Average Error: 29.6 → 0.1
Time: 23.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6613.323166361232:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2}}{N \cdot N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 6613.323166361232:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-1}{2}}{N \cdot N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \frac{1}{N}\right)\\

\end{array}
double f(double N) {
        double r2114821 = N;
        double r2114822 = 1.0;
        double r2114823 = r2114821 + r2114822;
        double r2114824 = log(r2114823);
        double r2114825 = log(r2114821);
        double r2114826 = r2114824 - r2114825;
        return r2114826;
}
↓
double f(double N) {
        double r2114827 = N;
        double r2114828 = 6613.323166361232;
        bool r2114829 = r2114827 <= r2114828;
        double r2114830 = 1.0;
        double r2114831 = r2114830 + r2114827;
        double r2114832 = r2114831 / r2114827;
        double r2114833 = sqrt(r2114832);
        double r2114834 = log(r2114833);
        double r2114835 = r2114834 + r2114834;
        double r2114836 = -0.5;
        double r2114837 = r2114827 * r2114827;
        double r2114838 = r2114836 / r2114837;
        double r2114839 = 0.3333333333333333;
        double r2114840 = r2114839 / r2114827;
        double r2114841 = r2114840 / r2114837;
        double r2114842 = r2114830 / r2114827;
        double r2114843 = r2114841 + r2114842;
        double r2114844 = r2114838 + r2114843;
        double r2114845 = r2114829 ? r2114835 : r2114844;
        return r2114845;
}

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 < 6613.323166361232

    1. Initial program 0.1

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

    2. Simplified0.1

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]
    3. Using strategy rm

    4. Applied log1p-udef0.1

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

    5. Applied diff-log0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied log-prod0.1

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

    if 6613.323166361232 < N

    1. Initial program 59.6

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

    2. Simplified59.6

      \[\leadsto \color{blue}{\mathsf{log1p}\left(N\right) - \log N}\]

    3. Taylor expanded around -inf 0.0

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

    4. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6613.323166361232:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{2}}{N \cdot N} + \left(\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))