Average Error: 29.8 → 0.1
Time: 6.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8716.005511358958756318315863609313964844:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8716.005511358958756318315863609313964844:\\
\;\;\;\;\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 r33578 = N;
        double r33579 = 1.0;
        double r33580 = r33578 + r33579;
        double r33581 = log(r33580);
        double r33582 = log(r33578);
        double r33583 = r33581 - r33582;
        return r33583;
}
↓
double f(double N) {
        double r33584 = N;
        double r33585 = 8716.005511358959;
        bool r33586 = r33584 <= r33585;
        double r33587 = 1.0;
        double r33588 = r33584 + r33587;
        double r33589 = r33588 / r33584;
        double r33590 = log(r33589);
        double r33591 = 1.0;
        double r33592 = 2.0;
        double r33593 = pow(r33584, r33592);
        double r33594 = r33591 / r33593;
        double r33595 = 0.3333333333333333;
        double r33596 = r33595 / r33584;
        double r33597 = 0.5;
        double r33598 = r33596 - r33597;
        double r33599 = r33594 * r33598;
        double r33600 = r33587 / r33584;
        double r33601 = r33599 + r33600;
        double r33602 = r33586 ? r33590 : r33601;
        return r33602;
}

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 < 8716.005511358959

    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 8716.005511358959 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8716.005511358958756318315863609313964844:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.3333333333333333148296162562473909929395}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019308 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))