Average Error: 29.0 → 0.1
Time: 15.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 3983.563244138594:\\ \;\;\;\;\log \left(1 + N\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3}}{\left(N \cdot N\right) \cdot N} + \left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 3983.563244138594:\\
\;\;\;\;\log \left(1 + N\right) - \log N\\

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

\end{array}
double f(double N) {
        double r818059 = N;
        double r818060 = 1.0;
        double r818061 = r818059 + r818060;
        double r818062 = log(r818061);
        double r818063 = log(r818059);
        double r818064 = r818062 - r818063;
        return r818064;
}
↓
double f(double N) {
        double r818065 = N;
        double r818066 = 3983.563244138594;
        bool r818067 = r818065 <= r818066;
        double r818068 = 1.0;
        double r818069 = r818068 + r818065;
        double r818070 = log(r818069);
        double r818071 = log(r818065);
        double r818072 = r818070 - r818071;
        double r818073 = 0.3333333333333333;
        double r818074 = r818065 * r818065;
        double r818075 = r818074 * r818065;
        double r818076 = r818073 / r818075;
        double r818077 = -0.5;
        double r818078 = r818077 / r818074;
        double r818079 = r818068 / r818065;
        double r818080 = r818078 + r818079;
        double r818081 = r818076 + r818080;
        double r818082 = r818067 ? r818072 : r818081;
        return r818082;
}

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 < 3983.563244138594

    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

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

    6. Applied log-div0.1

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

    if 3983.563244138594 < N

    1. Initial program 59.5

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

    3. Applied add-log-exp59.5

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

    4. Simplified59.3

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

    6. Applied add-exp-log59.3

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3983.563244138594:\\ \;\;\;\;\log \left(1 + N\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{3}}{\left(N \cdot N\right) \cdot N} + \left(\frac{\frac{-1}{2}}{N \cdot N} + \frac{1}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))