Average Error: 28.6 → 0.1
Time: 12.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8813.444147086946:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N \cdot N} + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8813.444147086946:\\
\;\;\;\;\log \left(1 + \frac{1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N \cdot N} + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r1063719 = N;
        double r1063720 = 1.0;
        double r1063721 = r1063719 + r1063720;
        double r1063722 = log(r1063721);
        double r1063723 = log(r1063719);
        double r1063724 = r1063722 - r1063723;
        return r1063724;
}
↓
double f(double N) {
        double r1063725 = N;
        double r1063726 = 8813.444147086946;
        bool r1063727 = r1063725 <= r1063726;
        double r1063728 = 1.0;
        double r1063729 = r1063728 / r1063725;
        double r1063730 = r1063728 + r1063729;
        double r1063731 = log(r1063730);
        double r1063732 = 0.3333333333333333;
        double r1063733 = r1063732 / r1063725;
        double r1063734 = 0.5;
        double r1063735 = r1063733 - r1063734;
        double r1063736 = r1063725 * r1063725;
        double r1063737 = r1063735 / r1063736;
        double r1063738 = r1063737 + r1063729;
        double r1063739 = r1063727 ? r1063731 : r1063738;
        return r1063739;
}

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 < 8813.444147086946

    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

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

    5. Taylor expanded around inf 0.1

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

    if 8813.444147086946 < N

    1. Initial program 59.5

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

    3. Applied add-log-exp59.5

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

    4. Simplified59.3

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

    5. Taylor expanded around inf 59.3

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8813.444147086946:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N} - \frac{1}{2}}{N \cdot N} + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))