Average Error: 29.6 → 0.1
Time: 29.9s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9516.15297930401:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9516.15297930401:\\
\;\;\;\;\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 r1865089 = N;
        double r1865090 = 1.0;
        double r1865091 = r1865089 + r1865090;
        double r1865092 = log(r1865091);
        double r1865093 = log(r1865089);
        double r1865094 = r1865092 - r1865093;
        return r1865094;
}
↓
double f(double N) {
        double r1865095 = N;
        double r1865096 = 9516.15297930401;
        bool r1865097 = r1865095 <= r1865096;
        double r1865098 = 1.0;
        double r1865099 = r1865098 + r1865095;
        double r1865100 = r1865099 / r1865095;
        double r1865101 = log(r1865100);
        double r1865102 = 0.3333333333333333;
        double r1865103 = r1865102 / r1865095;
        double r1865104 = r1865095 * r1865095;
        double r1865105 = r1865103 / r1865104;
        double r1865106 = r1865098 / r1865095;
        double r1865107 = 0.5;
        double r1865108 = r1865107 / r1865104;
        double r1865109 = r1865106 - r1865108;
        double r1865110 = r1865105 + r1865109;
        double r1865111 = r1865097 ? r1865101 : r1865110;
        return r1865111;
}

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 < 9516.15297930401

    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 9516.15297930401 < N

    1. Initial program 59.4

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9516.15297930401:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{3}}{N}}{N \cdot N} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))