Problem 312

C(C(C(10000))) mod 13^8 using period structure of C(n).

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 312
# C(C(C(10000))) mod 13^8 using period structure of C(n).

function modpow(base: i64, exp: i64, mod: i64) -> i64 {
    let mut r: i64 = 1
    let mut b: i64 = base % mod
    let mut e: i64 = exp
    while e > 0 {
        if e % 2 == 1 { r = (r * b) % mod }
        b = (b * b) % mod
        e = e / 2
    }
    return r
}

function C(n: i64, mod: i64) -> i64 {
    if n == 1 || n == 2 { return 1 }
    if n == 3 { return 8 % mod }
    # exponent = (3^(n-2) - 3) / 2
    # Need 12^exponent mod mod. Use Euler when gcd(12,mod)>1 carefully via CRT or direct.
    # For our mods (6*13^k or 13^8), use recursive C definition with modular mul when n small enough
    # after reductions n is small.
    let mut c1: i64 = 13824 % mod
    if n == 4 { return c1 }
    let mut i: i64 = 4
    while i < n {
        # cn = (3*c1)^3 mod mod
        let t: i64 = (3 * c1) % mod
        let t2: i64 = (t * t) % mod
        c1 = (t2 * t) % mod
        i = i + 1
    }
    return c1
}

function C_formula(n: i64, mod: i64) -> i64 {
    if n <= 2 { return 1 }
    if n == 3 { return 8 % mod }
    # For larger n after period reduction, recursive definition is fine if n is not huge.
    # After reductions: n around hundreds to millions — recursive loop OK for n~10^6? 
    # C(10000) needs 10000 iterations — fine.
    return C(n, mod)
}

function main() -> i32 {
    let x1: i64 = 10000 % (6 * 13 * 13)
    let x2: i64 = C_formula(x1, 6 * 13 * 13 * 13 * 13)
    let x3: i64 = C_formula(x2, 6 * 13 * 13 * 13 * 13 * 13 * 13)
    let x4: i64 = C_formula(x3, 13 * 13 * 13 * 13 * 13 * 13 * 13 * 13)
    printf("%lld\n", x4)
    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 modpow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod);
int64_t C_i64_i64(int64_t n, int64_t mod);
int64_t C_formula_i64_i64(int64_t n, int64_t mod);
int32_t main(void);

int64_t modpow_i64_i64_i64(int64_t base, int64_t exp, int64_t mod) {
    int64_t r = 1;
    int64_t b = FLOW_CHECKED_MOD((base), (mod));
    int64_t e = exp;
    while (e > 0) {
        if (FLOW_CHECKED_MOD((e), (2)) == 1) {
            r = FLOW_CHECKED_MOD(((r * b)), (mod));
        }
        b = FLOW_CHECKED_MOD(((b * b)), (mod));
        e = FLOW_CHECKED_DIV((e), (2));
    }
    return r;
}

int64_t C_i64_i64(int64_t n, int64_t mod) {
    if ((n == 1 || n == 2)) {
        return 1;
    }
    if (n == 3) {
        return FLOW_CHECKED_MOD((8), (mod));
    }
    int64_t c1 = FLOW_CHECKED_MOD((13824), (mod));
    if (n == 4) {
        return c1;
    }
    int64_t i = 4;
    while (i < n) {
        int64_t t = FLOW_CHECKED_MOD(((3 * c1)), (mod));
        int64_t t2 = FLOW_CHECKED_MOD(((t * t)), (mod));
        c1 = FLOW_CHECKED_MOD(((t2 * t)), (mod));
        i = (i + 1);
    }
    return c1;
}

int64_t C_formula_i64_i64(int64_t n, int64_t mod) {
    if (n <= 2) {
        return 1;
    }
    if (n == 3) {
        return FLOW_CHECKED_MOD((8), (mod));
    }
    return C_i64_i64(n, mod);
}

int32_t main(void) {
    int64_t x1 = FLOW_CHECKED_MOD((10000), (((6 * 13) * 13)));
    int64_t x2 = C_formula_i64_i64(x1, ((((6 * 13) * 13) * 13) * 13));
    int64_t x3 = C_formula_i64_i64(x2, ((((((6 * 13) * 13) * 13) * 13) * 13) * 13));
    int64_t x4 = C_formula_i64_i64(x3, (((((((13 * 13) * 13) * 13) * 13) * 13) * 13) * 13));
    printf("%lld\n", x4);
    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 @modpow(%arg0: i64, %arg1: i64, %arg2: i64) -> i64 {
    %0 = arith.constant 1 : 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.remsi %arg0, %arg2 : i64
    %5 = llvm.mlir.constant(1 : i64) : i64
    %6 = llvm.alloca %5 x i64 : (i64) -> !llvm.ptr
    llvm.store %4, %6 : i64, !llvm.ptr
    %7 = llvm.mlir.constant(1 : i64) : i64
    %8 = llvm.alloca %7 x i64 : (i64) -> !llvm.ptr
    llvm.store %arg1, %8 : i64, !llvm.ptr
    cf.br ^bb0
    ^bb0:
    %9 = llvm.load %8 : !llvm.ptr -> i64
    %10 = arith.constant 0 : i32
    %12 = arith.extsi %10 : i32 to i64
    %11 = arith.cmpi sgt, %9, %12 : i64
    cf.cond_br %11, ^bb1, ^bb2
    ^bb1:
      %13 = llvm.load %8 : !llvm.ptr -> i64
      %14 = arith.constant 2 : i32
      %16 = arith.extsi %14 : i32 to i64
      %15 = arith.remsi %13, %16 : i64
      %17 = arith.constant 1 : i32
      %19 = arith.extsi %17 : i32 to i64
      %18 = arith.cmpi eq, %15, %19 : i64
      cf.cond_br %18, ^bb3, ^bb4
      ^bb3:
        %20 = llvm.load %3 : !llvm.ptr -> i64
        %21 = llvm.load %6 : !llvm.ptr -> i64
        %22 = arith.muli %20, %21 : i64
        %23 = arith.remsi %22, %arg2 : i64
        llvm.store %23, %3 : i64, !llvm.ptr
        cf.br ^bb5
      ^bb4:
        cf.br ^bb5
      ^bb5:
      %24 = llvm.load %6 : !llvm.ptr -> i64
      %25 = llvm.load %6 : !llvm.ptr -> i64
      %26 = arith.muli %24, %25 : i64
      %27 = arith.remsi %26, %arg2 : i64
      llvm.store %27, %6 : i64, !llvm.ptr
      %28 = llvm.load %8 : !llvm.ptr -> i64
      %29 = arith.constant 2 : i32
      %31 = arith.extsi %29 : i32 to i64
      %30 = arith.divsi %28, %31 : i64
      llvm.store %30, %8 : i64, !llvm.ptr
      cf.br ^bb0
    ^bb2:
    %32 = llvm.load %3 : !llvm.ptr -> i64
    func.return %32 : i64
  }
  func.func @C(%arg0: i64, %arg1: i64) -> i64 {
    %33 = arith.constant 1 : i32
    %35 = arith.extsi %33 : i32 to i64
    %34 = arith.cmpi eq, %arg0, %35 : i64
    %36 = scf.if %34 -> (i1) {
      %37 = arith.constant true
      scf.yield %37 : i1
    } else {
      %38 = arith.constant 2 : i32
      %40 = arith.extsi %38 : i32 to i64
      %39 = arith.cmpi eq, %arg0, %40 : i64
      scf.yield %39 : i1
    }
    cf.cond_br %36, ^bb6, ^bb7
    ^bb6:
      %41 = arith.constant 1 : i32
      %42 = arith.extsi %41 : i32 to i64
      func.return %42 : i64
    ^bb7:
      cf.br ^bb8
    ^bb8:
    %43 = arith.constant 3 : i32
    %45 = arith.extsi %43 : i32 to i64
    %44 = arith.cmpi eq, %arg0, %45 : i64
    cf.cond_br %44, ^bb9, ^bb10
    ^bb9:
      %46 = arith.constant 8 : i32
      %48 = arith.extsi %46 : i32 to i64
      %47 = arith.remsi %48, %arg1 : i64
      func.return %47 : i64
    ^bb10:
      cf.br ^bb11
    ^bb11:
    %49 = arith.constant 13824 : i32
    %51 = arith.extsi %49 : i32 to i64
    %50 = arith.remsi %51, %arg1 : i64
    %52 = llvm.mlir.constant(1 : i64) : i64
    %53 = llvm.alloca %52 x i64 : (i64) -> !llvm.ptr
    llvm.store %50, %53 : i64, !llvm.ptr
    %54 = arith.constant 4 : i32
    %56 = arith.extsi %54 : i32 to i64
    %55 = arith.cmpi eq, %arg0, %56 : i64
    cf.cond_br %55, ^bb12, ^bb13
    ^bb12:
      %57 = llvm.load %53 : !llvm.ptr -> i64
      func.return %57 : i64
    ^bb13:
      cf.br ^bb14
    ^bb14:
    %58 = arith.constant 4 : i32
    %59 = arith.extsi %58 : i32 to i64
    %60 = llvm.mlir.constant(1 : i64) : i64
    %61 = llvm.alloca %60 x i64 : (i64) -> !llvm.ptr
    llvm.store %59, %61 : i64, !llvm.ptr
    cf.br ^bb15
    ^bb15:
    %62 = llvm.load %61 : !llvm.ptr -> i64
    %63 = arith.cmpi slt, %62, %arg0 : i64
    cf.cond_br %63, ^bb16, ^bb17
    ^bb16:
      %64 = arith.constant 3 : i32
      %65 = llvm.load %53 : !llvm.ptr -> i64
      %67 = arith.extsi %64 : i32 to i64
      %66 = arith.muli %67, %65 : i64
      %68 = arith.remsi %66, %arg1 : i64
      %69 = arith.muli %68, %68 : i64
      %70 = arith.remsi %69, %arg1 : i64
      %71 = arith.muli %70, %68 : i64
      %72 = arith.remsi %71, %arg1 : i64
      llvm.store %72, %53 : i64, !llvm.ptr
      %73 = llvm.load %61 : !llvm.ptr -> i64
      %74 = arith.constant 1 : i32
      %76 = arith.extsi %74 : i32 to i64
      %75 = arith.addi %73, %76 : i64
      llvm.store %75, %61 : i64, !llvm.ptr
      cf.br ^bb15
    ^bb17:
    %77 = llvm.load %53 : !llvm.ptr -> i64
    func.return %77 : i64
  }
  func.func @C_formula(%arg0: i64, %arg1: i64) -> i64 {
    %78 = arith.constant 2 : i32
    %80 = arith.extsi %78 : i32 to i64
    %79 = arith.cmpi sle, %arg0, %80 : i64
    cf.cond_br %79, ^bb18, ^bb19
    ^bb18:
      %81 = arith.constant 1 : i32
      %82 = arith.extsi %81 : i32 to i64
      func.return %82 : i64
    ^bb19:
      cf.br ^bb20
    ^bb20:
    %83 = arith.constant 3 : i32
    %85 = arith.extsi %83 : i32 to i64
    %84 = arith.cmpi eq, %arg0, %85 : i64
    cf.cond_br %84, ^bb21, ^bb22
    ^bb21:
      %86 = arith.constant 8 : i32
      %88 = arith.extsi %86 : i32 to i64
      %87 = arith.remsi %88, %arg1 : i64
      func.return %87 : i64
    ^bb22:
      cf.br ^bb23
    ^bb23:
    %89 = func.call @C(%arg0, %arg1) : (i64, i64) -> i64
    func.return %89 : i64
  }
  func.func @main() -> i32 {
    %90 = arith.constant 10000 : i32
    %91 = arith.constant 6 : i32
    %92 = arith.constant 13 : i32
    %93 = arith.muli %91, %92 : i32
    %94 = arith.constant 13 : i32
    %95 = arith.muli %93, %94 : i32
    %96 = arith.remsi %90, %95 : i32
    %97 = arith.extsi %96 : i32 to i64
    %99 = arith.constant 6 : i32
    %100 = arith.constant 13 : i32
    %101 = arith.muli %99, %100 : i32
    %102 = arith.constant 13 : i32
    %103 = arith.muli %101, %102 : i32
    %104 = arith.constant 13 : i32
    %105 = arith.muli %103, %104 : i32
    %106 = arith.constant 13 : i32
    %107 = arith.muli %105, %106 : i32
    %108 = arith.extsi %107 : i32 to i64
    %98 = func.call @C_formula(%97, %108) : (i64, i64) -> i64
    %110 = arith.constant 6 : i32
    %111 = arith.constant 13 : i32
    %112 = arith.muli %110, %111 : i32
    %113 = arith.constant 13 : i32
    %114 = arith.muli %112, %113 : i32
    %115 = arith.constant 13 : i32
    %116 = arith.muli %114, %115 : i32
    %117 = arith.constant 13 : i32
    %118 = arith.muli %116, %117 : i32
    %119 = arith.constant 13 : i32
    %120 = arith.muli %118, %119 : i32
    %121 = arith.constant 13 : i32
    %122 = arith.muli %120, %121 : i32
    %123 = arith.extsi %122 : i32 to i64
    %109 = func.call @C_formula(%98, %123) : (i64, i64) -> i64
    %125 = arith.constant 13 : i32
    %126 = arith.constant 13 : i32
    %127 = arith.muli %125, %126 : i32
    %128 = arith.constant 13 : i32
    %129 = arith.muli %127, %128 : i32
    %130 = arith.constant 13 : i32
    %131 = arith.muli %129, %130 : i32
    %132 = arith.constant 13 : i32
    %133 = arith.muli %131, %132 : i32
    %134 = arith.constant 13 : i32
    %135 = arith.muli %133, %134 : i32
    %136 = arith.constant 13 : i32
    %137 = arith.muli %135, %136 : i32
    %138 = arith.constant 13 : i32
    %139 = arith.muli %137, %138 : i32
    %140 = arith.extsi %139 : i32 to i64
    %124 = func.call @C_formula(%109, %140) : (i64, i64) -> i64
    %141 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %142 = llvm.call @printf(%141, %124) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %143 = arith.constant 0 : i32
    func.return %143 : i32
  }
}