Average Error: 29.8 → 0.1
Time: 3.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9562.6451805155593:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9562.6451805155593:\\
\;\;\;\;\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 r33633 = N;
        double r33634 = 1.0;
        double r33635 = r33633 + r33634;
        double r33636 = log(r33635);
        double r33637 = log(r33633);
        double r33638 = r33636 - r33637;
        return r33638;
}
↓
double f(double N) {
        double r33639 = N;
        double r33640 = 9562.64518051556;
        bool r33641 = r33639 <= r33640;
        double r33642 = 1.0;
        double r33643 = r33639 + r33642;
        double r33644 = r33643 / r33639;
        double r33645 = log(r33644);
        double r33646 = 1.0;
        double r33647 = 2.0;
        double r33648 = pow(r33639, r33647);
        double r33649 = r33646 / r33648;
        double r33650 = 0.3333333333333333;
        double r33651 = r33650 / r33639;
        double r33652 = 0.5;
        double r33653 = r33651 - r33652;
        double r33654 = r33649 * r33653;
        double r33655 = r33642 / r33639;
        double r33656 = r33654 + r33655;
        double r33657 = r33641 ? r33645 : r33656;
        return r33657;
}

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 < 9562.64518051556

    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 9562.64518051556 < N

    1. Initial program 59.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9562.6451805155593:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))