Average Error: 29.2 → 0.1
Time: 11.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4122.92847833978430571733042597770690918:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4122.92847833978430571733042597770690918:\\
\;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\

\mathbf{else}:\\
\;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\

\end{array}
double f(double N) {
        double r53769 = N;
        double r53770 = 1.0;
        double r53771 = r53769 + r53770;
        double r53772 = log(r53771);
        double r53773 = log(r53769);
        double r53774 = r53772 - r53773;
        return r53774;
}
↓
double f(double N) {
        double r53775 = N;
        double r53776 = 4122.928478339784;
        bool r53777 = r53775 <= r53776;
        double r53778 = 1.0;
        double r53779 = r53775 + r53778;
        double r53780 = log(r53779);
        double r53781 = 3.0;
        double r53782 = pow(r53780, r53781);
        double r53783 = cbrt(r53782);
        double r53784 = log(r53775);
        double r53785 = r53783 - r53784;
        double r53786 = 0.3333333333333333;
        double r53787 = pow(r53775, r53781);
        double r53788 = r53786 / r53787;
        double r53789 = 0.5;
        double r53790 = r53789 / r53775;
        double r53791 = r53778 - r53790;
        double r53792 = r53791 / r53775;
        double r53793 = r53788 + r53792;
        double r53794 = r53777 ? r53785 : r53793;
        return r53794;
}

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 < 4122.928478339784

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(N + 1\right) \cdot \log \left(N + 1\right)\right) \cdot \log \left(N + 1\right)}} - \log N\]

    4. Simplified0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\log \left(N + 1\right)\right)}^{3}}} - \log N\]

    if 4122.928478339784 < N

    1. Initial program 59.4

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4122.92847833978430571733042597770690918:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333148296162562473909929395}{{N}^{3}} + \frac{1 - \frac{0.5}{N}}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))