Problem 561

Divisor Pairs Q(N) counts 2-adic valuations of S((p_m#)^n) for n<=N, m=904961. Closed form via bit counts on floor(N/4) ranges.

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 561
# Divisor Pairs
# Q(N) counts 2-adic valuations of S((p_m#)^n) for n<=N, m=904961.
# Closed form via bit counts on floor(N/4) ranges.

function popcount(n0: i64) -> i64 {
    let mut n: i64 = n0
    let mut c: i64 = 0
    while n > 0 {
        c = c + (n & 1)
        n = n >> 1
    }
    return c
}

function Q(n: i64) -> i64 {
    let m: i64 = 904961
    let c: i64 = (n - 3) / 4 + 1
    let a: i64 = n / 4
    return (2 * a - popcount(a)) + m * (2 * c - popcount(c))
}

function main() -> i32 {
    printf("%lld\n", Q(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 popcount_i64(int64_t n0);
int64_t Q_i64(int64_t n);
int32_t main(void);

int64_t popcount_i64(int64_t n0) {
    int64_t n = n0;
    int64_t c = 0;
    while (n > 0) {
        c = (c + (n & 1));
        n = FLOW_CHECKED_SHR((n), (1));
    }
    return c;
}

int64_t Q_i64(int64_t n) {
    int64_t m = 904961;
    int64_t c = (FLOW_CHECKED_DIV(((n - 3)), (4)) + 1);
    int64_t a = FLOW_CHECKED_DIV((n), (4));
    return (((2 * a) - popcount_i64(a)) + (m * ((2 * c) - popcount_i64(c))));
}

int32_t main(void) {
    printf("%lld\n", Q_i64(1000000000000));
    return 0;
}

Generated MLIR

module {
  llvm.func @printf(!llvm.ptr, ...) -> i32
  llvm.mlir.global internal constant @str_0("%lld\n\00") {addr_space = 0 : i32} : !llvm.array<6 x i8>
  func.func @popcount(%arg0: i64) -> i64 {
    %0 = llvm.mlir.constant(1 : i64) : i64
    %1 = llvm.alloca %0 x i64 : (i64) -> !llvm.ptr
    llvm.store %arg0, %1 : i64, !llvm.ptr
    %2 = arith.constant 0 : i32
    %3 = arith.extsi %2 : i32 to i64
    %4 = llvm.mlir.constant(1 : i64) : i64
    %5 = llvm.alloca %4 x i64 : (i64) -> !llvm.ptr
    llvm.store %3, %5 : i64, !llvm.ptr
    cf.br ^bb0
    ^bb0:
    %6 = llvm.load %1 : !llvm.ptr -> i64
    %7 = arith.constant 0 : i32
    %9 = arith.extsi %7 : i32 to i64
    %8 = arith.cmpi sgt, %6, %9 : i64
    cf.cond_br %8, ^bb1, ^bb2
    ^bb1:
      %10 = llvm.load %5 : !llvm.ptr -> i64
      %11 = llvm.load %1 : !llvm.ptr -> i64
      %12 = arith.constant 1 : i32
      %14 = arith.extsi %12 : i32 to i64
      %13 = arith.andi %11, %14 : i64
      %15 = arith.addi %10, %13 : i64
      llvm.store %15, %5 : i64, !llvm.ptr
      %16 = llvm.load %1 : !llvm.ptr -> i64
      %17 = arith.constant 1 : i32
      %19 = arith.extsi %17 : i32 to i64
      %18 = arith.shrsi %16, %19 : i64
      llvm.store %18, %1 : i64, !llvm.ptr
      cf.br ^bb0
    ^bb2:
    %20 = llvm.load %5 : !llvm.ptr -> i64
    func.return %20 : i64
  }
  func.func @Q(%arg0: i64) -> i64 {
    %21 = arith.constant 904961 : i32
    %22 = arith.extsi %21 : i32 to i64
    %23 = arith.constant 3 : i32
    %25 = arith.extsi %23 : i32 to i64
    %24 = arith.subi %arg0, %25 : i64
    %26 = arith.constant 4 : i32
    %28 = arith.extsi %26 : i32 to i64
    %27 = arith.divsi %24, %28 : i64
    %29 = arith.constant 1 : i32
    %31 = arith.extsi %29 : i32 to i64
    %30 = arith.addi %27, %31 : i64
    %32 = arith.constant 4 : i32
    %34 = arith.extsi %32 : i32 to i64
    %33 = arith.divsi %arg0, %34 : i64
    %35 = arith.constant 2 : i32
    %37 = arith.extsi %35 : i32 to i64
    %36 = arith.muli %37, %33 : i64
    %38 = func.call @popcount(%33) : (i64) -> i64
    %39 = arith.subi %36, %38 : i64
    %40 = arith.constant 2 : i32
    %42 = arith.extsi %40 : i32 to i64
    %41 = arith.muli %42, %30 : i64
    %43 = func.call @popcount(%30) : (i64) -> i64
    %44 = arith.subi %41, %43 : i64
    %45 = arith.muli %22, %44 : i64
    %46 = arith.addi %39, %45 : i64
    func.return %46 : i64
  }
  func.func @main() -> i32 {
    %47 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %49 = arith.constant 995705032704 : i32
    %50 = arith.extsi %49 : i32 to i64
    %48 = func.call @Q(%50) : (i64) -> i64
    %51 = llvm.call @printf(%47, %48) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %52 = arith.constant 0 : i32
    func.return %52 : i32
  }
}