Problem 555

McCarthy 91 Function S(p,m) = sum over admissible (k,s) of SF(m,k,s). Closed form: sum_{d=1}^{p-1} sum_{s=d,2d,...}^{p-d} d*(2m+d-2s+1)/2

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 555
# McCarthy 91 Function
# S(p,m) = sum over admissible (k,s) of SF(m,k,s). Closed form:
# sum_{d=1}^{p-1} sum_{s=d,2d,...}^{p-d} d*(2m+d-2s+1)/2

function S(p: i64, m: i64) -> i64 {
    let mut total: i64 = 0
    let mut d: i64 = 1
    while d < p {
        let mut s: i64 = d
        while s <= p - d {
            total = total + (d * (2 * m + d - 2 * s + 1)) / 2
            s = s + d
        }
        d = d + 1
    }
    return total
}

function main() -> i32 {
    printf("%lld\n", S(1000000, 1000000))
    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 S_i64_i64(int64_t p, int64_t m);
int32_t main(void);

int64_t S_i64_i64(int64_t p, int64_t m) {
    int64_t total = 0;
    int64_t d = 1;
    while (d < p) {
        int64_t s = d;
        while (s <= (p - d)) {
            total = (total + FLOW_CHECKED_DIV(((d * ((((2 * m) + d) - (2 * s)) + 1))), (2)));
            s = (s + d);
        }
        d = (d + 1);
    }
    return total;
}

int32_t main(void) {
    printf("%lld\n", S_i64_i64(1000000, 1000000));
    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 @S(%arg0: i64, %arg1: i64) -> i64 {
    %0 = arith.constant 0 : i32
    %1 = arith.extsi %0 : i32 to i64
    %2 = llvm.mlir.constant(1 : i64) : i64
    %3 = llvm.alloca %2 x i64 : (i64) -> !llvm.ptr
    llvm.store %1, %3 : i64, !llvm.ptr
    %4 = arith.constant 1 : i32
    %5 = arith.extsi %4 : i32 to i64
    %6 = llvm.mlir.constant(1 : i64) : i64
    %7 = llvm.alloca %6 x i64 : (i64) -> !llvm.ptr
    llvm.store %5, %7 : i64, !llvm.ptr
    cf.br ^bb0
    ^bb0:
    %8 = llvm.load %7 : !llvm.ptr -> i64
    %9 = arith.cmpi slt, %8, %arg0 : i64
    cf.cond_br %9, ^bb1, ^bb2
    ^bb1:
      %10 = llvm.load %7 : !llvm.ptr -> i64
      %11 = llvm.mlir.constant(1 : i64) : i64
      %12 = llvm.alloca %11 x i64 : (i64) -> !llvm.ptr
      llvm.store %10, %12 : i64, !llvm.ptr
      cf.br ^bb3
      ^bb3:
      %13 = llvm.load %12 : !llvm.ptr -> i64
      %14 = llvm.load %7 : !llvm.ptr -> i64
      %15 = arith.subi %arg0, %14 : i64
      %16 = arith.cmpi sle, %13, %15 : i64
      cf.cond_br %16, ^bb4, ^bb5
      ^bb4:
        %17 = llvm.load %3 : !llvm.ptr -> i64
        %18 = llvm.load %7 : !llvm.ptr -> i64
        %19 = arith.constant 2 : i32
        %21 = arith.extsi %19 : i32 to i64
        %20 = arith.muli %21, %arg1 : i64
        %22 = llvm.load %7 : !llvm.ptr -> i64
        %23 = arith.addi %20, %22 : i64
        %24 = arith.constant 2 : i32
        %25 = llvm.load %12 : !llvm.ptr -> i64
        %27 = arith.extsi %24 : i32 to i64
        %26 = arith.muli %27, %25 : i64
        %28 = arith.subi %23, %26 : i64
        %29 = arith.constant 1 : i32
        %31 = arith.extsi %29 : i32 to i64
        %30 = arith.addi %28, %31 : i64
        %32 = arith.muli %18, %30 : i64
        %33 = arith.constant 2 : i32
        %35 = arith.extsi %33 : i32 to i64
        %34 = arith.divsi %32, %35 : i64
        %36 = arith.addi %17, %34 : i64
        llvm.store %36, %3 : i64, !llvm.ptr
        %37 = llvm.load %12 : !llvm.ptr -> i64
        %38 = llvm.load %7 : !llvm.ptr -> i64
        %39 = arith.addi %37, %38 : i64
        llvm.store %39, %12 : i64, !llvm.ptr
        cf.br ^bb3
      ^bb5:
      %40 = llvm.load %7 : !llvm.ptr -> i64
      %41 = arith.constant 1 : i32
      %43 = arith.extsi %41 : i32 to i64
      %42 = arith.addi %40, %43 : i64
      llvm.store %42, %7 : i64, !llvm.ptr
      cf.br ^bb0
    ^bb2:
    %44 = llvm.load %3 : !llvm.ptr -> i64
    func.return %44 : i64
  }
  func.func @main() -> i32 {
    %45 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %47 = arith.constant 1000000 : i32
    %48 = arith.constant 1000000 : i32
    %49 = arith.extsi %47 : i32 to i64
    %50 = arith.extsi %48 : i32 to i64
    %46 = func.call @S(%49, %50) : (i64, i64) -> i64
    %51 = llvm.call @printf(%45, %46) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %52 = arith.constant 0 : i32
    func.return %52 : i32
  }
}