Average Error: 29.2 → 0.1
Time: 10.7s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9550.567671573803:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N}}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9550.567671573803:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r754946 = N;
        double r754947 = 1.0;
        double r754948 = r754946 + r754947;
        double r754949 = log(r754948);
        double r754950 = log(r754946);
        double r754951 = r754949 - r754950;
        return r754951;
}
↓
double f(double N) {
        double r754952 = N;
        double r754953 = 9550.567671573803;
        bool r754954 = r754952 <= r754953;
        double r754955 = 1.0;
        double r754956 = r754955 + r754952;
        double r754957 = r754956 / r754952;
        double r754958 = log(r754957);
        double r754959 = r754955 / r754952;
        double r754960 = -0.5;
        double r754961 = r754952 * r754952;
        double r754962 = r754960 / r754961;
        double r754963 = r754959 + r754962;
        double r754964 = 0.3333333333333333;
        double r754965 = r754964 / r754952;
        double r754966 = r754965 / r754961;
        double r754967 = r754963 + r754966;
        double r754968 = r754954 ? r754958 : r754967;
        return r754968;
}

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 < 9550.567671573803

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Simplified0.1

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

    if 9550.567671573803 < N

    1. Initial program 59.6

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

    3. Applied add-log-exp59.6

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

    4. Simplified59.3

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

    6. Applied add-sqr-sqrt59.3

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

    7. Taylor expanded around inf 0.0

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

    8. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9550.567671573803:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))