# Project Euler 560
# Coprime Nim
# L(10^7,10^7) mod 10^9+7 via Grundy frequencies + FWHT XOR power.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
const MOD: i64 = 1000000007
function modpow(base0: i64, exp0: i64, mod: i64) -> i64 {
let mut result: i64 = 1
let mut base: i64 = base0 % mod
let mut exp: i64 = exp0
while exp > 0 {
if (exp & 1) != 0 {
result = ((result as i128) * (base as i128) % (mod as i128)) as i64
}
base = ((base as i128) * (base as i128) % (mod as i128)) as i64
exp = exp >> 1
}
return result
}
function fwht(a: ptr<i64>, n: i64) -> void {
let mut h: i64 = 1
while h < n {
let step: i64 = h << 1
let mut i: i64 = 0
while i < n {
let mut j: i64 = i
while j < i + h {
let x: i64 = a[j]
let y: i64 = a[j + h]
let mut s: i64 = x + y
if s >= MOD { s = s - MOD }
let mut d: i64 = x - y
if d < 0 { d = d + MOD }
a[j] = s
a[j + h] = d
j = j + 1
}
i = i + step
}
h = step
}
}
function compute_L(n: i64, k: i64) -> i64 {
let N: i64 = n - 1
let even_count: i64 = N / 2
let odd_len: i64 = (N + 1) / 2
let spf: ptr<i32> = calloc(odd_len, 4)
if spf == null { return -1 }
# counts of odd primes as spf; grow dynamically with cap
let mut cap: i64 = 800000
let counts: ptr<i32> = calloc(cap, 4)
if counts == null { free(spf); return -1 }
let mut P: i64 = 1 # counts[0] dummy; P = number of primes including 2
let limit: i64 = isqrt(N)
let i_end: i64 = (limit - 1) / 2
let mut i: i64 = 1
while i <= i_end {
if spf[i] == 0 {
let p: i32 = (2 * i + 1) as i32
spf[i] = p
if P >= cap { return -1 }
counts[P] = 1
P = P + 1
let start: i64 = ((p as i64) * (p as i64) - 1) / 2
let step: i64 = p as i64
let mut j: i64 = start
while j < odd_len {
if spf[j] == 0 {
spf[j] = p
counts[P - 1] = counts[P - 1] + 1
}
j = j + step
}
}
i = i + 1
}
i = 1
while i < odd_len {
if spf[i] == 0 {
if P >= cap { return -1 }
counts[P] = 1
P = P + 1
spf[i] = (2 * i + 1) as i32
}
i = i + 1
}
# M = next power of 2 >= P+1
let mut M: i64 = 1
while M < P + 1 {
M = M << 1
}
let a: ptr<i64> = calloc(M, 8)
if a == null { free(counts); free(spf); return -1 }
a[0] = even_count % MOD
a[1] = 1
let mut idx: i64 = 2
while idx <= P {
a[idx] = (counts[idx - 1] as i64) % MOD
idx = idx + 1
}
fwht(a, M)
let mut total: i64 = 0
idx = 0
while idx < M {
total = (total + modpow(a[idx], k, MOD)) % MOD
idx = idx + 1
}
total = (total * modpow(M, MOD - 2, MOD)) % MOD
free(a)
free(counts)
free(spf)
return total
}
function main() -> i32 {
printf("%lld\n", compute_L(10000000, 10000000))
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 modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod);
void fwht_ptr_i64_i64(int64_t* a, int64_t n);
int64_t compute_L_i64_i64(int64_t n, int64_t k);
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 modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod) {
int64_t result = 1;
int64_t base = FLOW_CHECKED_MOD((base0), (mod));
int64_t exp = exp0;
while (exp > 0) {
if ((exp & 1) != 0) {
result = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(result)) * ((__int128)(base)))), (((__int128)(mod))))));
}
base = ((int64_t)(FLOW_CHECKED_MOD(((((__int128)(base)) * ((__int128)(base)))), (((__int128)(mod))))));
exp = FLOW_CHECKED_SHR((exp), (1));
}
return result;
}
void fwht_ptr_i64_i64(int64_t* a, int64_t n) {
int64_t h = 1;
while (h < n) {
int64_t step = FLOW_CHECKED_SHL((h), (1));
int64_t i = 0;
while (i < n) {
int64_t j = i;
while (j < (i + h)) {
int64_t x = a[j];
int64_t y = a[(j + h)];
int64_t s = (x + y);
if (s >= MOD) {
s = (s - MOD);
}
int64_t d = (x - y);
if (d < 0) {
d = (d + MOD);
}
a[j] = s;
a[(j + h)] = d;
j = (j + 1);
}
i = (i + step);
}
h = step;
}
}
int64_t compute_L_i64_i64(int64_t n, int64_t k) {
int64_t N = (n - 1);
int64_t even_count = FLOW_CHECKED_DIV((N), (2));
int64_t odd_len = FLOW_CHECKED_DIV(((N + 1)), (2));
int32_t* spf = (int32_t*)(calloc(odd_len, 4));
if (spf == NULL) {
return (-1);
}
int64_t cap = 800000;
int32_t* counts = (int32_t*)(calloc(cap, 4));
if (counts == NULL) {
free(spf);
return (-1);
}
int64_t P = 1;
int64_t limit = isqrt_i64(N);
int64_t i_end = FLOW_CHECKED_DIV(((limit - 1)), (2));
int64_t i = 1;
while (i <= i_end) {
if (spf[i] == 0) {
int32_t p = ((int32_t)(((2 * i) + 1)));
spf[i] = p;
if (P >= cap) {
return (-1);
}
counts[P] = 1;
P = (P + 1);
int64_t start = FLOW_CHECKED_DIV((((((int64_t)(p)) * ((int64_t)(p))) - 1)), (2));
int64_t step = ((int64_t)(p));
int64_t j = start;
while (j < odd_len) {
if (spf[j] == 0) {
spf[j] = p;
counts[(P - 1)] = (counts[(P - 1)] + 1);
}
j = (j + step);
}
}
i = (i + 1);
}
i = 1;
while (i < odd_len) {
if (spf[i] == 0) {
if (P >= cap) {
return (-1);
}
counts[P] = 1;
P = (P + 1);
spf[i] = ((int32_t)(((2 * i) + 1)));
}
i = (i + 1);
}
int64_t M = 1;
while (M < (P + 1)) {
M = FLOW_CHECKED_SHL((M), (1));
}
int64_t* a = (int64_t*)(calloc(M, 8));
if (a == NULL) {
free(counts);
free(spf);
return (-1);
}
a[0] = FLOW_CHECKED_MOD((even_count), (MOD));
a[1] = 1;
int64_t idx = 2;
while (idx <= P) {
a[idx] = FLOW_CHECKED_MOD((((int64_t)(counts[(idx - 1)]))), (MOD));
idx = (idx + 1);
}
fwht_ptr_i64_i64(a, M);
int64_t total = 0;
idx = 0;
while (idx < M) {
total = FLOW_CHECKED_MOD(((total + modpow_i64_i64_i64(a[idx], k, MOD))), (MOD));
idx = (idx + 1);
}
total = FLOW_CHECKED_MOD(((total * modpow_i64_i64_i64(M, (MOD - 2), MOD))), (MOD));
free(a);
free(counts);
free(spf);
return total;
}
int32_t main(void) {
printf("%lld\n", compute_L_i64_i64(10000000, 10000000));
return 0;
}