Average Error: 28.9 → 0.1
Time: 26.6s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8114.657537075721:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8114.657537075721:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r822926 = N;
        double r822927 = 1.0;
        double r822928 = r822926 + r822927;
        double r822929 = log(r822928);
        double r822930 = log(r822926);
        double r822931 = r822929 - r822930;
        return r822931;
}
↓
double f(double N) {
        double r822932 = N;
        double r822933 = 8114.657537075721;
        bool r822934 = r822932 <= r822933;
        double r822935 = 1.0;
        double r822936 = r822935 + r822932;
        double r822937 = r822936 / r822932;
        double r822938 = log(r822937);
        double r822939 = 0.3333333333333333;
        double r822940 = r822939 / r822932;
        double r822941 = r822932 * r822932;
        double r822942 = r822940 / r822941;
        double r822943 = r822935 / r822932;
        double r822944 = 0.5;
        double r822945 = r822944 / r822941;
        double r822946 = r822943 - r822945;
        double r822947 = r822942 + r822946;
        double r822948 = r822934 ? r822938 : r822947;
        return r822948;
}

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 < 8114.657537075721

    1. Initial program 0.1

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

    3. Applied add-log-exp0.2

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

    4. Simplified0.1

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

    if 8114.657537075721 < N

    1. Initial program 59.5

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

    2. Taylor expanded around -inf 0.0

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

    3. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8114.657537075721:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))