Problem 970

Kangaroo Hopping over Sixes. H(n) = 2n + 2/3 + eps, where eps = 2*Re(e^{lam*n}/lam) is a tiny correction with lam = 1 + W_1(-1/e) (Lambert W, branch 1). For n = 10^6 the asymptotic formula is exact to far more digits than needed. Precomputed lam to 70 digits, f64 is sufficient for n=10^6.

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 970
# Kangaroo Hopping over Sixes.
# H(n) = 2n + 2/3 + eps, where eps = 2*Re(e^{lam*n}/lam) is a tiny
# correction with lam = 1 + W_1(-1/e) (Lambert W, branch 1).
# For n = 10^6 the asymptotic formula is exact to far more digits than needed.
# Precomputed lam to 70 digits, f64 is sufficient for n=10^6.

extern {
    function cos(x: f64) -> f64
    function log(x: f64) -> f64
    function exp(x: f64) -> f64
    function floor(x: f64) -> f64
    function fabs(x: f64) -> f64
}

function my_fmod(x: f64, y: f64) -> f64 {
    let q: f64 = floor(x / y)
    return x - q * y
}

function main() -> i32 {
    let lam_re: f64 = -2.088843015613043855957086716774947500545693741036729673239112544244607
    let lam_im: f64 = 7.461489285654254556906116612186415334509094993202209240934411391411877
    let lam_abs: f64 = 7.748360310659838754659859240216219375966049464762612291981467582861770
    let lam_arg: f64 = 1.843758551210239598129985611715645443454302414441256154232297011153285

    let n: f64 = 1000000.0
    let pi: f64 = 3.141592653589793238462643383279502884197169399375105820974944592307816
    let two_pi: f64 = 2.0 * pi
    let neg_pi: f64 = 0.0 - pi

    let ln10: f64 = log(10.0)

    let mut theta: f64 = lam_im * n - lam_arg
    theta = my_fmod(theta, two_pi)
    if theta > pi {
        theta = theta - two_pi
    } else {
        if theta < neg_pi {
            theta = theta + two_pi
        }
    }

    let c: f64 = cos(theta)

    let mut a: f64 = log(2.0 / lam_abs) / ln10
    a = a + lam_re * n / ln10
    a = a + log(fabs(c)) / ln10

    let L: i64 = (floor(0.0 - a) as i64)

    let mut delta: f64 = exp((a + (L as f64)) * ln10)
    if c < 0.0 {
        delta = 0.0 - delta
    }

    let s: f64 = 2.0 / 3.0 + delta
    let mut frac: f64 = s - floor(s)

    let mut result: i64 = 0
    let mut count: i32 = 0
    let mut i: i32 = 0
    while i < 200 && count < 8 {
        frac = frac * 10.0
        let d: i64 = (floor(frac) as i64)
        frac = frac - floor(frac)
        if d != 6 {
            result = result * 10 + d
            count = count + 1
        }
        i = i + 1
    }

    printf("%lld\n", result)
    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; }

double my_fmod_f64_f64(double x, double y);
int32_t main(void);






double my_fmod_f64_f64(double x, double y) {
    double q = floor((x / y));
    return (x - (q * y));
}

int32_t main(void) {
    double lam_re = (-2.088843015613043855957086716774947500545693741036729673239112544244607);
    double lam_im = 7.461489285654254556906116612186415334509094993202209240934411391411877;
    double lam_abs = 7.748360310659838754659859240216219375966049464762612291981467582861770;
    double lam_arg = 1.843758551210239598129985611715645443454302414441256154232297011153285;
    double n = 1000000.0;
    double pi = 3.141592653589793238462643383279502884197169399375105820974944592307816;
    double two_pi = (2.0 * pi);
    double neg_pi = (0.0 - pi);
    double ln10 = log(10.0);
    double theta = ((lam_im * n) - lam_arg);
    theta = my_fmod_f64_f64(theta, two_pi);
    if (theta > pi) {
        theta = (theta - two_pi);
    } else {
        if (theta < neg_pi) {
            theta = (theta + two_pi);
        }
    }
    double c = cos(theta);
    double a = (log((2.0 / lam_abs)) / ln10);
    a = (a + ((lam_re * n) / ln10));
    a = (a + (log(fabs(c)) / ln10));
    int64_t L = ((int64_t)(floor((0.0 - a))));
    double delta = exp(((a + ((double)(L))) * ln10));
    if (c < 0.0) {
        delta = (0.0 - delta);
    }
    double s = ((2.0 / 3.0) + delta);
    double frac = (s - floor(s));
    int64_t result = 0;
    int32_t count = 0;
    int32_t i = 0;
    while ((i < 200 && count < 8)) {
        frac = (frac * 10.0);
        int64_t d = ((int64_t)(floor(frac)));
        frac = (frac - floor(frac));
        if (d != 6) {
            result = ((result * 10) + d);
            count = (count + 1);
        }
        i = (i + 1);
    }
    printf("%lld\n", result);
    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 private @cos(f64) -> f64
  func.func private @log(f64) -> f64
  func.func private @exp(f64) -> f64
  func.func private @floor(f64) -> f64
  func.func private @fabs(f64) -> f64
  func.func @my_fmod(%arg0: f64, %arg1: f64) -> f64 {
    %1 = arith.divf %arg0, %arg1 : f64
    %0 = func.call @floor(%1) : (f64) -> f64
    %2 = arith.mulf %0, %arg1 : f64
    %3 = arith.subf %arg0, %2 : f64
    func.return %3 : f64
  }
  func.func @main() -> i32 {
    %4 = arith.constant 2.088843015613043855957086716774947500545693741036729673239112544244607 : f32
    %5 = arith.negf %4 : f32
    %6 = arith.extf %5 : f32 to f64
    %7 = arith.constant 7.461489285654254556906116612186415334509094993202209240934411391411877 : f32
    %8 = arith.extf %7 : f32 to f64
    %9 = arith.constant 7.748360310659838754659859240216219375966049464762612291981467582861770 : f32
    %10 = arith.extf %9 : f32 to f64
    %11 = arith.constant 1.843758551210239598129985611715645443454302414441256154232297011153285 : f32
    %12 = arith.extf %11 : f32 to f64
    %13 = arith.constant 1000000.0 : f32
    %14 = arith.extf %13 : f32 to f64
    %15 = arith.constant 3.141592653589793238462643383279502884197169399375105820974944592307816 : f32
    %16 = arith.extf %15 : f32 to f64
    %17 = arith.constant 2.0 : f32
    %19 = arith.extf %17 : f32 to f64
    %18 = arith.mulf %19, %16 : f64
    %20 = arith.constant 0.0 : f32
    %22 = arith.extf %20 : f32 to f64
    %21 = arith.subf %22, %16 : f64
    %23 = arith.constant 10.0 : f32
    %24 = math.log %23 : f32
    %25 = arith.extf %24 : f32 to f64
    %26 = arith.mulf %8, %14 : f64
    %27 = arith.subf %26, %12 : f64
    %28 = llvm.mlir.constant(1 : i64) : i64
    %29 = llvm.alloca %28 x f64 : (i64) -> !llvm.ptr
    llvm.store %27, %29 : f64, !llvm.ptr
    %31 = llvm.load %29 : !llvm.ptr -> f64
    %30 = func.call @my_fmod(%31, %18) : (f64, f64) -> f64
    llvm.store %30, %29 : f64, !llvm.ptr
    %32 = llvm.load %29 : !llvm.ptr -> f64
    %33 = arith.cmpf ogt, %32, %16 : f64
    cf.cond_br %33, ^bb0, ^bb1
    ^bb0:
      %34 = llvm.load %29 : !llvm.ptr -> f64
      %35 = arith.subf %34, %18 : f64
      llvm.store %35, %29 : f64, !llvm.ptr
      cf.br ^bb2
    ^bb1:
      %36 = llvm.load %29 : !llvm.ptr -> f64
      %37 = arith.cmpf olt, %36, %21 : f64
      cf.cond_br %37, ^bb3, ^bb4
      ^bb3:
        %38 = llvm.load %29 : !llvm.ptr -> f64
        %39 = arith.addf %38, %18 : f64
        llvm.store %39, %29 : f64, !llvm.ptr
        cf.br ^bb5
      ^bb4:
        cf.br ^bb5
      ^bb5:
      cf.br ^bb2
    ^bb2:
    %40 = llvm.load %29 : !llvm.ptr -> f64
    %41 = math.cos %40 : f64
    %42 = arith.constant 2.0 : f32
    %44 = arith.extf %42 : f32 to f64
    %43 = arith.divf %44, %10 : f64
    %45 = math.log %43 : f64
    %46 = arith.divf %45, %25 : f64
    %47 = llvm.mlir.constant(1 : i64) : i64
    %48 = llvm.alloca %47 x f64 : (i64) -> !llvm.ptr
    llvm.store %46, %48 : f64, !llvm.ptr
    %49 = llvm.load %48 : !llvm.ptr -> f64
    %50 = arith.mulf %6, %14 : f64
    %51 = arith.divf %50, %25 : f64
    %52 = arith.addf %49, %51 : f64
    llvm.store %52, %48 : f64, !llvm.ptr
    %53 = llvm.load %48 : !llvm.ptr -> f64
    %54 = math.absf %41 : f64
    %55 = math.log %54 : f64
    %56 = arith.divf %55, %25 : f64
    %57 = arith.addf %53, %56 : f64
    llvm.store %57, %48 : f64, !llvm.ptr
    %59 = arith.constant 0.0 : f32
    %60 = llvm.load %48 : !llvm.ptr -> f64
    %62 = arith.extf %59 : f32 to f64
    %61 = arith.subf %62, %60 : f64
    %58 = func.call @floor(%61) : (f64) -> f64
    %63 = arith.fptosi %58 : f64 to i64
    %64 = llvm.load %48 : !llvm.ptr -> f64
    %65 = arith.sitofp %63 : i64 to f64
    %66 = arith.addf %64, %65 : f64
    %67 = arith.mulf %66, %25 : f64
    %68 = math.exp %67 : f64
    %69 = llvm.mlir.constant(1 : i64) : i64
    %70 = llvm.alloca %69 x f64 : (i64) -> !llvm.ptr
    llvm.store %68, %70 : f64, !llvm.ptr
    %71 = arith.constant 0.0 : f32
    %73 = arith.extf %71 : f32 to f64
    %72 = arith.cmpf olt, %41, %73 : f64
    cf.cond_br %72, ^bb6, ^bb7
    ^bb6:
      %74 = arith.constant 0.0 : f32
      %75 = llvm.load %70 : !llvm.ptr -> f64
      %77 = arith.extf %74 : f32 to f64
      %76 = arith.subf %77, %75 : f64
      llvm.store %76, %70 : f64, !llvm.ptr
      cf.br ^bb8
    ^bb7:
      cf.br ^bb8
    ^bb8:
    %78 = arith.constant 2.0 : f32
    %79 = arith.constant 3.0 : f32
    %80 = arith.divf %78, %79 : f32
    %81 = llvm.load %70 : !llvm.ptr -> f64
    %83 = arith.extf %80 : f32 to f64
    %82 = arith.addf %83, %81 : f64
    %84 = func.call @floor(%82) : (f64) -> f64
    %85 = arith.subf %82, %84 : f64
    %86 = llvm.mlir.constant(1 : i64) : i64
    %87 = llvm.alloca %86 x f64 : (i64) -> !llvm.ptr
    llvm.store %85, %87 : f64, !llvm.ptr
    %88 = arith.constant 0 : i32
    %89 = arith.extsi %88 : i32 to i64
    %90 = llvm.mlir.constant(1 : i64) : i64
    %91 = llvm.alloca %90 x i64 : (i64) -> !llvm.ptr
    llvm.store %89, %91 : i64, !llvm.ptr
    %92 = arith.constant 0 : i32
    %93 = llvm.mlir.constant(1 : i64) : i64
    %94 = llvm.alloca %93 x i32 : (i64) -> !llvm.ptr
    llvm.store %92, %94 : i32, !llvm.ptr
    %95 = arith.constant 0 : i32
    %96 = llvm.mlir.constant(1 : i64) : i64
    %97 = llvm.alloca %96 x i32 : (i64) -> !llvm.ptr
    llvm.store %95, %97 : i32, !llvm.ptr
    cf.br ^bb9
    ^bb9:
    %98 = llvm.load %97 : !llvm.ptr -> i32
    %99 = arith.constant 200 : i32
    %100 = arith.cmpi slt, %98, %99 : i32
    %101 = scf.if %100 -> (i1) {
      %102 = llvm.load %94 : !llvm.ptr -> i32
      %103 = arith.constant 8 : i32
      %104 = arith.cmpi slt, %102, %103 : i32
      scf.yield %104 : i1
    } else {
      %105 = arith.constant false
      scf.yield %105 : i1
    }
    cf.cond_br %101, ^bb10, ^bb11
    ^bb10:
      %106 = llvm.load %87 : !llvm.ptr -> f64
      %107 = arith.constant 10.0 : f32
      %109 = arith.extf %107 : f32 to f64
      %108 = arith.mulf %106, %109 : f64
      llvm.store %108, %87 : f64, !llvm.ptr
      %111 = llvm.load %87 : !llvm.ptr -> f64
      %110 = func.call @floor(%111) : (f64) -> f64
      %112 = arith.fptosi %110 : f64 to i64
      %113 = llvm.load %87 : !llvm.ptr -> f64
      %115 = llvm.load %87 : !llvm.ptr -> f64
      %114 = func.call @floor(%115) : (f64) -> f64
      %116 = arith.subf %113, %114 : f64
      llvm.store %116, %87 : f64, !llvm.ptr
      %117 = arith.constant 6 : i32
      %119 = arith.extsi %117 : i32 to i64
      %118 = arith.cmpi ne, %112, %119 : i64
      cf.cond_br %118, ^bb12, ^bb13
      ^bb12:
        %120 = llvm.load %91 : !llvm.ptr -> i64
        %121 = arith.constant 10 : i32
        %123 = arith.extsi %121 : i32 to i64
        %122 = arith.muli %120, %123 : i64
        %124 = arith.addi %122, %112 : i64
        llvm.store %124, %91 : i64, !llvm.ptr
        %125 = llvm.load %94 : !llvm.ptr -> i32
        %126 = arith.constant 1 : i32
        %127 = arith.addi %125, %126 : i32
        llvm.store %127, %94 : i32, !llvm.ptr
        cf.br ^bb14
      ^bb13:
        cf.br ^bb14
      ^bb14:
      %128 = llvm.load %97 : !llvm.ptr -> i32
      %129 = arith.constant 1 : i32
      %130 = arith.addi %128, %129 : i32
      llvm.store %130, %97 : i32, !llvm.ptr
      cf.br ^bb9
    ^bb11:
    %131 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %132 = llvm.load %91 : !llvm.ptr -> i64
    %133 = llvm.call @printf(%131, %132) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %134 = arith.constant 0 : i32
    func.return %134 : i32
  }
}