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

↓

\[\begin{array}{l} \mathbf{if}\;N \le 3753.003344783202:\\ \;\;\;\;\left(\log \left({N}^{3} + {1}^{3}\right) - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.33333333333333337, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;N \le 3753.003344783202:\\
\;\;\;\;\left(\log \left({N}^{3} + {1}^{3}\right) - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right) - \log N\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.33333333333333337, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\

\end{array}

double f(double N) {
        double r40915 = N;
        double r40916 = 1.0;
        double r40917 = r40915 + r40916;
        double r40918 = log(r40917);
        double r40919 = log(r40915);
        double r40920 = r40918 - r40919;
        return r40920;
}

↓

double f(double N) {
        double r40921 = N;
        double r40922 = 3753.003344783202;
        bool r40923 = r40921 <= r40922;
        double r40924 = 3.0;
        double r40925 = pow(r40921, r40924);
        double r40926 = 1.0;
        double r40927 = pow(r40926, r40924);
        double r40928 = r40925 + r40927;
        double r40929 = log(r40928);
        double r40930 = r40921 * r40921;
        double r40931 = r40926 * r40926;
        double r40932 = r40921 * r40926;
        double r40933 = r40931 - r40932;
        double r40934 = r40930 + r40933;
        double r40935 = log(r40934);
        double r40936 = r40929 - r40935;
        double r40937 = log(r40921);
        double r40938 = r40936 - r40937;
        double r40939 = 0.5;
        double r40940 = -r40939;
        double r40941 = 1.0;
        double r40942 = 2.0;
        double r40943 = pow(r40921, r40942);
        double r40944 = r40941 / r40943;
        double r40945 = 0.33333333333333337;
        double r40946 = r40941 / r40925;
        double r40947 = r40926 / r40921;
        double r40948 = fma(r40945, r40946, r40947);
        double r40949 = fma(r40940, r40944, r40948);
        double r40950 = r40923 ? r40938 : r40949;
        return r40950;
}

Derivation

Split input into 2 regimes
if N < 3753.003344783202
1. Initial program 0.1
  \[\log \left(N + 1\right) - \log N\]
2. Using strategy rm
3. Applied flip3-+0.1
  \[\leadsto \log \color{blue}{\left(\frac{{N}^{3} + {1}^{3}}{N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)}\right)} - \log N\]
4. Applied log-div0.1
  \[\leadsto \color{blue}{\left(\log \left({N}^{3} + {1}^{3}\right) - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right)} - \log N\]
if 3753.003344783202 < N
1. Initial program 59.4
  \[\log \left(N + 1\right) - \log N\]
2. Using strategy rm
3. Applied flip3-+62.2
  \[\leadsto \log \color{blue}{\left(\frac{{N}^{3} + {1}^{3}}{N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)}\right)} - \log N\]
4. Applied log-div62.1
  \[\leadsto \color{blue}{\left(\log \left({N}^{3} + {1}^{3}\right) - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right)} - \log N\]
5. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\left(0.33333333333333337 \cdot \frac{1}{{N}^{3}} + 1 \cdot \frac{1}{N}\right) - 0.5 \cdot \frac{1}{{N}^{2}}}\]
6. Simplified0.0
  \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.33333333333333337, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;N \le 3753.003344783202:\\ \;\;\;\;\left(\log \left({N}^{3} + {1}^{3}\right) - \log \left(N \cdot N + \left(1 \cdot 1 - N \cdot 1\right)\right)\right) - \log N\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{1}{{N}^{2}}, \mathsf{fma}\left(0.33333333333333337, \frac{1}{{N}^{3}}, \frac{1}{N}\right)\right)\\ \end{array}\]

Error

Derivation

`if N < 3753.003344783202`

`if 3753.003344783202 < N`

Reproduce