Average Error: 29.2 → 0.1
Time: 15.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8915.293301236255501862615346908569335938:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N} - \frac{0.5}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8915.293301236255501862615346908569335938:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N} - \frac{0.5}{N \cdot N}\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r3041348 = N;
        double r3041349 = 1.0;
        double r3041350 = r3041348 + r3041349;
        double r3041351 = log(r3041350);
        double r3041352 = log(r3041348);
        double r3041353 = r3041351 - r3041352;
        return r3041353;
}
↓
double f(double N) {
        double r3041354 = N;
        double r3041355 = 8915.293301236256;
        bool r3041356 = r3041354 <= r3041355;
        double r3041357 = 1.0;
        double r3041358 = r3041357 + r3041354;
        double r3041359 = r3041358 / r3041354;
        double r3041360 = log(r3041359);
        double r3041361 = 0.3333333333333333;
        double r3041362 = r3041361 / r3041354;
        double r3041363 = r3041354 * r3041354;
        double r3041364 = r3041362 / r3041363;
        double r3041365 = 0.5;
        double r3041366 = r3041365 / r3041363;
        double r3041367 = r3041364 - r3041366;
        double r3041368 = r3041357 / r3041354;
        double r3041369 = r3041367 + r3041368;
        double r3041370 = r3041356 ? r3041360 : r3041369;
        return r3041370;
}

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 < 8915.293301236256

    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 8915.293301236256 < N

    1. Initial program 59.6

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N} - \frac{0.5}{N \cdot N}\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8915.293301236255501862615346908569335938:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{0.3333333333333333148296162562473909929395}{N}}{N \cdot N} - \frac{0.5}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))