Problem 401

SIGMA2(10^15) mod 10^9, where SIGMA2 = summatory of sigma_2.

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 401
# SIGMA2(10^15) mod 10^9, where SIGMA2 = summatory of sigma_2.

function mulmod(a: i64, b: i64, mod: i64) -> i64 {
    let x: i128 = (a % mod) as i128
    let y: i128 = (b % mod) as i128
    let r: i128 = (x * y) % (mod as i128)
    if r < (0 as i128) {
        return (r + (mod as i128)) as i64
    }
    return r as i64
}

# 1^2 + ... + n^2 = n(n+1)(2n+1)/6, computed mod `mod`.
function sum_sq_mod(n0: i64, mod: i64) -> i64 {
    if n0 <= 0 {
        return 0
    }
    let mut a: i64 = n0
    let mut b: i64 = n0 + 1
    let mut c: i64 = 2 * n0 + 1
    if a % 2 == 0 {
        a = a / 2
    } elif b % 2 == 0 {
        b = b / 2
    } else {
        c = c / 2
    }
    if a % 3 == 0 {
        a = a / 3
    } elif b % 3 == 0 {
        b = b / 3
    } else {
        c = c / 3
    }
    return mulmod(mulmod(a, b, mod), c, mod)
}

function main() -> i32 {
    let N: i64 = 1000000000000000
    let MOD: i64 = 1000000000
    let mut ans: i64 = 0
    let mut l: i64 = 1
    while l <= N {
        let q: i64 = N / l
        let r: i64 = N / q
        let mut block: i64 = sum_sq_mod(r, MOD) - sum_sq_mod(l - 1, MOD)
        if block < 0 {
            block = block + MOD
        }
        ans = ans + mulmod(q % MOD, block, MOD)
        if ans >= MOD {
            ans = ans % MOD
        }
        l = r + 1
    }
    printf("%lld\n", ans)
    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 mulmod_i64_i64_i64(int64_t a, int64_t b, int64_t mod);
int64_t sum_sq_mod_i64_i64(int64_t n0, int64_t mod);
int32_t main(void);

int64_t mulmod_i64_i64_i64(int64_t a, int64_t b, int64_t mod) {
    __int128 x = ((__int128)(FLOW_CHECKED_MOD((a), (mod))));
    __int128 y = ((__int128)(FLOW_CHECKED_MOD((b), (mod))));
    __int128 r = FLOW_CHECKED_MOD(((x * y)), (((__int128)(mod))));
    if (r < ((__int128)(0))) {
        return ((int64_t)((r + ((__int128)(mod)))));
    }
    return ((int64_t)(r));
}

int64_t sum_sq_mod_i64_i64(int64_t n0, int64_t mod) {
    if (n0 <= 0) {
        return 0;
    }
    int64_t a = n0;
    int64_t b = (n0 + 1);
    int64_t c = ((2 * n0) + 1);
    if (FLOW_CHECKED_MOD((a), (2)) == 0) {
        a = FLOW_CHECKED_DIV((a), (2));
    } else if (FLOW_CHECKED_MOD((b), (2)) == 0) {
        b = FLOW_CHECKED_DIV((b), (2));
    } else {
        c = FLOW_CHECKED_DIV((c), (2));
    }
    if (FLOW_CHECKED_MOD((a), (3)) == 0) {
        a = FLOW_CHECKED_DIV((a), (3));
    } else if (FLOW_CHECKED_MOD((b), (3)) == 0) {
        b = FLOW_CHECKED_DIV((b), (3));
    } else {
        c = FLOW_CHECKED_DIV((c), (3));
    }
    return mulmod_i64_i64_i64(mulmod_i64_i64_i64(a, b, mod), c, mod);
}

int32_t main(void) {
    int64_t N = 1000000000000000;
    int64_t MOD = 1000000000;
    int64_t ans = 0;
    int64_t l = 1;
    while (l <= N) {
        int64_t q = FLOW_CHECKED_DIV((N), (l));
        int64_t r = FLOW_CHECKED_DIV((N), (q));
        int64_t block = (sum_sq_mod_i64_i64(r, MOD) - sum_sq_mod_i64_i64((l - 1), MOD));
        if (block < 0) {
            block = (block + MOD);
        }
        ans = (ans + mulmod_i64_i64_i64(FLOW_CHECKED_MOD((q), (MOD)), block, MOD));
        if (ans >= MOD) {
            ans = FLOW_CHECKED_MOD((ans), (MOD));
        }
        l = (r + 1);
    }
    printf("%lld\n", ans);
    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 @mulmod(%arg0: i64, %arg1: i64, %arg2: i64) -> i64 {
    %0 = arith.remsi %arg0, %arg2 : i64
    %1 = arith.extsi %0 : i64 to i128
    %2 = arith.remsi %arg1, %arg2 : i64
    %3 = arith.extsi %2 : i64 to i128
    %5 = arith.trunci %1 : i128 to i64
    %6 = arith.trunci %3 : i128 to i64
    %4 = arith.muli %5, %6 : i64
    %7 = arith.extsi %arg2 : i64 to i128
    %9 = arith.trunci %7 : i128 to i64
    %8 = arith.remsi %4, %9 : i64
    %10 = arith.extsi %8 : i64 to i128
    %11 = arith.constant 0 : i32
    %12 = arith.extsi %11 : i32 to i128
    %14 = arith.trunci %10 : i128 to i64
    %15 = arith.trunci %12 : i128 to i64
    %13 = arith.cmpi slt, %14, %15 : i64
    cf.cond_br %13, ^bb0, ^bb1
    ^bb0:
      %16 = arith.extsi %arg2 : i64 to i128
      %18 = arith.trunci %10 : i128 to i64
      %19 = arith.trunci %16 : i128 to i64
      %17 = arith.addi %18, %19 : i64
      func.return %17 : i64
    ^bb1:
      cf.br ^bb2
    ^bb2:
    %20 = arith.trunci %10 : i128 to i64
    func.return %20 : i64
  }
  func.func @sum_sq_mod(%arg0: i64, %arg1: i64) -> i64 {
    %21 = arith.constant 0 : i32
    %23 = arith.extsi %21 : i32 to i64
    %22 = arith.cmpi sle, %arg0, %23 : i64
    cf.cond_br %22, ^bb3, ^bb4
    ^bb3:
      %24 = arith.constant 0 : i32
      %25 = arith.extsi %24 : i32 to i64
      func.return %25 : i64
    ^bb4:
      cf.br ^bb5
    ^bb5:
    %26 = llvm.mlir.constant(1 : i64) : i64
    %27 = llvm.alloca %26 x i64 : (i64) -> !llvm.ptr
    llvm.store %arg0, %27 : i64, !llvm.ptr
    %28 = arith.constant 1 : i32
    %30 = arith.extsi %28 : i32 to i64
    %29 = arith.addi %arg0, %30 : i64
    %31 = llvm.mlir.constant(1 : i64) : i64
    %32 = llvm.alloca %31 x i64 : (i64) -> !llvm.ptr
    llvm.store %29, %32 : i64, !llvm.ptr
    %33 = arith.constant 2 : i32
    %35 = arith.extsi %33 : i32 to i64
    %34 = arith.muli %35, %arg0 : i64
    %36 = arith.constant 1 : i32
    %38 = arith.extsi %36 : i32 to i64
    %37 = arith.addi %34, %38 : i64
    %39 = llvm.mlir.constant(1 : i64) : i64
    %40 = llvm.alloca %39 x i64 : (i64) -> !llvm.ptr
    llvm.store %37, %40 : i64, !llvm.ptr
    %41 = llvm.load %27 : !llvm.ptr -> i64
    %42 = arith.constant 2 : i32
    %44 = arith.extsi %42 : i32 to i64
    %43 = arith.remsi %41, %44 : i64
    %45 = arith.constant 0 : i32
    %47 = arith.extsi %45 : i32 to i64
    %46 = arith.cmpi eq, %43, %47 : i64
    cf.cond_br %46, ^bb6, ^bb7
    ^bb6:
      %48 = llvm.load %27 : !llvm.ptr -> i64
      %49 = arith.constant 2 : i32
      %51 = arith.extsi %49 : i32 to i64
      %50 = arith.divsi %48, %51 : i64
      llvm.store %50, %27 : i64, !llvm.ptr
      cf.br ^bb8
    ^bb7:
      %52 = llvm.load %32 : !llvm.ptr -> i64
      %53 = arith.constant 2 : i32
      %55 = arith.extsi %53 : i32 to i64
      %54 = arith.remsi %52, %55 : i64
      %56 = arith.constant 0 : i32
      %58 = arith.extsi %56 : i32 to i64
      %57 = arith.cmpi eq, %54, %58 : i64
      cf.cond_br %57, ^bb10, ^bb9
    ^bb10:
      %59 = llvm.load %32 : !llvm.ptr -> i64
      %60 = arith.constant 2 : i32
      %62 = arith.extsi %60 : i32 to i64
      %61 = arith.divsi %59, %62 : i64
      llvm.store %61, %32 : i64, !llvm.ptr
      cf.br ^bb8
    ^bb9:
      %63 = llvm.load %40 : !llvm.ptr -> i64
      %64 = arith.constant 2 : i32
      %66 = arith.extsi %64 : i32 to i64
      %65 = arith.divsi %63, %66 : i64
      llvm.store %65, %40 : i64, !llvm.ptr
      cf.br ^bb8
    ^bb8:
    %67 = llvm.load %27 : !llvm.ptr -> i64
    %68 = arith.constant 3 : i32
    %70 = arith.extsi %68 : i32 to i64
    %69 = arith.remsi %67, %70 : i64
    %71 = arith.constant 0 : i32
    %73 = arith.extsi %71 : i32 to i64
    %72 = arith.cmpi eq, %69, %73 : i64
    cf.cond_br %72, ^bb11, ^bb12
    ^bb11:
      %74 = llvm.load %27 : !llvm.ptr -> i64
      %75 = arith.constant 3 : i32
      %77 = arith.extsi %75 : i32 to i64
      %76 = arith.divsi %74, %77 : i64
      llvm.store %76, %27 : i64, !llvm.ptr
      cf.br ^bb13
    ^bb12:
      %78 = llvm.load %32 : !llvm.ptr -> i64
      %79 = arith.constant 3 : i32
      %81 = arith.extsi %79 : i32 to i64
      %80 = arith.remsi %78, %81 : i64
      %82 = arith.constant 0 : i32
      %84 = arith.extsi %82 : i32 to i64
      %83 = arith.cmpi eq, %80, %84 : i64
      cf.cond_br %83, ^bb15, ^bb14
    ^bb15:
      %85 = llvm.load %32 : !llvm.ptr -> i64
      %86 = arith.constant 3 : i32
      %88 = arith.extsi %86 : i32 to i64
      %87 = arith.divsi %85, %88 : i64
      llvm.store %87, %32 : i64, !llvm.ptr
      cf.br ^bb13
    ^bb14:
      %89 = llvm.load %40 : !llvm.ptr -> i64
      %90 = arith.constant 3 : i32
      %92 = arith.extsi %90 : i32 to i64
      %91 = arith.divsi %89, %92 : i64
      llvm.store %91, %40 : i64, !llvm.ptr
      cf.br ^bb13
    ^bb13:
    %95 = llvm.load %27 : !llvm.ptr -> i64
    %96 = llvm.load %32 : !llvm.ptr -> i64
    %94 = func.call @mulmod(%95, %96, %arg1) : (i64, i64, i64) -> i64
    %97 = llvm.load %40 : !llvm.ptr -> i64
    %93 = func.call @mulmod(%94, %97, %arg1) : (i64, i64, i64) -> i64
    func.return %93 : i64
  }
  func.func @main() -> i32 {
    %98 = arith.constant 999995705032704 : i32
    %99 = arith.extsi %98 : i32 to i64
    %100 = arith.constant 1000000000 : i32
    %101 = arith.extsi %100 : i32 to i64
    %102 = arith.constant 0 : i32
    %103 = arith.extsi %102 : i32 to i64
    %104 = llvm.mlir.constant(1 : i64) : i64
    %105 = llvm.alloca %104 x i64 : (i64) -> !llvm.ptr
    llvm.store %103, %105 : i64, !llvm.ptr
    %106 = arith.constant 1 : i32
    %107 = arith.extsi %106 : i32 to i64
    %108 = llvm.mlir.constant(1 : i64) : i64
    %109 = llvm.alloca %108 x i64 : (i64) -> !llvm.ptr
    llvm.store %107, %109 : i64, !llvm.ptr
    cf.br ^bb16
    ^bb16:
    %110 = llvm.load %109 : !llvm.ptr -> i64
    %111 = arith.cmpi sle, %110, %99 : i64
    cf.cond_br %111, ^bb17, ^bb18
    ^bb17:
      %112 = llvm.load %109 : !llvm.ptr -> i64
      %113 = arith.divsi %99, %112 : i64
      %114 = arith.divsi %99, %113 : i64
      %115 = func.call @sum_sq_mod(%114, %101) : (i64, i64) -> i64
      %117 = llvm.load %109 : !llvm.ptr -> i64
      %118 = arith.constant 1 : i32
      %120 = arith.extsi %118 : i32 to i64
      %119 = arith.subi %117, %120 : i64
      %116 = func.call @sum_sq_mod(%119, %101) : (i64, i64) -> i64
      %121 = arith.subi %115, %116 : i64
      %122 = llvm.mlir.constant(1 : i64) : i64
      %123 = llvm.alloca %122 x i64 : (i64) -> !llvm.ptr
      llvm.store %121, %123 : i64, !llvm.ptr
      %124 = llvm.load %123 : !llvm.ptr -> i64
      %125 = arith.constant 0 : i32
      %127 = arith.extsi %125 : i32 to i64
      %126 = arith.cmpi slt, %124, %127 : i64
      cf.cond_br %126, ^bb19, ^bb20
      ^bb19:
        %128 = llvm.load %123 : !llvm.ptr -> i64
        %129 = arith.addi %128, %101 : i64
        llvm.store %129, %123 : i64, !llvm.ptr
        cf.br ^bb21
      ^bb20:
        cf.br ^bb21
      ^bb21:
      %130 = llvm.load %105 : !llvm.ptr -> i64
      %132 = arith.remsi %113, %101 : i64
      %133 = llvm.load %123 : !llvm.ptr -> i64
      %131 = func.call @mulmod(%132, %133, %101) : (i64, i64, i64) -> i64
      %134 = arith.addi %130, %131 : i64
      llvm.store %134, %105 : i64, !llvm.ptr
      %135 = llvm.load %105 : !llvm.ptr -> i64
      %136 = arith.cmpi sge, %135, %101 : i64
      cf.cond_br %136, ^bb22, ^bb23
      ^bb22:
        %137 = llvm.load %105 : !llvm.ptr -> i64
        %138 = arith.remsi %137, %101 : i64
        llvm.store %138, %105 : i64, !llvm.ptr
        cf.br ^bb24
      ^bb23:
        cf.br ^bb24
      ^bb24:
      %139 = arith.constant 1 : i32
      %141 = arith.extsi %139 : i32 to i64
      %140 = arith.addi %114, %141 : i64
      llvm.store %140, %109 : i64, !llvm.ptr
      cf.br ^bb16
    ^bb18:
    %142 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %143 = llvm.load %105 : !llvm.ptr -> i64
    %144 = llvm.call @printf(%142, %143) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %145 = arith.constant 0 : i32
    func.return %145 : i32
  }
}