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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 7265.781170340218:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \log \left(\sqrt{e^{\log \left(\frac{N + 1}{N}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

double f(double N) {
        double r2374101 = N;
        double r2374102 = 1.0;
        double r2374103 = r2374101 + r2374102;
        double r2374104 = log(r2374103);
        double r2374105 = log(r2374101);
        double r2374106 = r2374104 - r2374105;
        return r2374106;
}

↓

double f(double N) {
        double r2374107 = N;
        double r2374108 = 7265.781170340218;
        bool r2374109 = r2374107 <= r2374108;
        double r2374110 = 0.5;
        double r2374111 = 1.0;
        double r2374112 = r2374107 + r2374111;
        double r2374113 = r2374112 / r2374107;
        double r2374114 = log(r2374113);
        double r2374115 = r2374110 * r2374114;
        double r2374116 = exp(r2374114);
        double r2374117 = sqrt(r2374116);
        double r2374118 = log(r2374117);
        double r2374119 = r2374115 + r2374118;
        double r2374120 = r2374111 / r2374107;
        double r2374121 = -0.5;
        double r2374122 = r2374107 * r2374107;
        double r2374123 = r2374121 / r2374122;
        double r2374124 = r2374120 + r2374123;
        double r2374125 = 0.3333333333333333;
        double r2374126 = r2374125 / r2374122;
        double r2374127 = r2374126 / r2374107;
        double r2374128 = r2374124 + r2374127;
        double r2374129 = r2374109 ? r2374119 : r2374128;
        return r2374129;
}

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

↓

\begin{array}{l}
\mathbf{if}\;N \le 7265.781170340218:\\
\;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \log \left(\sqrt{e^{\log \left(\frac{N + 1}{N}\right)}}\right)\\

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

\end{array}

Derivation

Split input into 2 regimes
if N < 7265.781170340218
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
  \[\leadsto \color{blue}{\log \left(\sqrt{\frac{N + 1}{N}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)}\]
7. Using strategy rm
8. Applied pow10.1
  \[\leadsto \log \left(\sqrt{\color{blue}{{\left(\frac{N + 1}{N}\right)}^{1}}}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]
9. Applied sqrt-pow10.1
  \[\leadsto \log \color{blue}{\left({\left(\frac{N + 1}{N}\right)}^{\left(\frac{1}{2}\right)}\right)} + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]
10. Applied log-pow0.1
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right)} + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]
11. Simplified0.1
  \[\leadsto \color{blue}{\frac{1}{2}} \cdot \log \left(\frac{N + 1}{N}\right) + \log \left(\sqrt{\frac{N + 1}{N}}\right)\]
12. Using strategy rm
13. Applied add-exp-log0.1
  \[\leadsto \frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \log \left(\sqrt{\color{blue}{e^{\log \left(\frac{N + 1}{N}\right)}}}\right)\]
if 7265.781170340218 < 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
  \[\leadsto \color{blue}{\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}}\]
Recombined 2 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 7265.781170340218:\\ \;\;\;\;\frac{1}{2} \cdot \log \left(\frac{N + 1}{N}\right) + \log \left(\sqrt{e^{\log \left(\frac{N + 1}{N}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{N} + \frac{\frac{-1}{2}}{N \cdot N}\right) + \frac{\frac{\frac{1}{3}}{N \cdot N}}{N}\\ \end{array}\]

Error

Derivation

`if N < 7265.781170340218`

`if 7265.781170340218 < N`

Reproduce