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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.333333333333333315}{N} \cdot \log \left(e^{\frac{\frac{1}{N}}{N}}\right) + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;N \le 13931.9619516847724:\\
\;\;\;\;\log \left(\frac{N + 1}{N}\right)\\

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

\end{array}

double f(double N) {
        double r46217 = N;
        double r46218 = 1.0;
        double r46219 = r46217 + r46218;
        double r46220 = log(r46219);
        double r46221 = log(r46217);
        double r46222 = r46220 - r46221;
        return r46222;
}

↓

double f(double N) {
        double r46223 = N;
        double r46224 = 13931.961951684772;
        bool r46225 = r46223 <= r46224;
        double r46226 = 1.0;
        double r46227 = r46223 + r46226;
        double r46228 = r46227 / r46223;
        double r46229 = log(r46228);
        double r46230 = 0.3333333333333333;
        double r46231 = r46230 / r46223;
        double r46232 = 1.0;
        double r46233 = r46232 / r46223;
        double r46234 = r46233 / r46223;
        double r46235 = exp(r46234);
        double r46236 = log(r46235);
        double r46237 = r46231 * r46236;
        double r46238 = r46226 / r46223;
        double r46239 = 0.5;
        double r46240 = r46239 / r46223;
        double r46241 = r46240 / r46223;
        double r46242 = r46238 - r46241;
        double r46243 = r46237 + r46242;
        double r46244 = r46225 ? r46229 : r46243;
        return r46244;
}

Derivation

Split input into 2 regimes
if N < 13931.961951684772
1. Initial program 0.1
  \[\log \left(N + 1\right) - \log N\]
2. Using strategy rm
3. Applied diff-log0.1
  \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)}\]
if 13931.961951684772 < 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-rgt-in0.0
  \[\leadsto \color{blue}{\left(\frac{0.333333333333333315}{N} \cdot \frac{1}{{N}^{2}} + \left(-0.5\right) \cdot \frac{1}{{N}^{2}}\right)} + \frac{1}{N}\]
7. Applied associate-+l+0.0
  \[\leadsto \color{blue}{\frac{0.333333333333333315}{N} \cdot \frac{1}{{N}^{2}} + \left(\left(-0.5\right) \cdot \frac{1}{{N}^{2}} + \frac{1}{N}\right)}\]
8. Simplified0.0
  \[\leadsto \frac{0.333333333333333315}{N} \cdot \frac{1}{{N}^{2}} + \color{blue}{\left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)}\]
9. Using strategy rm
10. Applied add-log-exp0.0
  \[\leadsto \frac{0.333333333333333315}{N} \cdot \color{blue}{\log \left(e^{\frac{1}{{N}^{2}}}\right)} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\]
11. Simplified0.0
  \[\leadsto \frac{0.333333333333333315}{N} \cdot \log \color{blue}{\left(e^{\frac{\frac{1}{N}}{N}}\right)} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.333333333333333315}{N} \cdot \log \left(e^{\frac{\frac{1}{N}}{N}}\right) + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

Error

Try it out

Derivation

`if N < 13931.961951684772`

`if 13931.961951684772 < N`

Reproduce

In
Out