Average Error: 29.7 → 0.1
Time: 23.5s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4627.67004360751616:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4627.67004360751616:\\
\;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\

\end{array}
double f(double N) {
        double r63903 = N;
        double r63904 = 1.0;
        double r63905 = r63903 + r63904;
        double r63906 = log(r63905);
        double r63907 = log(r63903);
        double r63908 = r63906 - r63907;
        return r63908;
}
↓
double f(double N) {
        double r63909 = N;
        double r63910 = 4627.670043607516;
        bool r63911 = r63909 <= r63910;
        double r63912 = 1.0;
        double r63913 = r63909 + r63912;
        double r63914 = log(r63913);
        double r63915 = 3.0;
        double r63916 = pow(r63914, r63915);
        double r63917 = cbrt(r63916);
        double r63918 = log(r63909);
        double r63919 = r63917 - r63918;
        double r63920 = r63912 / r63909;
        double r63921 = 0.5;
        double r63922 = r63909 * r63909;
        double r63923 = r63921 / r63922;
        double r63924 = r63920 - r63923;
        double r63925 = 0.3333333333333333;
        double r63926 = pow(r63909, r63915);
        double r63927 = r63925 / r63926;
        double r63928 = r63924 + r63927;
        double r63929 = r63911 ? r63919 : r63928;
        return r63929;
}

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 < 4627.670043607516

    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 4627.670043607516 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4627.67004360751616:\\ \;\;\;\;\sqrt[3]{{\left(\log \left(N + 1\right)\right)}^{3}} - \log N\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} - \frac{0.5}{N \cdot N}\right) + \frac{0.333333333333333315}{{N}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1.0)) (log N)))