Average Error: 29.8 → 0.1
Time: 11.0s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 11332.770353358372:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 11332.770353358372:\\
\;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\

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

\end{array}
double f(double N) {
        double r2197329 = N;
        double r2197330 = 1.0;
        double r2197331 = r2197329 + r2197330;
        double r2197332 = log(r2197331);
        double r2197333 = log(r2197329);
        double r2197334 = r2197332 - r2197333;
        return r2197334;
}
↓
double f(double N) {
        double r2197335 = N;
        double r2197336 = 11332.770353358372;
        bool r2197337 = r2197335 <= r2197336;
        double r2197338 = 1.0;
        double r2197339 = r2197338 + r2197335;
        double r2197340 = r2197339 / r2197335;
        double r2197341 = sqrt(r2197340);
        double r2197342 = log(r2197341);
        double r2197343 = r2197342 + r2197342;
        double r2197344 = 0.3333333333333333;
        double r2197345 = r2197335 * r2197335;
        double r2197346 = r2197335 * r2197345;
        double r2197347 = r2197344 / r2197346;
        double r2197348 = 0.5;
        double r2197349 = r2197348 / r2197345;
        double r2197350 = r2197347 - r2197349;
        double r2197351 = r2197338 / r2197335;
        double r2197352 = r2197350 + r2197351;
        double r2197353 = r2197337 ? r2197343 : r2197352;
        return r2197353;
}

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 < 11332.770353358372

    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-sqrt0.1

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

    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 11332.770353358372 < N

    1. Initial program 59.7

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 11332.770353358372:\\ \;\;\;\;\log \left(\sqrt{\frac{1 + N}{N}}\right) + \log \left(\sqrt{\frac{1 + N}{N}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{3}}{N \cdot \left(N \cdot N\right)} - \frac{\frac{1}{2}}{N \cdot N}\right) + \frac{1}{N}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  (- (log (+ N 1)) (log N)))