Average Error: 29.1 → 0.1
Time: 15.2s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 9233.861277203294:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{N}}{N \cdot N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 9233.861277203294:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r2433808 = N;
        double r2433809 = 1.0;
        double r2433810 = r2433808 + r2433809;
        double r2433811 = log(r2433810);
        double r2433812 = log(r2433808);
        double r2433813 = r2433811 - r2433812;
        return r2433813;
}
↓
double f(double N) {
        double r2433814 = N;
        double r2433815 = 9233.861277203294;
        bool r2433816 = r2433814 <= r2433815;
        double r2433817 = 1.0;
        double r2433818 = r2433817 + r2433814;
        double r2433819 = r2433818 / r2433814;
        double r2433820 = sqrt(r2433819);
        double r2433821 = log(r2433820);
        double r2433822 = r2433821 + r2433821;
        double r2433823 = r2433817 / r2433814;
        double r2433824 = -0.5;
        double r2433825 = 0.3333333333333333;
        double r2433826 = r2433825 / r2433814;
        double r2433827 = r2433824 + r2433826;
        double r2433828 = r2433814 * r2433814;
        double r2433829 = r2433827 / r2433828;
        double r2433830 = r2433823 + r2433829;
        double r2433831 = r2433816 ? r2433822 : r2433830;
        return r2433831;
}

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 < 9233.861277203294

    1. Initial program 0.1

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

    3. Applied diff-log0.1

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

    5. Applied add-sqr-sqrt0.2

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

    6. Applied log-prod0.1

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

    if 9233.861277203294 < N

    1. Initial program 59.5

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 9233.861277203294:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} + \frac{\frac{-1}{2} + \frac{\frac{1}{3}}{N}}{N \cdot N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))