Average Error: 29.6 → 0.1
Time: 15.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 8850.986780357693:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{3}}{N} + \frac{-1}{2}\right) \cdot \frac{\frac{1}{N}}{N} + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 8850.986780357693:\\
\;\;\;\;\log \left(\frac{1 + N}{N}\right)\\

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

\end{array}
double f(double N) {
        double r941805 = N;
        double r941806 = 1.0;
        double r941807 = r941805 + r941806;
        double r941808 = log(r941807);
        double r941809 = log(r941805);
        double r941810 = r941808 - r941809;
        return r941810;
}
↓
double f(double N) {
        double r941811 = N;
        double r941812 = 8850.986780357693;
        bool r941813 = r941811 <= r941812;
        double r941814 = 1.0;
        double r941815 = r941814 + r941811;
        double r941816 = r941815 / r941811;
        double r941817 = log(r941816);
        double r941818 = 0.3333333333333333;
        double r941819 = r941818 / r941811;
        double r941820 = -0.5;
        double r941821 = r941819 + r941820;
        double r941822 = r941814 / r941811;
        double r941823 = r941822 / r941811;
        double r941824 = r941821 * r941823;
        double r941825 = r941824 + r941822;
        double r941826 = r941813 ? r941817 : r941825;
        return r941826;
}

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 < 8850.986780357693

    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 8850.986780357693 < 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

      \[\leadsto \color{blue}{\frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)} + \left(\frac{1}{N} - \frac{\frac{1}{2}}{N \cdot N}\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp0.1

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

    6. Taylor expanded around 0 0.0

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

    7. Simplified0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 8850.986780357693:\\ \;\;\;\;\log \left(\frac{1 + N}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{3}}{N} + \frac{-1}{2}\right) \cdot \frac{\frac{1}{N}}{N} + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))