Average Error: 29.5 → 0.1
Time: 16.1s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 4842.291926013775:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{N}\right) + \log \left(\sqrt{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{1}{3}}{N} + \frac{-1}{2}}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 4842.291926013775:\\
\;\;\;\;\log \left(\frac{\sqrt{1 + N}}{N}\right) + \log \left(\sqrt{1 + N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{N} + \frac{\frac{\frac{1}{3}}{N} + \frac{-1}{2}}{N \cdot N}\\

\end{array}
double f(double N) {
        double r2496733 = N;
        double r2496734 = 1.0;
        double r2496735 = r2496733 + r2496734;
        double r2496736 = log(r2496735);
        double r2496737 = log(r2496733);
        double r2496738 = r2496736 - r2496737;
        return r2496738;
}
↓
double f(double N) {
        double r2496739 = N;
        double r2496740 = 4842.291926013775;
        bool r2496741 = r2496739 <= r2496740;
        double r2496742 = 1.0;
        double r2496743 = r2496742 + r2496739;
        double r2496744 = sqrt(r2496743);
        double r2496745 = r2496744 / r2496739;
        double r2496746 = log(r2496745);
        double r2496747 = log(r2496744);
        double r2496748 = r2496746 + r2496747;
        double r2496749 = r2496742 / r2496739;
        double r2496750 = 0.3333333333333333;
        double r2496751 = r2496750 / r2496739;
        double r2496752 = -0.5;
        double r2496753 = r2496751 + r2496752;
        double r2496754 = r2496739 * r2496739;
        double r2496755 = r2496753 / r2496754;
        double r2496756 = r2496749 + r2496755;
        double r2496757 = r2496741 ? r2496748 : r2496756;
        return r2496757;
}

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 < 4842.291926013775

    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 *-un-lft-identity0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied times-frac0.1

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

    9. Applied log-prod0.1

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

    10. Simplified0.1

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

    if 4842.291926013775 < 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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 4842.291926013775:\\ \;\;\;\;\log \left(\frac{\sqrt{1 + N}}{N}\right) + \log \left(\sqrt{1 + N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{\frac{1}{3}}{N} + \frac{-1}{2}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))