# Project Euler 435
# sum_{x=0..100} F_n(x) mod 15! for n=10^15.
# F_n(x) = (f_n x^{n+2} + f_{n+1} x^{n+1} - x) / (x^2 + x - 1)
function modmul(a: i64, b: i64, mod: i64) -> i64 {
let r: i128 = ((a % mod) as i128) * ((b % mod) as i128) % (mod as i128)
if r < (0 as i128) { return (r + (mod as i128)) as i64 }
return r as i64
}
function modpow(base0: i64, exp0: i64, mod: i64) -> i64 {
if mod == 1 { return 0 }
let mut r: i64 = 1
let mut b: i64 = base0 % mod
if b < 0 { b = b + mod }
let mut e: i64 = exp0
while e > 0 {
if e % 2 == 1 { r = modmul(r, b, mod) }
b = modmul(b, b, mod)
e = e / 2
}
return r
}
# Returns F_n mod m via doubling (matrix [[1,1],[1,0]]^n).
function fib_mod(n0: i64, mod: i64) -> i64 {
if n0 == 0 { return 0 }
if mod == 1 { return 0 }
# compute (F_n, F_{n+1})
let mut a: i64 = 0
let mut b: i64 = 1
# process bits of n from high to low after leading 1
let mut bit: i32 = 62
while bit >= 0 {
if ((n0 >> bit) & 1) != 0 { break }
bit = bit - 1
}
# start with F_1=1, F_2=1 for the leading 1
a = 1
b = 1
bit = bit - 1
while bit >= 0 {
# double: (F_k, F_{k+1}) -> (F_{2k}, F_{2k+1})
let f2k: i64 = modmul(a, modmul(2 * b - a + mod, 1, mod), mod)
# F_{2k} = F_k*(2F_{k+1}-F_k)
let t: i64 = (2 * b - a) % mod
if t < 0 { t = t + mod }
let F2k: i64 = modmul(a, t, mod)
let F2k1: i64 = (modmul(a, a, mod) + modmul(b, b, mod)) % mod
a = F2k
b = F2k1
if ((n0 >> bit) & 1) != 0 {
# (F, Fnext) -> (Fnext, F+Fnext)
let na: i64 = b
let nb: i64 = (a + b) % mod
a = na
b = nb
}
bit = bit - 1
}
return a
}
function F_poly(n: i64, x: i64, mod: i64) -> i64 {
if x == 0 { return 0 }
let den: i64 = x * x + x - 1
let big: i64 = mod * den
let fn: i64 = fib_mod(n, big)
let fn1: i64 = fib_mod(n + 1, big)
let mut num: i64 = modmul(fn, modpow(x, n + 2, big), big)
num = (num + modmul(fn1, modpow(x, n + 1, big), big)) % big
num = (num - x) % big
if num < 0 { num = num + big }
return num / den
}
function main() -> i32 {
let MOD: i64 = 1307674368000
let N: i64 = 1000000000000000
# sanity: F_7(11)
let check: i64 = F_poly(7, 11, 1000000000000)
if check != 268357683 {
printf("check failed %lld\n", check)
return 1
}
let mut total: i64 = 0
let mut x: i64 = 0
while x <= 100 {
total = (total + F_poly(N, x, MOD)) % MOD
x = x + 1
}
printf("%lld\n", total)
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 modmul_i64_i64_i64(int64_t a, int64_t b, int64_t mod);
int64_t modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod);
int64_t fib_mod_i64_i64(int64_t n0, int64_t mod);
int64_t F_poly_i64_i64_i64(int64_t n, int64_t x, int64_t mod);
int32_t main(void);
int64_t modmul_i64_i64_i64(int64_t a, int64_t b, int64_t mod) {
__int128 r = FLOW_CHECKED_MOD(((((__int128)(FLOW_CHECKED_MOD((a), (mod)))) * ((__int128)(FLOW_CHECKED_MOD((b), (mod)))))), (((__int128)(mod))));
if (r < ((__int128)(0))) {
return ((int64_t)((r + ((__int128)(mod)))));
}
return ((int64_t)(r));
}
int64_t modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod) {
if (mod == 1) {
return 0;
}
int64_t r = 1;
int64_t b = FLOW_CHECKED_MOD((base0), (mod));
if (b < 0) {
b = (b + mod);
}
int64_t e = exp0;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
r = modmul_i64_i64_i64(r, b, mod);
}
b = modmul_i64_i64_i64(b, b, mod);
e = FLOW_CHECKED_DIV((e), (2));
}
return r;
}
int64_t fib_mod_i64_i64(int64_t n0, int64_t mod) {
if (n0 == 0) {
return 0;
}
if (mod == 1) {
return 0;
}
int64_t a = 0;
int64_t b = 1;
int32_t bit = 62;
while (bit >= 0) {
if ((FLOW_CHECKED_SHR((n0), (bit)) & 1) != 0) {
break;
}
bit = (bit - 1);
}
a = 1;
b = 1;
bit = (bit - 1);
while (bit >= 0) {
int64_t f2k = modmul_i64_i64_i64(a, modmul_i64_i64_i64((((2 * b) - a) + mod), 1, mod), mod);
int64_t t = FLOW_CHECKED_MOD((((2 * b) - a)), (mod));
if (t < 0) {
t = (t + mod);
}
int64_t F2k = modmul_i64_i64_i64(a, t, mod);
int64_t F2k1 = FLOW_CHECKED_MOD(((modmul_i64_i64_i64(a, a, mod) + modmul_i64_i64_i64(b, b, mod))), (mod));
a = F2k;
b = F2k1;
if ((FLOW_CHECKED_SHR((n0), (bit)) & 1) != 0) {
int64_t na = b;
int64_t nb = FLOW_CHECKED_MOD(((a + b)), (mod));
a = na;
b = nb;
}
bit = (bit - 1);
}
return a;
}
int64_t F_poly_i64_i64_i64(int64_t n, int64_t x, int64_t mod) {
if (x == 0) {
return 0;
}
int64_t den = (((x * x) + x) - 1);
int64_t big = (mod * den);
int64_t fn = fib_mod_i64_i64(n, big);
int64_t fn1 = fib_mod_i64_i64((n + 1), big);
int64_t num = modmul_i64_i64_i64(fn, modpow_i64_i64_i64(x, (n + 2), big), big);
num = FLOW_CHECKED_MOD(((num + modmul_i64_i64_i64(fn1, modpow_i64_i64_i64(x, (n + 1), big), big))), (big));
num = FLOW_CHECKED_MOD(((num - x)), (big));
if (num < 0) {
num = (num + big);
}
return FLOW_CHECKED_DIV((num), (den));
}
int32_t main(void) {
int64_t MOD = 1307674368000;
int64_t N = 1000000000000000;
int64_t check = F_poly_i64_i64_i64(7, 11, 1000000000000);
if (check != 268357683) {
printf("check failed %lld\n", check);
return 1;
}
int64_t total = 0;
int64_t x = 0;
while (x <= 100) {
total = FLOW_CHECKED_MOD(((total + F_poly_i64_i64_i64(N, x, MOD))), (MOD));
x = (x + 1);
}
printf("%lld\n", total);
return 0;
}