Average Error: 29.1 → 0.1
Time: 4.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9256.959046884599956683814525604248046875:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9256.959046884599956683814525604248046875:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r47359 = N;
        double r47360 = 1.0;
        double r47361 = r47359 + r47360;
        double r47362 = log(r47361);
        double r47363 = log(r47359);
        double r47364 = r47362 - r47363;
        return r47364;
}
↓
double f(double N) {
        double r47365 = N;
        double r47366 = 9256.9590468846;
        bool r47367 = r47365 <= r47366;
        double r47368 = 1.0;
        double r47369 = r47365 + r47368;
        double r47370 = r47369 / r47365;
        double r47371 = log(r47370);
        double r47372 = 1.0;
        double r47373 = 2.0;
        double r47374 = pow(r47365, r47373);
        double r47375 = r47372 / r47374;
        double r47376 = 0.3333333333333333;
        double r47377 = r47376 / r47365;
        double r47378 = 0.5;
        double r47379 = r47377 - r47378;
        double r47380 = r47375 * r47379;
        double r47381 = r47368 / r47365;
        double r47382 = r47380 + r47381;
        double r47383 = r47367 ? r47371 : r47382;
        return r47383;
}

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 < 9256.9590468846

    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 9256.9590468846 < N

    1. Initial program 59.7

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9256.959046884599956683814525604248046875:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019346 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))