Problem 770

Delphi Flip: g(1.9999), the smallest n where A guarantees at least 1.9999 g. F(n) = 2 / (1 + p_n), where p_n = C(2n,n) / 4^n. Condition F(n) >= X is equivalent to p_n <= (2 - X) / X. Uses lgamma for small n and Stirling series for huge n.

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 770
# Delphi Flip: g(1.9999), the smallest n where A guarantees at least 1.9999 g.
# F(n) = 2 / (1 + p_n), where p_n = C(2n,n) / 4^n.
# Condition F(n) >= X is equivalent to p_n <= (2 - X) / X.
# Uses lgamma for small n and Stirling series for huge n.

extern {
    function log(x: f64) -> f64
    function lgamma(x: f64) -> f64
}

const PI: f64 = 3.14159265358979323846

# ln p_n via Stirling expansion for huge n:
# ln p_n = -1/2 ln(pi n) - 1/(8n) + 1/(192 n^3) - 1/(640 n^5)
function ln_p_stirling(n: f64) -> f64 {
    let inv: f64 = 1.0 / n
    let inv2: f64 = inv * inv
    let inv3: f64 = inv2 * inv
    let inv5: f64 = inv3 * inv2
    return -0.5 * log(PI * n) - 0.125 * inv + (1.0 / 192.0) * inv3 - (1.0 / 640.0) * inv5
}

# ln(C(2n,n)/4^n) via lgamma. Fine for small n where no cancellation occurs.
function ln_p_lgamma(n: i64) -> f64 {
    let nf: f64 = n as f64
    return lgamma(2.0 * nf + 1.0) - 2.0 * lgamma(nf + 1.0) - 2.0 * nf * log(2.0)
}

function g_for_fraction(x_num: i64, x_den: i64) -> i64 {
    # r = (2 - X) / X = (2*x_den - x_num) / x_num
    let r_num: i64 = 2 * x_den - x_num
    let r_den: i64 = x_num

    let r: f64 = (r_num as f64) / (r_den as f64)
    let ln_r: f64 = log(r_num as f64) - log(r_den as f64)

    # p_n ~ 1/sqrt(pi n)  =>  n ~ 1/(pi r^2)
    let n_est: i64 = (1.0 / (PI * r * r)) as i64

    if n_est < 20000 {
        let mut n: i64 = 0
        while ln_p_lgamma(n) > ln_r {
            n = n + 1
        }
        return n
    }

    # Huge n: walk in log-space using the Stirling expansion.
    let mut n: i64 = n_est - 10
    if n < 1 { n = 1 }
    while ln_p_stirling(n as f64) > ln_r {
        n = n + 1
    }
    while n > 1 && ln_p_stirling((n - 1) as f64) <= ln_r {
        n = n - 1
    }
    return n
}

function main() -> i32 {
    printf("%lld\n", g_for_fraction(19999, 10000))
    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 lgamma(double x);
double ln_p_stirling_f64(double n);
double ln_p_lgamma_i64(int64_t n);
int64_t g_for_fraction_i64_i64(int64_t x_num, int64_t x_den);
int32_t main(void);

static const double PI = 3.14159265358979323846;



double ln_p_stirling_f64(double n) {
    double inv = (1.0 / n);
    double inv2 = (inv * inv);
    double inv3 = (inv2 * inv);
    double inv5 = (inv3 * inv2);
    return (((((-0.5) * log((PI * n))) - (0.125 * inv)) + ((1.0 / 192.0) * inv3)) - ((1.0 / 640.0) * inv5));
}

double ln_p_lgamma_i64(int64_t n) {
    double nf = ((double)(n));
    return ((lgamma(((2.0 * nf) + 1.0)) - (2.0 * lgamma((nf + 1.0)))) - ((2.0 * nf) * log(2.0)));
}

int64_t g_for_fraction_i64_i64(int64_t x_num, int64_t x_den) {
    int64_t r_num = ((2 * x_den) - x_num);
    int64_t r_den = x_num;
    double r = (((double)(r_num)) / ((double)(r_den)));
    double ln_r = (log(((double)(r_num))) - log(((double)(r_den))));
    int64_t n_est = ((int64_t)((1.0 / ((PI * r) * r))));
    if (n_est < 20000) {
        int64_t n = 0;
        while (ln_p_lgamma_i64(n) > ln_r) {
            n = (n + 1);
        }
        return n;
    }
    int64_t n = (n_est - 10);
    if (n < 1) {
        n = 1;
    }
    while (ln_p_stirling_f64(((double)(n))) > ln_r) {
        n = (n + 1);
    }
    while ((n > 1 && ln_p_stirling_f64(((double)((n - 1)))) <= ln_r)) {
        n = (n - 1);
    }
    return n;
}

int32_t main(void) {
    printf("%lld\n", g_for_fraction_i64_i64(19999, 10000));
    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 @log(f64) -> f64
  func.func private @lgamma(f64) -> f64
  // Constant: PI
  llvm.mlir.global internal constant @PI(3.14159265358979323846 : f64) : f64
  func.func @ln_p_stirling(%arg0: f64) -> f64 {
    %0 = arith.constant 1.0 : f32
    %2 = arith.extf %0 : f32 to f64
    %1 = arith.divf %2, %arg0 : f64
    %3 = arith.mulf %1, %1 : f64
    %4 = arith.mulf %3, %1 : f64
    %5 = arith.mulf %4, %3 : f64
    %6 = arith.constant 0.5 : f32
    %7 = arith.negf %6 : f32
    %8 = llvm.mlir.addressof @PI : !llvm.ptr
    %9 = llvm.load %8 : !llvm.ptr -> f64
    %10 = arith.mulf %9, %arg0 : f64
    %11 = math.log %10 : f64
    %13 = arith.extf %7 : f32 to f64
    %12 = arith.mulf %13, %11 : f64
    %14 = arith.constant 0.125 : f32
    %16 = arith.extf %14 : f32 to f64
    %15 = arith.mulf %16, %1 : f64
    %17 = arith.subf %12, %15 : f64
    %18 = arith.constant 1.0 : f32
    %19 = arith.constant 192.0 : f32
    %20 = arith.divf %18, %19 : f32
    %22 = arith.extf %20 : f32 to f64
    %21 = arith.mulf %22, %4 : f64
    %23 = arith.addf %17, %21 : f64
    %24 = arith.constant 1.0 : f32
    %25 = arith.constant 640.0 : f32
    %26 = arith.divf %24, %25 : f32
    %28 = arith.extf %26 : f32 to f64
    %27 = arith.mulf %28, %5 : f64
    %29 = arith.subf %23, %27 : f64
    func.return %29 : f64
  }
  func.func @ln_p_lgamma(%arg0: i64) -> f64 {
    %30 = arith.sitofp %arg0 : i64 to f64
    %32 = arith.constant 2.0 : f32
    %34 = arith.extf %32 : f32 to f64
    %33 = arith.mulf %34, %30 : f64
    %35 = arith.constant 1.0 : f32
    %37 = arith.extf %35 : f32 to f64
    %36 = arith.addf %33, %37 : f64
    %31 = func.call @lgamma(%36) : (f64) -> f64
    %38 = arith.constant 2.0 : f32
    %40 = arith.constant 1.0 : f32
    %42 = arith.extf %40 : f32 to f64
    %41 = arith.addf %30, %42 : f64
    %39 = func.call @lgamma(%41) : (f64) -> f64
    %44 = arith.extf %38 : f32 to f64
    %43 = arith.mulf %44, %39 : f64
    %45 = arith.subf %31, %43 : f64
    %46 = arith.constant 2.0 : f32
    %48 = arith.extf %46 : f32 to f64
    %47 = arith.mulf %48, %30 : f64
    %49 = arith.constant 2.0 : f32
    %50 = math.log %49 : f32
    %51 = arith.mulf %47, %50 : f64
    %52 = arith.subf %45, %51 : f64
    func.return %52 : f64
  }
  func.func @g_for_fraction(%arg0: i64, %arg1: i64) -> i64 {
    %53 = arith.constant 2 : i32
    %55 = arith.extsi %53 : i32 to i64
    %54 = arith.muli %55, %arg1 : i64
    %56 = arith.subi %54, %arg0 : i64
    %57 = arith.sitofp %56 : i64 to f64
    %58 = arith.sitofp %arg0 : i64 to f64
    %59 = arith.divf %57, %58 : f64
    %60 = arith.sitofp %56 : i64 to f64
    %61 = math.log %60 : f64
    %62 = arith.sitofp %arg0 : i64 to f64
    %63 = math.log %62 : f64
    %64 = arith.subf %61, %63 : f64
    %65 = arith.constant 1.0 : f32
    %66 = llvm.mlir.addressof @PI : !llvm.ptr
    %67 = llvm.load %66 : !llvm.ptr -> f64
    %68 = arith.mulf %67, %59 : f64
    %69 = arith.mulf %68, %59 : f64
    %71 = arith.extf %65 : f32 to f64
    %70 = arith.divf %71, %69 : f64
    %72 = arith.fptosi %70 : f64 to i64
    %73 = arith.constant 20000 : i32
    %75 = arith.extsi %73 : i32 to i64
    %74 = arith.cmpi slt, %72, %75 : i64
    cf.cond_br %74, ^bb0, ^bb1
    ^bb0:
      %76 = arith.constant 0 : i32
      %77 = arith.extsi %76 : i32 to i64
      %78 = llvm.mlir.constant(1 : i64) : i64
      %79 = llvm.alloca %78 x i64 : (i64) -> !llvm.ptr
      llvm.store %77, %79 : i64, !llvm.ptr
      cf.br ^bb3
      ^bb3:
      %81 = llvm.load %79 : !llvm.ptr -> i64
      %80 = func.call @ln_p_lgamma(%81) : (i64) -> f64
      %82 = arith.cmpf ogt, %80, %64 : f64
      cf.cond_br %82, ^bb4, ^bb5
      ^bb4:
        %83 = llvm.load %79 : !llvm.ptr -> i64
        %84 = arith.constant 1 : i32
        %86 = arith.extsi %84 : i32 to i64
        %85 = arith.addi %83, %86 : i64
        llvm.store %85, %79 : i64, !llvm.ptr
        cf.br ^bb3
      ^bb5:
      %87 = llvm.load %79 : !llvm.ptr -> i64
      func.return %87 : i64
    ^bb1:
      cf.br ^bb2
    ^bb2:
    %88 = arith.constant 10 : i32
    %90 = arith.extsi %88 : i32 to i64
    %89 = arith.subi %72, %90 : i64
    %91 = llvm.mlir.constant(1 : i64) : i64
    %92 = llvm.alloca %91 x i64 : (i64) -> !llvm.ptr
    llvm.store %89, %92 : i64, !llvm.ptr
    %93 = llvm.load %92 : !llvm.ptr -> i64
    %94 = arith.constant 1 : i32
    %96 = arith.extsi %94 : i32 to i64
    %95 = arith.cmpi slt, %93, %96 : i64
    cf.cond_br %95, ^bb6, ^bb7
    ^bb6:
      %97 = arith.constant 1 : i32
      %98 = arith.extsi %97 : i32 to i64
      llvm.store %98, %92 : i64, !llvm.ptr
      cf.br ^bb8
    ^bb7:
      cf.br ^bb8
    ^bb8:
    cf.br ^bb9
    ^bb9:
    %100 = llvm.load %92 : !llvm.ptr -> i64
    %101 = arith.sitofp %100 : i64 to f64
    %99 = func.call @ln_p_stirling(%101) : (f64) -> f64
    %102 = arith.cmpf ogt, %99, %64 : f64
    cf.cond_br %102, ^bb10, ^bb11
    ^bb10:
      %103 = llvm.load %92 : !llvm.ptr -> i64
      %104 = arith.constant 1 : i32
      %106 = arith.extsi %104 : i32 to i64
      %105 = arith.addi %103, %106 : i64
      llvm.store %105, %92 : i64, !llvm.ptr
      cf.br ^bb9
    ^bb11:
    cf.br ^bb12
    ^bb12:
    %107 = llvm.load %92 : !llvm.ptr -> i64
    %108 = arith.constant 1 : i32
    %110 = arith.extsi %108 : i32 to i64
    %109 = arith.cmpi sgt, %107, %110 : i64
    %111 = scf.if %109 -> (i1) {
      %113 = llvm.load %92 : !llvm.ptr -> i64
      %114 = arith.constant 1 : i32
      %116 = arith.extsi %114 : i32 to i64
      %115 = arith.subi %113, %116 : i64
      %117 = arith.sitofp %115 : i64 to f64
      %112 = func.call @ln_p_stirling(%117) : (f64) -> f64
      %118 = arith.cmpf ole, %112, %64 : f64
      scf.yield %118 : i1
    } else {
      %119 = arith.constant false
      scf.yield %119 : i1
    }
    cf.cond_br %111, ^bb13, ^bb14
    ^bb13:
      %120 = llvm.load %92 : !llvm.ptr -> i64
      %121 = arith.constant 1 : i32
      %123 = arith.extsi %121 : i32 to i64
      %122 = arith.subi %120, %123 : i64
      llvm.store %122, %92 : i64, !llvm.ptr
      cf.br ^bb12
    ^bb14:
    %124 = llvm.load %92 : !llvm.ptr -> i64
    func.return %124 : i64
  }
  func.func @main() -> i32 {
    %125 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %127 = arith.constant 19999 : i32
    %128 = arith.constant 10000 : i32
    %129 = arith.extsi %127 : i32 to i64
    %130 = arith.extsi %128 : i32 to i64
    %126 = func.call @g_for_fraction(%129, %130) : (i64, i64) -> i64
    %131 = llvm.call @printf(%125, %126) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, i64) -> i32
    %132 = arith.constant 0 : i32
    func.return %132 : i32
  }
}