Average Error: 29.4 → 0.0
Time: 22.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7453.229305032585:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7453.229305032585:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r1567079 = N;
        double r1567080 = 1.0;
        double r1567081 = r1567079 + r1567080;
        double r1567082 = log(r1567081);
        double r1567083 = log(r1567079);
        double r1567084 = r1567082 - r1567083;
        return r1567084;
}
↓
double f(double N) {
        double r1567085 = N;
        double r1567086 = 7453.229305032585;
        bool r1567087 = r1567085 <= r1567086;
        double r1567088 = 1.0;
        double r1567089 = r1567088 + r1567085;
        double r1567090 = r1567089 / r1567085;
        double r1567091 = log(r1567090);
        double r1567092 = r1567088 / r1567085;
        double r1567093 = -0.5;
        double r1567094 = r1567085 * r1567085;
        double r1567095 = r1567093 / r1567094;
        double r1567096 = r1567092 + r1567095;
        double r1567097 = 0.3333333333333333;
        double r1567098 = r1567097 / r1567094;
        double r1567099 = r1567098 / r1567085;
        double r1567100 = r1567096 + r1567099;
        double r1567101 = r1567087 ? r1567091 : r1567100;
        return r1567101;
}

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 < 7453.229305032585

    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 7453.229305032585 < 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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7453.229305032585:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))