Problem 405

f(10^(10^18)) mod 17^7 via closed form + Euler totient.

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Performance comparison

Flow source

# Project Euler 405
# f(10^(10^18)) mod 17^7 via closed form + Euler totient.

function modpow(base0: i64, exp0: i64, mod: i64) -> i64 {
    let mut r: i64 = 1
    let mut b: i64 = base0 % mod
    if b < 0 { b = b + mod }
    let mut e: i64 = exp0
    while e > 0 {
        if e % 2 == 1 {
            let t: i128 = (r as i128) * (b as i128) % (mod as i128)
            r = t as i64
        }
        let t2: i128 = (b as i128) * (b as i128) % (mod as i128)
        b = t2 as i64
        e = e / 2
    }
    return r
}

function modinv(x: i64, mod: i64, phi: i64) -> i64 {
    return modpow(x, phi - 1, mod)
}

function main() -> i32 {
    # MOD = 17^7
    let MOD: i64 = 410338673
    let PHI: i64 = MOD - MOD / 17
    # e = 10^(10^18) mod PHI
    let e: i64 = modpow(10, 1000000000000000000, PHI)
    let inv15: i64 = modinv(MOD - 15, MOD, PHI)
    let inv3: i64 = modinv(MOD - 3, MOD, PHI)
    let inv5: i64 = modinv(5, MOD, PHI)
    let t1: i64 = (modpow(MOD - 1, e, MOD) * inv15) % MOD
    let t2: i64 = (((modpow(2, e, MOD) * 4) % MOD) * inv3) % MOD
    let t3: i64 = (((2 * inv5) % MOD) * modpow(4, e, MOD)) % MOD
    let ans: i64 = (t1 + t2 + t3 + 1) % MOD
    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 modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod);
int64_t modinv_i64_i64_i64(int64_t x, int64_t mod, int64_t phi);
int32_t main(void);

int64_t modpow_i64_i64_i64(int64_t base0, int64_t exp0, int64_t mod) {
    int64_t r = 1;
    int64_t b = FLOW_CHECKED_MOD((base0), (mod));
    if (b < 0) {
        b = (b + mod);
    }
    int64_t e = exp0;
    while (e > 0) {
        if (FLOW_CHECKED_MOD((e), (2)) == 1) {
            __int128 t = FLOW_CHECKED_MOD(((((__int128)(r)) * ((__int128)(b)))), (((__int128)(mod))));
            r = ((int64_t)(t));
        }
        __int128 t2 = FLOW_CHECKED_MOD(((((__int128)(b)) * ((__int128)(b)))), (((__int128)(mod))));
        b = ((int64_t)(t2));
        e = FLOW_CHECKED_DIV((e), (2));
    }
    return r;
}

int64_t modinv_i64_i64_i64(int64_t x, int64_t mod, int64_t phi) {
    return modpow_i64_i64_i64(x, (phi - 1), mod);
}

int32_t main(void) {
    int64_t MOD = 410338673;
    int64_t PHI = (MOD - FLOW_CHECKED_DIV((MOD), (17)));
    int64_t e = modpow_i64_i64_i64(10, 1000000000000000000, PHI);
    int64_t inv15 = modinv_i64_i64_i64((MOD - 15), MOD, PHI);
    int64_t inv3 = modinv_i64_i64_i64((MOD - 3), MOD, PHI);
    int64_t inv5 = modinv_i64_i64_i64(5, MOD, PHI);
    int64_t t1 = FLOW_CHECKED_MOD(((modpow_i64_i64_i64((MOD - 1), e, MOD) * inv15)), (MOD));
    int64_t t2 = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((modpow_i64_i64_i64(2, e, MOD) * 4)), (MOD)) * inv3)), (MOD));
    int64_t t3 = FLOW_CHECKED_MOD(((FLOW_CHECKED_MOD(((2 * inv5)), (MOD)) * modpow_i64_i64_i64(4, e, MOD))), (MOD));
    int64_t ans = FLOW_CHECKED_MOD(((((t1 + t2) + t3) + 1)), (MOD));
    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 @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.load %6 : !llvm.ptr -> i64
    %8 = arith.constant 0 : i32
    %10 = arith.extsi %8 : i32 to i64
    %9 = arith.cmpi slt, %7, %10 : i64
    cf.cond_br %9, ^bb0, ^bb1
    ^bb0:
      %11 = llvm.load %6 : !llvm.ptr -> i64
      %12 = arith.addi %11, %arg2 : i64
      llvm.store %12, %6 : i64, !llvm.ptr
      cf.br ^bb2
    ^bb1:
      cf.br ^bb2
    ^bb2:
    %13 = llvm.mlir.constant(1 : i64) : i64
    %14 = llvm.alloca %13 x i64 : (i64) -> !llvm.ptr
    llvm.store %arg1, %14 : i64, !llvm.ptr
    cf.br ^bb3
    ^bb3:
    %15 = llvm.load %14 : !llvm.ptr -> i64
    %16 = arith.constant 0 : i32
    %18 = arith.extsi %16 : i32 to i64
    %17 = arith.cmpi sgt, %15, %18 : i64
    cf.cond_br %17, ^bb4, ^bb5
    ^bb4:
      %19 = llvm.load %14 : !llvm.ptr -> i64
      %20 = arith.constant 2 : i32
      %22 = arith.extsi %20 : i32 to i64
      %21 = arith.remsi %19, %22 : i64
      %23 = arith.constant 1 : i32
      %25 = arith.extsi %23 : i32 to i64
      %24 = arith.cmpi eq, %21, %25 : i64
      cf.cond_br %24, ^bb6, ^bb7
      ^bb6:
        %26 = llvm.load %3 : !llvm.ptr -> i64
        %27 = arith.extsi %26 : i64 to i128
        %28 = llvm.load %6 : !llvm.ptr -> i64
        %29 = arith.extsi %28 : i64 to i128
        %31 = arith.trunci %27 : i128 to i64
        %32 = arith.trunci %29 : i128 to i64
        %30 = arith.muli %31, %32 : i64
        %33 = arith.extsi %arg2 : i64 to i128
        %35 = arith.trunci %33 : i128 to i64
        %34 = arith.remsi %30, %35 : i64
        %36 = arith.extsi %34 : i64 to i128
        %37 = arith.trunci %36 : i128 to i64
        llvm.store %37, %3 : i64, !llvm.ptr
        cf.br ^bb8
      ^bb7:
        cf.br ^bb8
      ^bb8:
      %38 = llvm.load %6 : !llvm.ptr -> i64
      %39 = arith.extsi %38 : i64 to i128
      %40 = llvm.load %6 : !llvm.ptr -> i64
      %41 = arith.extsi %40 : i64 to i128
      %43 = arith.trunci %39 : i128 to i64
      %44 = arith.trunci %41 : i128 to i64
      %42 = arith.muli %43, %44 : i64
      %45 = arith.extsi %arg2 : i64 to i128
      %47 = arith.trunci %45 : i128 to i64
      %46 = arith.remsi %42, %47 : i64
      %48 = arith.extsi %46 : i64 to i128
      %49 = arith.trunci %48 : i128 to i64
      llvm.store %49, %6 : i64, !llvm.ptr
      %50 = llvm.load %14 : !llvm.ptr -> i64
      %51 = arith.constant 2 : i32
      %53 = arith.extsi %51 : i32 to i64
      %52 = arith.divsi %50, %53 : i64
      llvm.store %52, %14 : i64, !llvm.ptr
      cf.br ^bb3
    ^bb5:
    %54 = llvm.load %3 : !llvm.ptr -> i64
    func.return %54 : i64
  }
  func.func @modinv(%arg0: i64, %arg1: i64, %arg2: i64) -> i64 {
    %56 = arith.constant 1 : i32
    %58 = arith.extsi %56 : i32 to i64
    %57 = arith.subi %arg2, %58 : i64
    %55 = func.call @modpow(%arg0, %57, %arg1) : (i64, i64, i64) -> i64
    func.return %55 : i64
  }
  func.func @main() -> i32 {
    %59 = arith.constant 410338673 : i32
    %60 = arith.extsi %59 : i32 to i64
    %61 = arith.constant 17 : i32
    %63 = arith.extsi %61 : i32 to i64
    %62 = arith.divsi %60, %63 : i64
    %64 = arith.subi %60, %62 : i64
    %66 = arith.constant 10 : i32
    %67 = arith.constant 999999995705032704 : i32
    %68 = arith.extsi %66 : i32 to i64
    %69 = arith.extsi %67 : i32 to i64
    %65 = func.call @modpow(%68, %69, %64) : (i64, i64, i64) -> i64
    %71 = arith.constant 15 : i32
    %73 = arith.extsi %71 : i32 to i64
    %72 = arith.subi %60, %73 : i64
    %70 = func.call @modinv(%72, %60, %64) : (i64, i64, i64) -> i64
    %75 = arith.constant 3 : i32
    %77 = arith.extsi %75 : i32 to i64
    %76 = arith.subi %60, %77 : i64
    %74 = func.call @modinv(%76, %60, %64) : (i64, i64, i64) -> i64
    %79 = arith.constant 5 : i32
    %80 = arith.extsi %79 : i32 to i64
    %78 = func.call @modinv(%80, %60, %64) : (i64, i64, i64) -> i64
    %82 = arith.constant 1 : i32
    %84 = arith.extsi %82 : i32 to i64
    %83 = arith.subi %60, %84 : i64
    %81 = func.call @modpow(%83, %65, %60) : (i64, i64, i64) -> i64
    %85 = arith.muli %81, %70 : i64
    %86 = arith.remsi %85, %60 : i64
    %88 = arith.constant 2 : i32
    %89 = arith.extsi %88 : i32 to i64
    %87 = func.call @modpow(%89, %65, %60) : (i64, i64, i64) -> i64
    %90 = arith.constant 4 : i32
    %92 = arith.extsi %90 : i32 to i64
    %91 = arith.muli %87, %92 : i64
    %93 = arith.remsi %91, %60 : i64
    %94 = arith.muli %93, %74 : i64
    %95 = arith.remsi %94, %60 : i64
    %96 = arith.constant 2 : i32
    %98 = arith.extsi %96 : i32 to i64
    %97 = arith.muli %98, %78 : i64
    %99 = arith.remsi %97, %60 : i64
    %101 = arith.constant 4 : i32
    %102 = arith.extsi %101 : i32 to i64
    %100 = func.call @modpow(%102, %65, %60) : (i64, i64, i64) -> i64
    %103 = arith.muli %99, %100 : i64
    %104 = arith.remsi %103, %60 : i64
    %105 = arith.addi %86, %95 : i64
    %106 = arith.addi %105, %104 : i64
    %107 = arith.constant 1 : i32
    %109 = arith.extsi %107 : i32 to i64
    %108 = arith.addi %106, %109 : i64
    %110 = arith.remsi %108, %60 : i64
    %111 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %112 = llvm.call @printf(%111, %110) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %113 = arith.constant 0 : i32
    func.return %113 : i32
  }
}