Average Error: 29.2 → 0.0
Time: 25.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9504.960256628374:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)} + \left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9504.960256628374:\\
\;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\

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

\end{array}
double f(double N) {
        double r806263 = N;
        double r806264 = 1.0;
        double r806265 = r806263 + r806264;
        double r806266 = log(r806265);
        double r806267 = log(r806263);
        double r806268 = r806266 - r806267;
        return r806268;
}
↓
double f(double N) {
        double r806269 = N;
        double r806270 = 9504.960256628374;
        bool r806271 = r806269 <= r806270;
        double r806272 = 1.0;
        double r806273 = r806272 + r806269;
        double r806274 = r806269 / r806273;
        double r806275 = log(r806274);
        double r806276 = -r806275;
        double r806277 = 0.3333333333333333;
        double r806278 = r806269 * r806269;
        double r806279 = r806269 * r806278;
        double r806280 = r806277 / r806279;
        double r806281 = r806272 / r806269;
        double r806282 = -0.5;
        double r806283 = r806282 / r806278;
        double r806284 = r806281 + r806283;
        double r806285 = r806280 + r806284;
        double r806286 = r806271 ? r806276 : r806285;
        return r806286;
}

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 < 9504.960256628374

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\frac{1 + N}{N}\right)}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.1

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

    7. Applied *-un-lft-identity0.1

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

    8. Applied distribute-lft-out0.1

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

    9. Applied associate-/l*0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{\frac{N}{1 + N}}\right)}\]
    10. Using strategy rm

    11. Applied log-rec0.1

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

    if 9504.960256628374 < N

    1. Initial program 59.6

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

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9504.960256628374:\\ \;\;\;\;-\log \left(\frac{N}{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)} + \left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))