Average Error: 29.5 → 0.1
Time: 3.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7441.5629236213317:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7441.5629236213317:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r35719 = N;
        double r35720 = 1.0;
        double r35721 = r35719 + r35720;
        double r35722 = log(r35721);
        double r35723 = log(r35719);
        double r35724 = r35722 - r35723;
        return r35724;
}
↓
double f(double N) {
        double r35725 = N;
        double r35726 = 7441.562923621332;
        bool r35727 = r35725 <= r35726;
        double r35728 = 1.0;
        double r35729 = r35725 + r35728;
        double r35730 = r35729 / r35725;
        double r35731 = log(r35730);
        double r35732 = 1.0;
        double r35733 = 2.0;
        double r35734 = pow(r35725, r35733);
        double r35735 = r35732 / r35734;
        double r35736 = 0.3333333333333333;
        double r35737 = r35736 / r35725;
        double r35738 = 0.5;
        double r35739 = r35737 - r35738;
        double r35740 = r35735 * r35739;
        double r35741 = r35728 / r35725;
        double r35742 = r35740 + r35741;
        double r35743 = r35727 ? r35731 : r35742;
        return r35743;
}

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 < 7441.562923621332

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    if 7441.562923621332 < N

    1. Initial program 59.3

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7441.5629236213317:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))