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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{{N}^{2}}}\right) \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

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

↓

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

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

\end{array}

double f(double N) {
        double r41091 = N;
        double r41092 = 1.0;
        double r41093 = r41091 + r41092;
        double r41094 = log(r41093);
        double r41095 = log(r41091);
        double r41096 = r41094 - r41095;
        return r41096;
}

↓

double f(double N) {
        double r41097 = N;
        double r41098 = 13931.961951684772;
        bool r41099 = r41097 <= r41098;
        double r41100 = 1.0;
        double r41101 = r41097 + r41100;
        double r41102 = r41101 / r41097;
        double r41103 = log(r41102);
        double r41104 = 1.0;
        double r41105 = 2.0;
        double r41106 = pow(r41097, r41105);
        double r41107 = r41104 / r41106;
        double r41108 = exp(r41107);
        double r41109 = log(r41108);
        double r41110 = 0.3333333333333333;
        double r41111 = r41110 / r41097;
        double r41112 = r41109 * r41111;
        double r41113 = r41100 / r41097;
        double r41114 = 0.5;
        double r41115 = r41114 / r41097;
        double r41116 = r41115 / r41097;
        double r41117 = r41113 - r41116;
        double r41118 = r41112 + r41117;
        double r41119 = r41099 ? r41103 : r41118;
        return r41119;
}

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-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
  \[\leadsto \frac{1}{{N}^{2}} \cdot \frac{0.333333333333333315}{N} + \color{blue}{\left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)}\]
9. Using strategy rm
10. Applied add-log-exp0.0
  \[\leadsto \color{blue}{\log \left(e^{\frac{1}{{N}^{2}}}\right)} \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 13931.9619516847724:\\ \;\;\;\;\log \left(\frac{N + 1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{{N}^{2}}}\right) \cdot \frac{0.333333333333333315}{N} + \left(\frac{1}{N} - \frac{\frac{0.5}{N}}{N}\right)\\ \end{array}\]

Error

Try it out

Derivation

`if N < 13931.961951684772`

`if 13931.961951684772 < N`

Reproduce

In
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