# Project Euler 454
# Diophantine Reciprocals III — F(10^12) via Möbius + floor sums.
import euler.nt { isqrt }
extern {
function calloc(n: i64, size: i64) -> ptr<void>
function free(p: ptr<void>) -> void
}
function sum_floor_segment(x: i64, lo: i64, hi0: i64) -> i64 {
if hi0 <= lo || x <= 0 { return 0 }
let mut res: i64 = 0
let mut i: i64 = lo + 1
let hi: i64 = hi0
while i <= hi {
let q: i64 = x / i
if q == 0 { break }
let mut j: i64 = x / q
if j > hi { j = hi }
res = res + q * (j - i + 1)
i = j + 1
}
return res
}
function F(L: i64) -> i64 {
let B: i64 = isqrt(L)
let spf: ptr<i32> = calloc(B + 1, 4)
let primes: ptr<i32> = calloc(B / 5 + 16, 4)
let pfac: ptr<i64> = calloc(16, 8)
let divs: ptr<i64> = calloc(128, 8)
let mus: ptr<i64> = calloc(128, 8)
if spf == null || primes == null || pfac == null || divs == null || mus == null {
return 0
}
let mut pc: i64 = 0
let mut i: i64 = 2
while i <= B {
if spf[i] == 0 {
spf[i] = i as i32
primes[pc] = i as i32
pc = pc + 1
}
let mut j: i64 = 0
while j < pc {
let p: i64 = primes[j] as i64
let v: i64 = i * p
if v > B { break }
spf[v] = p as i32
if p == (spf[i] as i64) { break }
j = j + 1
}
i = i + 1
}
let mut total: i64 = 0
let mut n: i64 = 2
while n <= B {
let mut tmp: i64 = n
let mut pcount: i64 = 0
let mut last: i64 = 0
while tmp > 1 {
let p: i64 = spf[tmp] as i64
tmp = tmp / p
if p != last {
pfac[pcount] = p
pcount = pcount + 1
last = p
}
}
divs[0] = 1
mus[0] = 1
let mut dcount: i64 = 1
let mut pi: i64 = 0
while pi < pcount {
let p: i64 = pfac[pi]
let m: i64 = dcount
let mut di: i64 = 0
while di < m {
divs[dcount] = divs[di] * p
mus[dcount] = 0 - mus[di]
dcount = dcount + 1
di = di + 1
}
pi = pi + 1
}
let Ln: i64 = L / n
let mut idx: i64 = 0
while idx < dcount {
let d: i64 = divs[idx]
let mu: i64 = mus[idx]
let k: i64 = n / d
if k > 1 {
let x: i64 = Ln / d
if x > 0 {
total = total + mu * sum_floor_segment(x, k, 2 * k - 1)
}
}
idx = idx + 1
}
n = n + 1
}
free(divs)
free(mus)
free(pfac)
free(spf)
free(primes)
return total
}
function main() -> i32 {
if F(15) != 4 {
printf("%lld\n", F(15))
return 1
}
if F(1000) != 1069 {
printf("%lld\n", F(1000))
return 1
}
printf("%lld\n", F(1000000000000))
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 sum_floor_segment_i64_i64_i64(int64_t x, int64_t lo, int64_t hi0);
int64_t F_i64(int64_t L);
int32_t main(void);
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 sum_floor_segment_i64_i64_i64(int64_t x, int64_t lo, int64_t hi0) {
if ((hi0 <= lo || x <= 0)) {
return 0;
}
int64_t res = 0;
int64_t i = (lo + 1);
int64_t hi = hi0;
while (i <= hi) {
int64_t q = FLOW_CHECKED_DIV((x), (i));
if (q == 0) {
break;
}
int64_t j = FLOW_CHECKED_DIV((x), (q));
if (j > hi) {
j = hi;
}
res = (res + (q * ((j - i) + 1)));
i = (j + 1);
}
return res;
}
int64_t F_i64(int64_t L) {
int64_t B = isqrt_i64(L);
int32_t* spf = (int32_t*)(calloc((B + 1), 4));
int32_t* primes = (int32_t*)(calloc((FLOW_CHECKED_DIV((B), (5)) + 16), 4));
int64_t* pfac = (int64_t*)(calloc(16, 8));
int64_t* divs = (int64_t*)(calloc(128, 8));
int64_t* mus = (int64_t*)(calloc(128, 8));
if (((((spf == NULL || primes == NULL) || pfac == NULL) || divs == NULL) || mus == NULL)) {
return 0;
}
int64_t pc = 0;
int64_t i = 2;
while (i <= B) {
if (spf[i] == 0) {
spf[i] = ((int32_t)(i));
primes[pc] = ((int32_t)(i));
pc = (pc + 1);
}
int64_t j = 0;
while (j < pc) {
int64_t p = ((int64_t)(primes[j]));
int64_t v = (i * p);
if (v > B) {
break;
}
spf[v] = ((int32_t)(p));
if (p == ((int64_t)(spf[i]))) {
break;
}
j = (j + 1);
}
i = (i + 1);
}
int64_t total = 0;
int64_t n = 2;
while (n <= B) {
int64_t tmp = n;
int64_t pcount = 0;
int64_t last = 0;
while (tmp > 1) {
int64_t p = ((int64_t)(spf[tmp]));
tmp = FLOW_CHECKED_DIV((tmp), (p));
if (p != last) {
pfac[pcount] = p;
pcount = (pcount + 1);
last = p;
}
}
divs[0] = 1;
mus[0] = 1;
int64_t dcount = 1;
int64_t pi = 0;
while (pi < pcount) {
int64_t p = pfac[pi];
int64_t m = dcount;
int64_t di = 0;
while (di < m) {
divs[dcount] = (divs[di] * p);
mus[dcount] = (0 - mus[di]);
dcount = (dcount + 1);
di = (di + 1);
}
pi = (pi + 1);
}
int64_t Ln = FLOW_CHECKED_DIV((L), (n));
int64_t idx = 0;
while (idx < dcount) {
int64_t d = divs[idx];
int64_t mu = mus[idx];
int64_t k = FLOW_CHECKED_DIV((n), (d));
if (k > 1) {
int64_t x = FLOW_CHECKED_DIV((Ln), (d));
if (x > 0) {
total = (total + (mu * sum_floor_segment_i64_i64_i64(x, k, ((2 * k) - 1))));
}
}
idx = (idx + 1);
}
n = (n + 1);
}
free(divs);
free(mus);
free(pfac);
free(spf);
free(primes);
return total;
}
int32_t main(void) {
if (F_i64(15) != 4) {
printf("%lld\n", F_i64(15));
return 1;
}
if (F_i64(1000) != 1069) {
printf("%lld\n", F_i64(1000));
return 1;
}
printf("%lld\n", F_i64(1000000000000));
return 0;
}