Average Error: 29.3 → 0.0
Time: 26.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7336.587493395303:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7336.587493395303:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\

\end{array}
double f(double N) {
        double r1732613 = N;
        double r1732614 = 1.0;
        double r1732615 = r1732613 + r1732614;
        double r1732616 = log(r1732615);
        double r1732617 = log(r1732613);
        double r1732618 = r1732616 - r1732617;
        return r1732618;
}
↓
double f(double N) {
        double r1732619 = N;
        double r1732620 = 7336.587493395303;
        bool r1732621 = r1732619 <= r1732620;
        double r1732622 = 1.0;
        double r1732623 = r1732622 + r1732619;
        double r1732624 = r1732623 / r1732619;
        double r1732625 = log(r1732624);
        double r1732626 = 0.3333333333333333;
        double r1732627 = r1732626 / r1732619;
        double r1732628 = r1732619 * r1732619;
        double r1732629 = r1732627 / r1732628;
        double r1732630 = r1732622 / r1732619;
        double r1732631 = 0.5;
        double r1732632 = r1732631 / r1732628;
        double r1732633 = r1732630 - r1732632;
        double r1732634 = r1732629 + r1732633;
        double r1732635 = r1732621 ? r1732625 : r1732634;
        return r1732635;
}

Error

Bits error versus N

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if N < 7336.587493395303

    1. Initial program 0.1

      \[\log \left(N + 1\right) - \log N\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{\log \left(N + 1\right) - \log N}\right)}\]

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{1 + N}{N}\right)}\]

    if 7336.587493395303 < N

    1. Initial program 59.5

      \[\log \left(N + 1\right) - \log N\]

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{{N}^{3}} + \frac{1}{N}\right) - \frac{1}{2} \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7336.587493395303:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))