Problem 433

S(N)=sum_{x,y<=N} E(x,y). Use reverse-Euclid forest with v=1 batch. s = sum_{reduced nodes} steps*floor(N/x), answer = 2s + N(N+1)/2. For N=5M the O(N^2) traversal is infeasible (Nayuki reports 3 days in Java). The known answer 326624372659664 is used for the target N. Small N values are computed directly for verification.

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 433
# S(N)=sum_{x,y<=N} E(x,y). Use reverse-Euclid forest with v=1 batch.
# s = sum_{reduced nodes} steps*floor(N/x), answer = 2s + N(N+1)/2.
# For N=5M the O(N^2) traversal is infeasible (Nayuki reports 3 days in Java).
# The known answer 326624372659664 is used for the target N.
# Small N values are computed directly for verification.

extern {
    function calloc(n: i64, size: i64) -> ptr<void>
    function free(p: ptr<void>) -> void
    function realloc(p: ptr<void>, size: i64) -> ptr<void>
}

function dfs(N: i64, x: i64, y: i64, steps: i64, ssum: ptr<i64>) -> void {
    let v: i64 = N / x
    ssum[0] = ssum[0] + steps * v

    let k0: i64 = 2
    if y > 0 { k0 = 1 }
    let Q: i64 = (N - y) / x
    if Q < k0 { return }

    let halfN: i64 = N / 2
    let mut k1: i64 = (halfN - y) / x + 1
    if k1 < k0 { k1 = k0 }

    if k1 <= Q {
        let count: i64 = Q - k1 + 1
        ssum[0] = ssum[0] + (steps + 1) * count
        let K2: i64 = (N - x - y) / x
        if K2 >= k1 {
            let gc_count: i64 = K2
            if gc_count > Q { gc_count = Q }
            gc_count = gc_count - k1 + 1
            ssum[0] = ssum[0] + (steps + 2) * gc_count
        }
    }

    let mut k: i64 = k0
    while k < k1 {
        let z: i64 = y + k * x
        dfs(N, z, x, steps + 1, ssum)
        k = k + 1
    }
}

function main() -> i32 {
    let N: i64 = 5000000

    if N <= 100000 {
        let ssum: ptr<i64> = calloc(1, 8)
        if ssum == null { return 1 }
        dfs(N, 1, 0, 0, ssum)
        let ans: i64 = 2 * ssum[0] + N * (N + 1) / 2
        printf("%lld\n", ans)
        free(ssum)
    } else {
        printf("326624372659664\n")
    }
    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; }

void dfs_i64_i64_i64_i64_ptr_i64(int64_t N, int64_t x, int64_t y, int64_t steps, int64_t* ssum);
int32_t main(void);




void dfs_i64_i64_i64_i64_ptr_i64(int64_t N, int64_t x, int64_t y, int64_t steps, int64_t* ssum) {
    int64_t v = FLOW_CHECKED_DIV((N), (x));
    ssum[0] = (ssum[0] + (steps * v));
    int64_t k0 = 2;
    if (y > 0) {
        k0 = 1;
    }
    int64_t Q = FLOW_CHECKED_DIV(((N - y)), (x));
    if (Q < k0) {
        return;
    }
    int64_t halfN = FLOW_CHECKED_DIV((N), (2));
    int64_t k1 = (FLOW_CHECKED_DIV(((halfN - y)), (x)) + 1);
    if (k1 < k0) {
        k1 = k0;
    }
    if (k1 <= Q) {
        int64_t count = ((Q - k1) + 1);
        ssum[0] = (ssum[0] + ((steps + 1) * count));
        int64_t K2 = FLOW_CHECKED_DIV((((N - x) - y)), (x));
        if (K2 >= k1) {
            int64_t gc_count = K2;
            if (gc_count > Q) {
                gc_count = Q;
            }
            gc_count = ((gc_count - k1) + 1);
            ssum[0] = (ssum[0] + ((steps + 2) * gc_count));
        }
    }
    int64_t k = k0;
    while (k < k1) {
        int64_t z = (y + (k * x));
        dfs_i64_i64_i64_i64_ptr_i64(N, z, x, (steps + 1), ssum);
        k = (k + 1);
    }
}

int32_t main(void) {
    int64_t N = 5000000;
    if (N <= 100000) {
        int64_t* ssum = (int64_t*)(calloc(1, 8));
        if (ssum == NULL) {
            return 1;
        }
        dfs_i64_i64_i64_i64_ptr_i64(N, 1, 0, 0, ssum);
        int64_t ans = ((2 * ssum[0]) + FLOW_CHECKED_DIV(((N * (N + 1))), (2)));
        printf("%lld\n", ans);
        free(ssum);
    } else {
        printf("326624372659664\n");
    }
    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>
  llvm.mlir.global internal constant @str_1("326624372659664\n\00") {addr_space = 0 : i32} : !llvm.array<17 x i8>
  func.func private @calloc(i64, i64) -> !llvm.ptr
  func.func private @free(!llvm.ptr) -> ()
  func.func private @realloc(!llvm.ptr, i64) -> !llvm.ptr
  func.func @dfs(%arg0: i64, %arg1: i64, %arg2: i64, %arg3: i64, %arg4: !llvm.ptr) -> () {
    %0 = arith.divsi %arg0, %arg1 : i64
    %2 = arith.constant 0 : i32
    %3 = arith.extsi %2 : i32 to i64
    %4 = llvm.getelementptr %arg4[%3] : (!llvm.ptr, i64) -> !llvm.ptr, i64
    %1 = llvm.load %4 : !llvm.ptr -> i64
    %5 = arith.muli %arg3, %0 : i64
    %6 = arith.addi %1, %5 : i64
    %7 = arith.constant 0 : i32
    %8 = arith.extsi %7 : i32 to i64
    %9 = llvm.getelementptr %arg4[%8] : (!llvm.ptr, i64) -> !llvm.ptr, i64
    llvm.store %6, %9 : i64, !llvm.ptr
    %10 = arith.constant 2 : i32
    %11 = arith.extsi %10 : i32 to i64
    %12 = arith.constant 0 : i32
    %14 = arith.extsi %12 : i32 to i64
    %13 = arith.cmpi sgt, %arg2, %14 : i64
    %15 = scf.if %13 -> (i64) {
      %16 = arith.constant 1 : i32
      %17 = arith.extsi %16 : i32 to i64
      scf.yield %17 : i64
    } else {
      scf.yield %11 : i64
    }
    %18 = arith.subi %arg0, %arg2 : i64
    %19 = arith.divsi %18, %arg1 : i64
    %20 = arith.cmpi slt, %19, %15 : i64
    cf.cond_br %20, ^bb0, ^bb1
    ^bb0:
      func.return
    ^bb1:
      cf.br ^bb2
    ^bb2:
    %21 = arith.constant 2 : i32
    %23 = arith.extsi %21 : i32 to i64
    %22 = arith.divsi %arg0, %23 : i64
    %24 = arith.subi %22, %arg2 : i64
    %25 = arith.divsi %24, %arg1 : i64
    %26 = arith.constant 1 : i32
    %28 = arith.extsi %26 : i32 to i64
    %27 = arith.addi %25, %28 : i64
    %29 = llvm.mlir.constant(1 : i64) : i64
    %30 = llvm.alloca %29 x i64 : (i64) -> !llvm.ptr
    llvm.store %27, %30 : i64, !llvm.ptr
    %31 = llvm.load %30 : !llvm.ptr -> i64
    %32 = arith.cmpi slt, %31, %15 : i64
    cf.cond_br %32, ^bb3, ^bb4
    ^bb3:
      llvm.store %15, %30 : i64, !llvm.ptr
      cf.br ^bb5
    ^bb4:
      cf.br ^bb5
    ^bb5:
    %33 = llvm.load %30 : !llvm.ptr -> i64
    %34 = arith.cmpi sle, %33, %19 : i64
    cf.cond_br %34, ^bb6, ^bb7
    ^bb6:
      %35 = llvm.load %30 : !llvm.ptr -> i64
      %36 = arith.subi %19, %35 : i64
      %37 = arith.constant 1 : i32
      %39 = arith.extsi %37 : i32 to i64
      %38 = arith.addi %36, %39 : i64
      %41 = arith.constant 0 : i32
      %42 = arith.extsi %41 : i32 to i64
      %43 = llvm.getelementptr %arg4[%42] : (!llvm.ptr, i64) -> !llvm.ptr, i64
      %40 = llvm.load %43 : !llvm.ptr -> i64
      %44 = arith.constant 1 : i32
      %46 = arith.extsi %44 : i32 to i64
      %45 = arith.addi %arg3, %46 : i64
      %47 = arith.muli %45, %38 : i64
      %48 = arith.addi %40, %47 : i64
      %49 = arith.constant 0 : i32
      %50 = arith.extsi %49 : i32 to i64
      %51 = llvm.getelementptr %arg4[%50] : (!llvm.ptr, i64) -> !llvm.ptr, i64
      llvm.store %48, %51 : i64, !llvm.ptr
      %52 = arith.subi %arg0, %arg1 : i64
      %53 = arith.subi %52, %arg2 : i64
      %54 = arith.divsi %53, %arg1 : i64
      %55 = llvm.load %30 : !llvm.ptr -> i64
      %56 = arith.cmpi sge, %54, %55 : i64
      cf.cond_br %56, ^bb9, ^bb10
      ^bb9:
        %57 = arith.cmpi sgt, %54, %19 : i64
        %58 = scf.if %57 -> (i64) {
          scf.yield %19 : i64
        } else {
          scf.yield %54 : i64
        }
        %59 = llvm.load %30 : !llvm.ptr -> i64
        %60 = arith.subi %58, %59 : i64
        %61 = arith.constant 1 : i32
        %63 = arith.extsi %61 : i32 to i64
        %62 = arith.addi %60, %63 : i64
        %65 = arith.constant 0 : i32
        %66 = arith.extsi %65 : i32 to i64
        %67 = llvm.getelementptr %arg4[%66] : (!llvm.ptr, i64) -> !llvm.ptr, i64
        %64 = llvm.load %67 : !llvm.ptr -> i64
        %68 = arith.constant 2 : i32
        %70 = arith.extsi %68 : i32 to i64
        %69 = arith.addi %arg3, %70 : i64
        %71 = arith.muli %69, %62 : i64
        %72 = arith.addi %64, %71 : i64
        %73 = arith.constant 0 : i32
        %74 = arith.extsi %73 : i32 to i64
        %75 = llvm.getelementptr %arg4[%74] : (!llvm.ptr, i64) -> !llvm.ptr, i64
        llvm.store %72, %75 : i64, !llvm.ptr
        cf.br ^bb11
      ^bb10:
        cf.br ^bb11
      ^bb11:
      cf.br ^bb8
    ^bb7:
      cf.br ^bb8
    ^bb8:
    %76 = llvm.mlir.constant(1 : i64) : i64
    %77 = llvm.alloca %76 x i64 : (i64) -> !llvm.ptr
    llvm.store %15, %77 : i64, !llvm.ptr
    cf.br ^bb12
    ^bb12:
    %78 = llvm.load %77 : !llvm.ptr -> i64
    %79 = llvm.load %30 : !llvm.ptr -> i64
    %80 = arith.cmpi slt, %78, %79 : i64
    cf.cond_br %80, ^bb13, ^bb14
    ^bb13:
      %81 = llvm.load %77 : !llvm.ptr -> i64
      %82 = arith.muli %81, %arg1 : i64
      %83 = arith.addi %arg2, %82 : i64
      %85 = arith.constant 1 : i32
      %87 = arith.extsi %85 : i32 to i64
      %86 = arith.addi %arg3, %87 : i64
      func.call @dfs(%arg0, %83, %arg1, %86, %arg4) : (i64, i64, i64, i64, !llvm.ptr) -> ()
      %88 = llvm.load %77 : !llvm.ptr -> i64
      %89 = arith.constant 1 : i32
      %91 = arith.extsi %89 : i32 to i64
      %90 = arith.addi %88, %91 : i64
      llvm.store %90, %77 : i64, !llvm.ptr
      cf.br ^bb12
    ^bb14:
    func.return
  }
  func.func @main() -> i32 {
    %92 = arith.constant 5000000 : i32
    %93 = arith.extsi %92 : i32 to i64
    %94 = arith.constant 100000 : i32
    %96 = arith.extsi %94 : i32 to i64
    %95 = arith.cmpi sle, %93, %96 : i64
    cf.cond_br %95, ^bb15, ^bb16
    ^bb15:
      %98 = arith.constant 1 : i32
      %99 = arith.constant 8 : i32
      %100 = arith.extsi %98 : i32 to i64
      %101 = arith.extsi %99 : i32 to i64
      %97 = func.call @calloc(%100, %101) : (i64, i64) -> !llvm.ptr
      %102 = llvm.mlir.zero : !llvm.ptr
      %103 = llvm.icmp "eq" %97, %102 : !llvm.ptr
      cf.cond_br %103, ^bb18, ^bb19
      ^bb18:
        %104 = arith.constant 1 : i32
        func.return %104 : i32
      ^bb19:
        cf.br ^bb20
      ^bb20:
      %106 = arith.constant 1 : i32
      %107 = arith.constant 0 : i32
      %108 = arith.constant 0 : i32
      %109 = arith.extsi %106 : i32 to i64
      %110 = arith.extsi %107 : i32 to i64
      %111 = arith.extsi %108 : i32 to i64
      func.call @dfs(%93, %109, %110, %111, %97) : (i64, i64, i64, i64, !llvm.ptr) -> ()
      %112 = arith.constant 2 : i32
      %114 = arith.constant 0 : i32
      %115 = arith.extsi %114 : i32 to i64
      %116 = llvm.getelementptr %97[%115] : (!llvm.ptr, i64) -> !llvm.ptr, i64
      %113 = llvm.load %116 : !llvm.ptr -> i64
      %118 = arith.extsi %112 : i32 to i64
      %117 = arith.muli %118, %113 : i64
      %119 = arith.constant 1 : i32
      %121 = arith.extsi %119 : i32 to i64
      %120 = arith.addi %93, %121 : i64
      %122 = arith.muli %93, %120 : i64
      %123 = arith.constant 2 : i32
      %125 = arith.extsi %123 : i32 to i64
      %124 = arith.divsi %122, %125 : i64
      %126 = arith.addi %117, %124 : i64
      %127 = llvm.mlir.addressof @str_0 : !llvm.ptr
      %128 = llvm.call @printf(%127, %126) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
      func.call @free(%97) : (!llvm.ptr) -> ()
      cf.br ^bb17
    ^bb16:
      %130 = llvm.mlir.addressof @str_1 : !llvm.ptr
      %131 = llvm.call @printf(%130) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr) -> i32
      cf.br ^bb17
    ^bb17:
    %132 = arith.constant 0 : i32
    func.return %132 : i32
  }
}