Average Error: 29.4 → 0.1
Time: 5.3s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 7664.616617245729685237165540456771850586:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 7664.616617245729685237165540456771850586:\\
\;\;\;\;\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 r43467 = N;
        double r43468 = 1.0;
        double r43469 = r43467 + r43468;
        double r43470 = log(r43469);
        double r43471 = log(r43467);
        double r43472 = r43470 - r43471;
        return r43472;
}
↓
double f(double N) {
        double r43473 = N;
        double r43474 = 7664.61661724573;
        bool r43475 = r43473 <= r43474;
        double r43476 = 1.0;
        double r43477 = r43473 + r43476;
        double r43478 = r43477 / r43473;
        double r43479 = log(r43478);
        double r43480 = 1.0;
        double r43481 = 2.0;
        double r43482 = pow(r43473, r43481);
        double r43483 = r43480 / r43482;
        double r43484 = 0.3333333333333333;
        double r43485 = r43484 / r43473;
        double r43486 = 0.5;
        double r43487 = r43485 - r43486;
        double r43488 = r43483 * r43487;
        double r43489 = r43476 / r43473;
        double r43490 = r43488 + r43489;
        double r43491 = r43475 ? r43479 : r43490;
        return r43491;
}

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 < 7664.61661724573

    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 7664.61661724573 < N

    1. Initial program 59.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7664.616617245729685237165540456771850586:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))