Problem 722

Slowly Converging Series: evaluate E_15(q) = sum sigma_15(n) q^n at q = 1 - 2^-25. Mellin inversion of sum sigma_k(n) e^{-nt} against Gamma(s) zeta(s) zeta(s-k) t^{-s} leaves only the residues at s = k+1, s = 1 and s = 0 (all others sit on trivial zeros of zeta); the remainder is exponentially small, of order e^{-4 pi^2 / t}. For k = 15: E_15(e^{-t}) = 15! zeta(16) / t^16 + zeta(0) zeta(-15) = 15! zeta(16) / t^16 - 3617/16320. With t = -ln(1 - 2^-25) write t = 2^-25 * u, u = 1 + x/2 + x^2/3 + ... so t^16 = u^16 * 2^-400 and everything stays inside f64 range. The closed form reproduces the three check values E_1(1-1/2^4), E_3(1-1/2^8) and E_7(1-1/2^15) from the statement.

View problem on Project Euler

Performance comparison

Flow source

# Project Euler 722
# Slowly Converging Series: evaluate E_15(q) = sum sigma_15(n) q^n at
# q = 1 - 2^-25.
#
# Mellin inversion of sum sigma_k(n) e^{-nt} against Gamma(s) zeta(s)
# zeta(s-k) t^{-s} leaves only the residues at s = k+1, s = 1 and s = 0
# (all others sit on trivial zeros of zeta); the remainder is
# exponentially small, of order e^{-4 pi^2 / t}.  For k = 15:
#   E_15(e^{-t}) = 15! zeta(16) / t^16 + zeta(0) zeta(-15)
#               = 15! zeta(16) / t^16 - 3617/16320.
# With t = -ln(1 - 2^-25) write t = 2^-25 * u, u = 1 + x/2 + x^2/3 + ...
# so t^16 = u^16 * 2^-400 and everything stays inside f64 range.
# The closed form reproduces the three check values E_1(1-1/2^4),
# E_3(1-1/2^8) and E_7(1-1/2^15) from the statement.

function main() -> i32 {
    let x: f64 = 1.0 / 33554432.0
    let u: f64 = 1.0 + x / 2.0 + x * x / 3.0 + x * x * x / 4.0

    # zeta(16) by direct summation, smallest terms first
    let mut z16: f64 = 0.0
    let mut n: i64 = 60
    while n >= 1 {
        let nf: f64 = n as f64
        let n2: f64 = nf * nf
        let n4: f64 = n2 * n2
        let n8: f64 = n4 * n4
        z16 = z16 + 1.0 / (n8 * n8)
        n = n - 1
    }

    let mut fact15: f64 = 1.0
    let mut i: i64 = 2
    while i <= 15 {
        fact15 = fact15 * (i as f64)
        i = i + 1
    }

    let mut p2_400: f64 = 1.0
    let mut j: i64 = 0
    while j < 400 {
        p2_400 = p2_400 * 2.0
        j = j + 1
    }

    # u^16 by repeated squaring
    let mut upow: f64 = u
    let mut s: i64 = 0
    while s < 4 {
        upow = upow * upow
        s = s + 1
    }

    let e15: f64 = fact15 * z16 * p2_400 / upow - 3617.0 / 16320.0

    # normalize to mantissa in [1, 10) with a decimal exponent
    let mut mant: f64 = e15
    let mut ex: i64 = 0
    let tenp22: f64 = 10000000000000000000000.0
    while mant >= tenp22 {
        mant = mant / tenp22
        ex = ex + 22
    }
    while mant >= 10.0 {
        mant = mant / 10.0
        ex = ex + 1
    }
    while mant < 1.0 {
        mant = mant * 10.0
        ex = ex - 1
    }
    printf("%.12fe%lld\n", mant, ex)
    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; }

int32_t main(void);

int32_t main(void) {
    double x = (1.0 / 33554432.0);
    double u = (((1.0 + (x / 2.0)) + ((x * x) / 3.0)) + (((x * x) * x) / 4.0));
    double z16 = 0.0;
    int64_t n = 60;
    while (n >= 1) {
        double nf = ((double)(n));
        double n2 = (nf * nf);
        double n4 = (n2 * n2);
        double n8 = (n4 * n4);
        z16 = (z16 + (1.0 / (n8 * n8)));
        n = (n - 1);
    }
    double fact15 = 1.0;
    int64_t i = 2;
    while (i <= 15) {
        fact15 = (fact15 * ((double)(i)));
        i = (i + 1);
    }
    double p2_400 = 1.0;
    int64_t j = 0;
    while (j < 400) {
        p2_400 = (p2_400 * 2.0);
        j = (j + 1);
    }
    double upow = u;
    int64_t s = 0;
    while (s < 4) {
        upow = (upow * upow);
        s = (s + 1);
    }
    double e15 = ((((fact15 * z16) * p2_400) / upow) - (3617.0 / 16320.0));
    double mant = e15;
    int64_t ex = 0;
    double tenp22 = 10000000000000000000000.0;
    while (mant >= tenp22) {
        mant = (mant / tenp22);
        ex = (ex + 22);
    }
    while (mant >= 10.0) {
        mant = (mant / 10.0);
        ex = (ex + 1);
    }
    while (mant < 1.0) {
        mant = (mant * 10.0);
        ex = (ex - 1);
    }
    printf("%.12fe%lld\n", mant, ex);
    return 0;
}

Generated MLIR

module {
  llvm.func @printf(!llvm.ptr, ...) -> i32
  llvm.mlir.global internal constant @str_0("%.12fe%lld\n\00") {addr_space = 0 : i32} : !llvm.array<12 x i8>
  func.func @main() -> i32 {
    %0 = arith.constant 1.0 : f32
    %1 = arith.constant 33554432.0 : f32
    %2 = arith.divf %0, %1 : f32
    %3 = arith.extf %2 : f32 to f64
    %4 = arith.constant 1.0 : f32
    %5 = arith.constant 2.0 : f32
    %7 = arith.extf %5 : f32 to f64
    %6 = arith.divf %3, %7 : f64
    %9 = arith.extf %4 : f32 to f64
    %8 = arith.addf %9, %6 : f64
    %10 = arith.mulf %3, %3 : f64
    %11 = arith.constant 3.0 : f32
    %13 = arith.extf %11 : f32 to f64
    %12 = arith.divf %10, %13 : f64
    %14 = arith.addf %8, %12 : f64
    %15 = arith.mulf %3, %3 : f64
    %16 = arith.mulf %15, %3 : f64
    %17 = arith.constant 4.0 : f32
    %19 = arith.extf %17 : f32 to f64
    %18 = arith.divf %16, %19 : f64
    %20 = arith.addf %14, %18 : f64
    %21 = arith.constant 0.0 : f32
    %22 = arith.extf %21 : f32 to f64
    %23 = llvm.mlir.constant(1 : i64) : i64
    %24 = llvm.alloca %23 x f64 : (i64) -> !llvm.ptr
    llvm.store %22, %24 : f64, !llvm.ptr
    %25 = arith.constant 60 : i32
    %26 = arith.extsi %25 : i32 to i64
    %27 = llvm.mlir.constant(1 : i64) : i64
    %28 = llvm.alloca %27 x i64 : (i64) -> !llvm.ptr
    llvm.store %26, %28 : i64, !llvm.ptr
    cf.br ^bb0
    ^bb0:
    %29 = llvm.load %28 : !llvm.ptr -> i64
    %30 = arith.constant 1 : i32
    %32 = arith.extsi %30 : i32 to i64
    %31 = arith.cmpi sge, %29, %32 : i64
    cf.cond_br %31, ^bb1, ^bb2
    ^bb1:
      %33 = llvm.load %28 : !llvm.ptr -> i64
      %34 = arith.sitofp %33 : i64 to f64
      %35 = arith.mulf %34, %34 : f64
      %36 = arith.mulf %35, %35 : f64
      %37 = arith.mulf %36, %36 : f64
      %38 = llvm.load %24 : !llvm.ptr -> f64
      %39 = arith.constant 1.0 : f32
      %40 = arith.mulf %37, %37 : f64
      %42 = arith.extf %39 : f32 to f64
      %41 = arith.divf %42, %40 : f64
      %43 = arith.addf %38, %41 : f64
      llvm.store %43, %24 : f64, !llvm.ptr
      %44 = llvm.load %28 : !llvm.ptr -> i64
      %45 = arith.constant 1 : i32
      %47 = arith.extsi %45 : i32 to i64
      %46 = arith.subi %44, %47 : i64
      llvm.store %46, %28 : i64, !llvm.ptr
      cf.br ^bb0
    ^bb2:
    %48 = arith.constant 1.0 : f32
    %49 = arith.extf %48 : f32 to f64
    %50 = llvm.mlir.constant(1 : i64) : i64
    %51 = llvm.alloca %50 x f64 : (i64) -> !llvm.ptr
    llvm.store %49, %51 : f64, !llvm.ptr
    %52 = arith.constant 2 : i32
    %53 = arith.extsi %52 : i32 to i64
    %54 = llvm.mlir.constant(1 : i64) : i64
    %55 = llvm.alloca %54 x i64 : (i64) -> !llvm.ptr
    llvm.store %53, %55 : i64, !llvm.ptr
    cf.br ^bb3
    ^bb3:
    %56 = llvm.load %55 : !llvm.ptr -> i64
    %57 = arith.constant 15 : i32
    %59 = arith.extsi %57 : i32 to i64
    %58 = arith.cmpi sle, %56, %59 : i64
    cf.cond_br %58, ^bb4, ^bb5
    ^bb4:
      %60 = llvm.load %51 : !llvm.ptr -> f64
      %61 = llvm.load %55 : !llvm.ptr -> i64
      %62 = arith.sitofp %61 : i64 to f64
      %63 = arith.mulf %60, %62 : f64
      llvm.store %63, %51 : f64, !llvm.ptr
      %64 = llvm.load %55 : !llvm.ptr -> i64
      %65 = arith.constant 1 : i32
      %67 = arith.extsi %65 : i32 to i64
      %66 = arith.addi %64, %67 : i64
      llvm.store %66, %55 : i64, !llvm.ptr
      cf.br ^bb3
    ^bb5:
    %68 = arith.constant 1.0 : f32
    %69 = arith.extf %68 : f32 to f64
    %70 = llvm.mlir.constant(1 : i64) : i64
    %71 = llvm.alloca %70 x f64 : (i64) -> !llvm.ptr
    llvm.store %69, %71 : f64, !llvm.ptr
    %72 = arith.constant 0 : i32
    %73 = arith.extsi %72 : i32 to i64
    %74 = llvm.mlir.constant(1 : i64) : i64
    %75 = llvm.alloca %74 x i64 : (i64) -> !llvm.ptr
    llvm.store %73, %75 : i64, !llvm.ptr
    cf.br ^bb6
    ^bb6:
    %76 = llvm.load %75 : !llvm.ptr -> i64
    %77 = arith.constant 400 : i32
    %79 = arith.extsi %77 : i32 to i64
    %78 = arith.cmpi slt, %76, %79 : i64
    cf.cond_br %78, ^bb7, ^bb8
    ^bb7:
      %80 = llvm.load %71 : !llvm.ptr -> f64
      %81 = arith.constant 2.0 : f32
      %83 = arith.extf %81 : f32 to f64
      %82 = arith.mulf %80, %83 : f64
      llvm.store %82, %71 : f64, !llvm.ptr
      %84 = llvm.load %75 : !llvm.ptr -> i64
      %85 = arith.constant 1 : i32
      %87 = arith.extsi %85 : i32 to i64
      %86 = arith.addi %84, %87 : i64
      llvm.store %86, %75 : i64, !llvm.ptr
      cf.br ^bb6
    ^bb8:
    %88 = llvm.mlir.constant(1 : i64) : i64
    %89 = llvm.alloca %88 x f64 : (i64) -> !llvm.ptr
    llvm.store %20, %89 : f64, !llvm.ptr
    %90 = arith.constant 0 : i32
    %91 = arith.extsi %90 : i32 to i64
    %92 = llvm.mlir.constant(1 : i64) : i64
    %93 = llvm.alloca %92 x i64 : (i64) -> !llvm.ptr
    llvm.store %91, %93 : i64, !llvm.ptr
    cf.br ^bb9
    ^bb9:
    %94 = llvm.load %93 : !llvm.ptr -> i64
    %95 = arith.constant 4 : i32
    %97 = arith.extsi %95 : i32 to i64
    %96 = arith.cmpi slt, %94, %97 : i64
    cf.cond_br %96, ^bb10, ^bb11
    ^bb10:
      %98 = llvm.load %89 : !llvm.ptr -> f64
      %99 = llvm.load %89 : !llvm.ptr -> f64
      %100 = arith.mulf %98, %99 : f64
      llvm.store %100, %89 : f64, !llvm.ptr
      %101 = llvm.load %93 : !llvm.ptr -> i64
      %102 = arith.constant 1 : i32
      %104 = arith.extsi %102 : i32 to i64
      %103 = arith.addi %101, %104 : i64
      llvm.store %103, %93 : i64, !llvm.ptr
      cf.br ^bb9
    ^bb11:
    %105 = llvm.load %51 : !llvm.ptr -> f64
    %106 = llvm.load %24 : !llvm.ptr -> f64
    %107 = arith.mulf %105, %106 : f64
    %108 = llvm.load %71 : !llvm.ptr -> f64
    %109 = arith.mulf %107, %108 : f64
    %110 = llvm.load %89 : !llvm.ptr -> f64
    %111 = arith.divf %109, %110 : f64
    %112 = arith.constant 3617.0 : f32
    %113 = arith.constant 16320.0 : f32
    %114 = arith.divf %112, %113 : f32
    %116 = arith.extf %114 : f32 to f64
    %115 = arith.subf %111, %116 : f64
    %117 = llvm.mlir.constant(1 : i64) : i64
    %118 = llvm.alloca %117 x f64 : (i64) -> !llvm.ptr
    llvm.store %115, %118 : f64, !llvm.ptr
    %119 = arith.constant 0 : i32
    %120 = arith.extsi %119 : i32 to i64
    %121 = llvm.mlir.constant(1 : i64) : i64
    %122 = llvm.alloca %121 x i64 : (i64) -> !llvm.ptr
    llvm.store %120, %122 : i64, !llvm.ptr
    %123 = arith.constant 10000000000000000000000.0 : f32
    %124 = arith.extf %123 : f32 to f64
    cf.br ^bb12
    ^bb12:
    %125 = llvm.load %118 : !llvm.ptr -> f64
    %126 = arith.cmpf oge, %125, %124 : f64
    cf.cond_br %126, ^bb13, ^bb14
    ^bb13:
      %127 = llvm.load %118 : !llvm.ptr -> f64
      %128 = arith.divf %127, %124 : f64
      llvm.store %128, %118 : f64, !llvm.ptr
      %129 = llvm.load %122 : !llvm.ptr -> i64
      %130 = arith.constant 22 : i32
      %132 = arith.extsi %130 : i32 to i64
      %131 = arith.addi %129, %132 : i64
      llvm.store %131, %122 : i64, !llvm.ptr
      cf.br ^bb12
    ^bb14:
    cf.br ^bb15
    ^bb15:
    %133 = llvm.load %118 : !llvm.ptr -> f64
    %134 = arith.constant 10.0 : f32
    %136 = arith.extf %134 : f32 to f64
    %135 = arith.cmpf oge, %133, %136 : f64
    cf.cond_br %135, ^bb16, ^bb17
    ^bb16:
      %137 = llvm.load %118 : !llvm.ptr -> f64
      %138 = arith.constant 10.0 : f32
      %140 = arith.extf %138 : f32 to f64
      %139 = arith.divf %137, %140 : f64
      llvm.store %139, %118 : f64, !llvm.ptr
      %141 = llvm.load %122 : !llvm.ptr -> i64
      %142 = arith.constant 1 : i32
      %144 = arith.extsi %142 : i32 to i64
      %143 = arith.addi %141, %144 : i64
      llvm.store %143, %122 : i64, !llvm.ptr
      cf.br ^bb15
    ^bb17:
    cf.br ^bb18
    ^bb18:
    %145 = llvm.load %118 : !llvm.ptr -> f64
    %146 = arith.constant 1.0 : f32
    %148 = arith.extf %146 : f32 to f64
    %147 = arith.cmpf olt, %145, %148 : f64
    cf.cond_br %147, ^bb19, ^bb20
    ^bb19:
      %149 = llvm.load %118 : !llvm.ptr -> f64
      %150 = arith.constant 10.0 : f32
      %152 = arith.extf %150 : f32 to f64
      %151 = arith.mulf %149, %152 : f64
      llvm.store %151, %118 : f64, !llvm.ptr
      %153 = llvm.load %122 : !llvm.ptr -> i64
      %154 = arith.constant 1 : i32
      %156 = arith.extsi %154 : i32 to i64
      %155 = arith.subi %153, %156 : i64
      llvm.store %155, %122 : i64, !llvm.ptr
      cf.br ^bb18
    ^bb20:
    %157 = llvm.mlir.addressof @str_0 : !llvm.ptr
    %158 = llvm.load %118 : !llvm.ptr -> f64
    %159 = llvm.load %122 : !llvm.ptr -> i64
    %160 = llvm.call @printf(%157, %158, %159) vararg(!llvm.func<i32 (ptr, ...)>) : (!llvm.ptr, f64, i64) -> i32
    %161 = arith.constant 0 : i32
    func.return %161 : i32
  }
}