Average Error: 29.6 → 0.1
Time: 13.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7771.4619801305835:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{\frac{1}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7771.4619801305835:\\
\;\;\;\;\log \left(1 + \frac{1}{N}\right)\\

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

\end{array}
double f(double N) {
        double r2435542 = N;
        double r2435543 = 1.0;
        double r2435544 = r2435542 + r2435543;
        double r2435545 = log(r2435544);
        double r2435546 = log(r2435542);
        double r2435547 = r2435545 - r2435546;
        return r2435547;
}
↓
double f(double N) {
        double r2435548 = N;
        double r2435549 = 7771.4619801305835;
        bool r2435550 = r2435548 <= r2435549;
        double r2435551 = 1.0;
        double r2435552 = r2435551 / r2435548;
        double r2435553 = r2435551 + r2435552;
        double r2435554 = log(r2435553);
        double r2435555 = 0.5;
        double r2435556 = 0.3333333333333333;
        double r2435557 = r2435556 / r2435548;
        double r2435558 = r2435555 - r2435557;
        double r2435559 = r2435552 / r2435548;
        double r2435560 = r2435558 * r2435559;
        double r2435561 = r2435552 - r2435560;
        double r2435562 = r2435550 ? r2435554 : r2435561;
        return r2435562;
}

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 < 7771.4619801305835

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    4. Taylor expanded around 0 0.1

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

    if 7771.4619801305835 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7771.4619801305835:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right) \cdot \frac{\frac{1}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))