(((void*)(invp)));
free(((void*)(fac)));
free(((void*)(H)));
printf("%lld\n", ans);
return 0;
}