Problem 713

L(N)=sum_m T(N,m) via floor-block Dirichlet summation.

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 713
# L(N)=sum_m T(N,m) via floor-block Dirichlet summation.

function L(N: i64) -> i64 {
    if N < 2 { return 0 }
    let mut total: i64 = 0
    let mut k: i64 = 1
    let k_end: i64 = N - 1
    while k <= k_end {
        let q: i64 = N / k
        let mut k_max: i64 = N / q
        if k_max > k_end { k_max = k_end }
        let cnt: i64 = k_max - k + 1
        let sum_k: i64 = (k + k_max) * cnt / 2
        total = total + cnt * N * q - (q * (q + 1) / 2) * sum_k
        k = k_max + 1
    }
    return total
}

function main() -> i32 {
    printf("%lld\n", L(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 L_i64(int64_t N);
int32_t main(void);

int64_t L_i64(int64_t N) {
    if (N < 2) {
        return 0;
    }
    int64_t total = 0;
    int64_t k = 1;
    int64_t k_end = (N - 1);
    while (k <= k_end) {
        int64_t q = FLOW_CHECKED_DIV((N), (k));
        int64_t k_max = FLOW_CHECKED_DIV((N), (q));
        if (k_max > k_end) {
            k_max = k_end;
        }
        int64_t cnt = ((k_max - k) + 1);
        int64_t sum_k = FLOW_CHECKED_DIV((((k + k_max) * cnt)), (2));
        total = ((total + ((cnt * N) * q)) - (FLOW_CHECKED_DIV(((q * (q + 1))), (2)) * sum_k));
        k = (k_max + 1);
    }
    return total;
}

int32_t main(void) {
    printf("%lld\n", L_i64(10000000));
    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 @L(%arg0: i64) -> i64 {
    %0 = arith.constant 2 : i32
    %2 = arith.extsi %0 : i32 to i64
    %1 = arith.cmpi slt, %arg0, %2 : i64
    cf.cond_br %1, ^bb0, ^bb1
    ^bb0:
      %3 = arith.constant 0 : i32
      %4 = arith.extsi %3 : i32 to i64
      func.return %4 : i64
    ^bb1:
      cf.br ^bb2
    ^bb2:
    %5 = arith.constant 0 : i32
    %6 = arith.extsi %5 : i32 to i64
    %7 = llvm.mlir.constant(1 : i64) : i64
    %8 = llvm.alloca %7 x i64 : (i64) -> !llvm.ptr
    llvm.store %6, %8 : i64, !llvm.ptr
    %9 = arith.constant 1 : i32
    %10 = arith.extsi %9 : i32 to i64
    %11 = llvm.mlir.constant(1 : i64) : i64
    %12 = llvm.alloca %11 x i64 : (i64) -> !llvm.ptr
    llvm.store %10, %12 : i64, !llvm.ptr
    %13 = arith.constant 1 : i32
    %15 = arith.extsi %13 : i32 to i64
    %14 = arith.subi %arg0, %15 : i64
    cf.br ^bb3
    ^bb3:
    %16 = llvm.load %12 : !llvm.ptr -> i64
    %17 = arith.cmpi sle, %16, %14 : i64
    cf.cond_br %17, ^bb4, ^bb5
    ^bb4:
      %18 = llvm.load %12 : !llvm.ptr -> i64
      %19 = arith.divsi %arg0, %18 : i64
      %20 = arith.divsi %arg0, %19 : i64
      %21 = llvm.mlir.constant(1 : i64) : i64
      %22 = llvm.alloca %21 x i64 : (i64) -> !llvm.ptr
      llvm.store %20, %22 : i64, !llvm.ptr
      %23 = llvm.load %22 : !llvm.ptr -> i64
      %24 = arith.cmpi sgt, %23, %14 : i64
      cf.cond_br %24, ^bb6, ^bb7
      ^bb6:
        llvm.store %14, %22 : i64, !llvm.ptr
        cf.br ^bb8
      ^bb7:
        cf.br ^bb8
      ^bb8:
      %25 = llvm.load %22 : !llvm.ptr -> i64
      %26 = llvm.load %12 : !llvm.ptr -> i64
      %27 = arith.subi %25, %26 : i64
      %28 = arith.constant 1 : i32
      %30 = arith.extsi %28 : i32 to i64
      %29 = arith.addi %27, %30 : i64
      %31 = llvm.load %12 : !llvm.ptr -> i64
      %32 = llvm.load %22 : !llvm.ptr -> i64
      %33 = arith.addi %31, %32 : i64
      %34 = arith.muli %33, %29 : i64
      %35 = arith.constant 2 : i32
      %37 = arith.extsi %35 : i32 to i64
      %36 = arith.divsi %34, %37 : i64
      %38 = llvm.load %8 : !llvm.ptr -> i64
      %39 = arith.muli %29, %arg0 : i64
      %40 = arith.muli %39, %19 : i64
      %41 = arith.addi %38, %40 : i64
      %42 = arith.constant 1 : i32
      %44 = arith.extsi %42 : i32 to i64
      %43 = arith.addi %19, %44 : i64
      %45 = arith.muli %19, %43 : i64
      %46 = arith.constant 2 : i32
      %48 = arith.extsi %46 : i32 to i64
      %47 = arith.divsi %45, %48 : i64
      %49 = arith.muli %47, %36 : i64
      %50 = arith.subi %41, %49 : i64
      llvm.store %50, %8 : i64, !llvm.ptr
      %51 = llvm.load %22 : !llvm.ptr -> i64
      %52 = arith.constant 1 : i32
      %54 = arith.extsi %52 : i32 to i64
      %53 = arith.addi %51, %54 : i64
      llvm.store %53, %12 : i64, !llvm.ptr
      cf.br ^bb3
    ^bb5:
    %55 = llvm.load %8 : !llvm.ptr -> i64
    func.return %55 : i64
  }
  func.func @main() -> i32 {
    %56 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %58 = arith.constant 10000000 : i32
    %59 = arith.extsi %58 : i32 to i64
    %57 = func.call @L(%59) : (i64) -> i64
    %60 = llvm.call @printf(%56, %57) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %61 = arith.constant 0 : i32
    func.return %61 : i32
  }
}