# Project Euler 657
# Incomplete Words: I(10^7, 10^12) mod 1e9+7.
# Inclusion-exclusion: sum_{s=1..alpha} (-1)^{s+1} C(alpha,s) * geom_sum(alpha-s, n)
import euler.nt { mod_pow }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const MOD: i64 = 1000000007
# Return sum_{k=0..n} r^k mod mod
function geom_sum(r: i64, n: i64, mod: i64) -> i64 {
if n < 0 { return 0 }
let r_mod: i64 = r % mod
if r_mod == 1 { return (n + 1) % mod }
if r_mod == 0 { return 1 }
let rn1: i64 = mod_pow(r_mod, n + 1, mod)
let num: i64 = (rn1 - 1 + mod) % mod
let den: i64 = mod_pow((r_mod - 1 + mod) % mod, mod - 2, mod)
let t: i128 = (num as i128) * (den as i128) % (mod as i128)
return t as i64
}
function main() -> i32 {
let alpha: i64 = 10000000
let n: i64 = 1000000000000
# Precompute modular inverses 1..alpha using linear recurrence
let inv: ptr<i64> = calloc(alpha + 1, 8)
if inv == null { return 1 }
inv[1] = 1
for i in 2..(alpha + 1) {
inv[i] = (MOD - (MOD / i) * inv[MOD % i] % MOD) % MOD
}
# Effective exponent: (n+1) mod (MOD-1) by Fermat's little theorem
let exp: i64 = (n + 1) % (MOD - 1)
# Precompute r^exp mod MOD for r=0..alpha-1
# r=0: 0^exp = 0 (exp > 0)
# r=1: 1
# For r >= 2: compute via mod_pow
# But 10^7 mod_pow calls is too slow. Instead, note that
# r^exp = r^(exp) and we can compute all at once using:
# For each r, r^exp = product of r^(2^i) for set bits of exp.
# With exp ~ 10^12 % 10^9 ~ 999994005, that's still ~30 bits.
# Better: use baby-step giant-step or just compute in batches.
# Actually, let's just compute r^exp for each r directly.
# 10^7 * 30 multiplications = 3*10^8, should be ~3 seconds in C.
let pow_table: ptr<i64> = calloc(alpha, 8)
if pow_table == null { return 1 }
pow_table[0] = 0 # 0^exp = 0
if alpha > 1 { pow_table[1] = 1 }
for r in 2..alpha {
let mut result: i64 = 1
let mut base: i64 = r
let mut e: i64 = exp
while e > 0 {
if (e & 1) == 1 {
let t: i128 = (result as i128) * (base as i128) % (MOD as i128)
result = t as i64
}
let t2: i128 = (base as i128) * (base as i128) % (MOD as i128)
base = t2 as i64
e = e / 2
}
pow_table[r] = result
}
let mut res: i64 = 0
let mut comb: i64 = 1 # C(alpha, 0)
for s in 1..(alpha + 1) {
# C(alpha,s) from C(alpha,s-1)
let t1: i128 = (comb as i128) * ((alpha - s + 1) as i128) % (MOD as i128)
comb = (t1 * (inv[s] as i128) % (MOD as i128)) as i64
let r: i64 = alpha - s
# geom_sum(r, n) = (r^(n+1) - 1) / (r - 1) for r >= 2
# = (n+1) for r = 1
# = 1 for r = 0
let mut gs: i64 = 0
if r == 0 {
gs = 1
} else {
if r == 1 {
gs = (n + 1) % MOD
} else {
let rn1: i64 = pow_table[r]
let num: i64 = (rn1 - 1 + MOD) % MOD
let den: i64 = inv[r - 1]
let t: i128 = (num as i128) * (den as i128) % (MOD as i128)
gs = t as i64
}
}
let t2: i128 = (comb as i128) * (gs as i128) % (MOD as i128)
if s % 2 == 1 {
res = (res + (t2 as i64)) % MOD
} else {
res = (res - (t2 as i64) + MOD) % MOD
}
}
printf("%lld\n", (res % MOD + MOD) % MOD)
free(pow_table)
free(inv)
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 gcd_i64_i64(int64_t a0, int64_t b0);
int64_t lcm_i64_i64(int64_t a, int64_t b);
int64_t isqrt_i64(int64_t n);
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod);
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
bool is_prime_i64(int64_t n);
int64_t geom_sum_i64_i64_i64(int64_t r, int64_t n, int64_t mod);
int32_t main(void);
static const int64_t MOD = 1000000007;
int64_t gcd_i64_i64(int64_t a0, int64_t b0) {
int64_t a = a0;
int64_t b = b0;
while (b != 0) {
int64_t t = FLOW_CHECKED_MOD((a), (b));
a = b;
b = t;
}
return a;
}
int64_t lcm_i64_i64(int64_t a, int64_t b) {
if ((a == 0 || b == 0)) {
return 0;
}
return (FLOW_CHECKED_DIV((a), (gcd_i64_i64(a, b))) * b);
}
int64_t isqrt_i64(int64_t n) {
if (n < 2) {
return n;
}
int64_t x = n;
int64_t y = FLOW_CHECKED_DIV(((x + 1)), (2));
while (y < x) {
x = y;
y = FLOW_CHECKED_DIV(((x + FLOW_CHECKED_DIV((n), (x)))), (2));
}
return x;
}
int64_t mulmod_i64_i64_i64(int64_t a0, int64_t b0, int64_t mod) {
int64_t a = FLOW_CHECKED_MOD((a0), (mod));
int64_t b = FLOW_CHECKED_MOD((b0), (mod));
int64_t result = 0;
while (b > 0) {
if (FLOW_CHECKED_MOD((b), (2)) == 1) {
result = FLOW_CHECKED_MOD(((result + a)), (mod));
}
a = FLOW_CHECKED_MOD(((a * 2)), (mod));
b = FLOW_CHECKED_DIV((b), (2));
}
return result;
}
int64_t mod_pow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
if (mod == 1) {
return 0;
}
int64_t result = 1;
int64_t b = FLOW_CHECKED_MOD((base), (mod));
int64_t e = exp;
while (e > 0) {
if (FLOW_CHECKED_MOD((e), (2)) == 1) {
result = mulmod_i64_i64_i64(result, b, mod);
}
b = mulmod_i64_i64_i64(b, b, mod);
e = FLOW_CHECKED_DIV((e), (2));
}
return result;
}
bool is_prime_i64(int64_t n) {
if (n < 2) {
return 0;
}
if (n < 4) {
return 1;
}
if ((FLOW_CHECKED_MOD((n), (2)) == 0 || FLOW_CHECKED_MOD((n), (3)) == 0)) {
return 0;
}
int64_t i = 5;
while ((i * i) <= n) {
if ((FLOW_CHECKED_MOD((n), (i)) == 0 || FLOW_CHECKED_MOD((n), ((i + 2))) == 0)) {
return 0;
}
i = (i + 6);
}
return 1;
}
int64_t geom_sum_i64_i64_i64(int64_t r, int64_t n, int64_t mod) {
if (n < 0) {
return 0;
}
int64_t r_mod = FLOW_CHECKED_MOD((r), (mod));
if (r_mod == 1) {
return FLOW_CHECKED_MOD(((n + 1)), (mod));
}
if (r_mod == 0) {
return 1;
}
int64_t rn1 = mod_pow_i64_i64_i64(r_mod, (n + 1), mod);
int64_t num = FLOW_CHECKED_MOD((((rn1 - 1) + mod)), (mod));
int64_t den = mod_pow_i64_i64_i64(FLOW_CHECKED_MOD((((r_mod - 1) + mod)), (mod)), (mod - 2), mod);
__int128 t = FLOW_CHECKED_MOD(((((__int128)(num)) * ((__int128)(den)))), (((__int128)(mod))));
return ((int64_t)(t));
}
int32_t main(void) {
int64_t alpha = 10000000;
int64_t n = 1000000000000;
int64_t* inv = (int64_t*)(calloc((alpha + 1), 8));
if (inv == NULL) {
return 1;
}
inv[1] = 1;
int32_t __flow_step_1 = 1;
for (int32_t i = 2; (2 <= (alpha + 1)) ? i < (alpha + 1) : i > (alpha + 1); i += (2 <= (alpha + 1)) ? 1 : -1) {
inv[i] = FLOW_CHECKED_MOD(((MOD - FLOW_CHECKED_MOD(((FLOW_CHECKED_DIV((MOD), (i)) * inv[FLOW_CHECKED_MOD((MOD), (i))])), (MOD)))), (MOD));
}
int64_t exp = FLOW_CHECKED_MOD(((n + 1)), ((MOD - 1)));
int64_t* pow_table = (int64_t*)(calloc(alpha, 8));
if (pow_table == NULL) {
return 1;
}
pow_table[0] = 0;
if (alpha > 1) {
pow_table[1] = 1;
}
int32_t __flow_step_2 = 1;
for (int32_t r = 2; (2 <= alpha) ? r < alpha : r > alpha; r += (2 <= alpha) ? 1 : -1) {
int64_t result = 1;
int64_t base = r;
int64_t e = exp;
while (e > 0) {
if ((e & 1) == 1) {
__int128 t = FLOW_CHECKED_MOD(((((__int128)(result)) * ((__int128)(base)))), (((__int128)(MOD))));
result = ((int64_t)(t));
}
__int128 t2 = FLOW_CHECKED_MOD(((((__int128)(base)) * ((__int128)(base)))), (((__int128)(MOD))));
base = ((int64_t)(t2));
e = FLOW_CHECKED_DIV((e), (2));
}
pow_table[r] = result;
}
int64_t res = 0;
int64_t comb = 1;
int32_t __flow_step_3 = 1;
for (int32_t s = 1; (1 <= (alpha + 1)) ? s < (alpha + 1) : s > (alpha + 1); s += (1 <= (alpha + 1)) ? 1 : -1) {
__int128 t1 = FLOW_CHECKED_MOD(((((__int128)(comb)) * ((__int128)(((alpha - s) + 1))))), (((__int128)(MOD))));
comb = ((int64_t)(FLOW_CHECKED_MOD(((t1 * ((__int128)(inv[s])))), (((__int128)(MOD))))));
int64_t r = (alpha - s);
int64_t gs = 0;
if (r == 0) {
gs = 1;
} else {
if (r == 1) {
gs = FLOW_CHECKED_MOD(((n + 1)), (MOD));
} else {
int64_t rn1 = pow_table[r];
int64_t num = FLOW_CHECKED_MOD((((rn1 - 1) + MOD)), (MOD));
int64_t den = inv[(r - 1)];
__int128 t = FLOW_CHECKED_MOD(((((__int128)(num)) * ((__int128)(den)))), (((__int128)(MOD))));
gs = ((int64_t)(t));
}
}
__int128 t2 = FLOW_CHECKED_MOD(((((__int128)(comb)) * ((__int128)(gs)))), (((__int128)(MOD))));
if (FLOW_CHECKED_MOD((s), (2)) == 1) {
res = FLOW_CHECKED_MOD(((res + ((int64_t)(t2)))), (MOD));
} else {
res = FLOW_CHECKED_MOD((((res - ((int64_t)(t2))) + MOD)), (MOD));
}
}
printf("%lld\n", FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD((res), (MOD)) + MOD)), (MOD)));
free(pow_table);
free(inv);
return 0;
}