Average Error: 29.0 → 0.1
Time: 3.8s
Precision: 64
\[\log \left(N + 1\right) - \log N\]
↓
\[\begin{array}{l} \mathbf{if}\;N \le 6620.0699010986755:\\ \;\;\;\;\log \left(\left(N + 1\right) \cdot \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]
\log \left(N + 1\right) - \log N
↓
\begin{array}{l}
\mathbf{if}\;N \le 6620.0699010986755:\\
\;\;\;\;\log \left(\left(N + 1\right) \cdot \frac{1}{N}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\

\end{array}
double f(double N) {
        double r47296 = N;
        double r47297 = 1.0;
        double r47298 = r47296 + r47297;
        double r47299 = log(r47298);
        double r47300 = log(r47296);
        double r47301 = r47299 - r47300;
        return r47301;
}
↓
double f(double N) {
        double r47302 = N;
        double r47303 = 6620.0699010986755;
        bool r47304 = r47302 <= r47303;
        double r47305 = 1.0;
        double r47306 = r47302 + r47305;
        double r47307 = 1.0;
        double r47308 = r47307 / r47302;
        double r47309 = r47306 * r47308;
        double r47310 = log(r47309);
        double r47311 = 2.0;
        double r47312 = pow(r47302, r47311);
        double r47313 = r47307 / r47312;
        double r47314 = 0.3333333333333333;
        double r47315 = r47314 / r47302;
        double r47316 = r47313 * r47315;
        double r47317 = r47305 / r47302;
        double r47318 = 0.5;
        double r47319 = r47318 / r47302;
        double r47320 = r47319 / r47302;
        double r47321 = r47317 - r47320;
        double r47322 = r47316 + r47321;
        double r47323 = r47304 ? r47310 : r47322;
        return r47323;
}

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 < 6620.0699010986755

    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 div-inv0.1

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

    if 6620.0699010986755 < N

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]

    3. Simplified0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \left(\frac{0.333333333333333315}{N} - 0.5\right) + \frac{1}{N}}\]
    4. Using strategy rm

    5. Applied sub-neg0.0

      \[\leadsto \frac{1}{{N}^{2}} \cdot \color{blue}{\left(\frac{0.333333333333333315}{N} + \left(-0.5\right)\right)} + \frac{1}{N}\]

    6. Applied distribute-lft-in0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \frac{1}{{N}^{2}} \cdot \left(-0.5\right)\right)} + \frac{1}{N}\]

    7. Applied associate-+l+0.0

      \[\leadsto \color{blue}{\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{{N}^{2}} \cdot \left(-0.5\right) + \frac{1}{N}\right)}\]

    8. Simplified0.0

      \[\leadsto \frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \color{blue}{\left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;N \le 6620.0699010986755:\\ \;\;\;\;\log \left(\left(N + 1\right) \cdot \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1)) (log N)))