Average Error: 29.3 → 0.1
Time: 5.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9218.49831353843729:\\ \;\;\;\;\log \left(\frac{N \cdot N - 1 \cdot 1}{N \cdot \left(N - 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9218.49831353843729:\\
\;\;\;\;\log \left(\frac{N \cdot N - 1 \cdot 1}{N \cdot \left(N - 1\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\

\end{array}
double f(double N) {
        double r52580 = N;
        double r52581 = 1.0;
        double r52582 = r52580 + r52581;
        double r52583 = log(r52582);
        double r52584 = log(r52580);
        double r52585 = r52583 - r52584;
        return r52585;
}
↓
double f(double N) {
        double r52586 = N;
        double r52587 = 9218.498313538437;
        bool r52588 = r52586 <= r52587;
        double r52589 = r52586 * r52586;
        double r52590 = 1.0;
        double r52591 = r52590 * r52590;
        double r52592 = r52589 - r52591;
        double r52593 = r52586 - r52590;
        double r52594 = r52586 * r52593;
        double r52595 = r52592 / r52594;
        double r52596 = log(r52595);
        double r52597 = 1.0;
        double r52598 = 2.0;
        double r52599 = pow(r52586, r52598);
        double r52600 = r52597 / r52599;
        double r52601 = 0.3333333333333333;
        double r52602 = r52601 / r52586;
        double r52603 = 0.5;
        double r52604 = r52602 - r52603;
        double r52605 = r52600 * r52604;
        double r52606 = r52590 / r52586;
        double r52607 = r52605 + r52606;
        double r52608 = r52588 ? r52596 : r52607;
        return r52608;
}

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 < 9218.498313538437

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    5. Applied flip-+0.1

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

    6. Applied associate-/l/0.1

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

    if 9218.498313538437 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9218.49831353843729:\\ \;\;\;\;\log \left(\frac{N \cdot N - 1 \cdot 1}{N \cdot \left(N - 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))