Average Error: 29.1 → 0.1
Time: 20.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4585.464779854794:\\ \;\;\;\;\log \left(1 + N\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{1}{N}}{N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4585.464779854794:\\
\;\;\;\;\log \left(1 + N\right) - \log N\\

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

\end{array}
double f(double N) {
        double r2450406 = N;
        double r2450407 = 1.0;
        double r2450408 = r2450406 + r2450407;
        double r2450409 = log(r2450408);
        double r2450410 = log(r2450406);
        double r2450411 = r2450409 - r2450410;
        return r2450411;
}
↓
double f(double N) {
        double r2450412 = N;
        double r2450413 = 4585.464779854794;
        bool r2450414 = r2450412 <= r2450413;
        double r2450415 = 1.0;
        double r2450416 = r2450415 + r2450412;
        double r2450417 = log(r2450416);
        double r2450418 = log(r2450412);
        double r2450419 = r2450417 - r2450418;
        double r2450420 = r2450415 / r2450412;
        double r2450421 = r2450420 / r2450412;
        double r2450422 = 0.5;
        double r2450423 = 0.3333333333333333;
        double r2450424 = r2450423 / r2450412;
        double r2450425 = r2450422 - r2450424;
        double r2450426 = r2450421 * r2450425;
        double r2450427 = r2450420 - r2450426;
        double r2450428 = r2450414 ? r2450419 : r2450427;
        return r2450428;
}

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 < 4585.464779854794

    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. Using strategy rm

    5. Applied add-exp-log0.1

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

    6. Applied add-exp-log0.1

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

    7. Applied div-exp0.1

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

    8. Applied rem-log-exp0.1

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

    if 4585.464779854794 < N

    1. Initial program 59.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4585.464779854794:\\ \;\;\;\;\log \left(1 + N\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{\frac{1}{N}}{N} \cdot \left(\frac{1}{2} - \frac{\frac{1}{3}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))