\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0666762070372178:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9575491063907275:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]

double f(double x) {
        double r18249884 = x;
        double r18249885 = r18249884 * r18249884;
        double r18249886 = 1.0;
        double r18249887 = r18249885 + r18249886;
        double r18249888 = sqrt(r18249887);
        double r18249889 = r18249884 + r18249888;
        double r18249890 = log(r18249889);
        return r18249890;
}

↓

double f(double x) {
        double r18249891 = x;
        double r18249892 = -1.0666762070372178;
        bool r18249893 = r18249891 <= r18249892;
        double r18249894 = 0.125;
        double r18249895 = r18249894 / r18249891;
        double r18249896 = r18249895 / r18249891;
        double r18249897 = r18249896 / r18249891;
        double r18249898 = -0.5;
        double r18249899 = r18249898 / r18249891;
        double r18249900 = r18249897 + r18249899;
        double r18249901 = 0.0625;
        double r18249902 = 5.0;
        double r18249903 = pow(r18249891, r18249902);
        double r18249904 = r18249901 / r18249903;
        double r18249905 = r18249900 - r18249904;
        double r18249906 = log(r18249905);
        double r18249907 = 0.9575491063907275;
        bool r18249908 = r18249891 <= r18249907;
        double r18249909 = 0.075;
        double r18249910 = r18249903 * r18249909;
        double r18249911 = r18249891 * r18249891;
        double r18249912 = -0.16666666666666666;
        double r18249913 = r18249911 * r18249912;
        double r18249914 = r18249891 * r18249913;
        double r18249915 = r18249891 + r18249914;
        double r18249916 = r18249910 + r18249915;
        double r18249917 = 0.5;
        double r18249918 = r18249917 / r18249891;
        double r18249919 = -0.125;
        double r18249920 = r18249891 * r18249911;
        double r18249921 = r18249919 / r18249920;
        double r18249922 = r18249891 + r18249921;
        double r18249923 = r18249918 + r18249922;
        double r18249924 = r18249891 + r18249923;
        double r18249925 = log(r18249924);
        double r18249926 = r18249908 ? r18249916 : r18249925;
        double r18249927 = r18249893 ? r18249906 : r18249926;
        return r18249927;
}

\log \left(x + \sqrt{x \cdot x + 1}\right)

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.0666762070372178:\\
\;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.9575491063907275:\\
\;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\

\end{array}

Original	52.2
Target	44.0
Herbie	0.2

Derivation

Split input into 3 regimes
if x < -1.0666762070372178
1. Initial program 61.8
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around -inf 0.2
  \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]
3. Simplified0.2
  \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{-1}{2}}{x} + \frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)}\]
if -1.0666762070372178 < x < 0.9575491063907275
1. Initial program 58.5
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around 0 0.2
  \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]
3. Simplified0.2
  \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + x\right) + {x}^{5} \cdot \frac{3}{40}}\]
if 0.9575491063907275 < x
1. Initial program 30.0
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around inf 0.2
  \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\right)\]
3. Simplified0.2
  \[\leadsto \log \left(x + \color{blue}{\left(\left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + x\right) + \frac{\frac{1}{2}}{x}\right)}\right)\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0666762070372178:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x} + \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9575491063907275:\\ \;\;\;\;{x}^{5} \cdot \frac{3}{40} + \left(x + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{1}{2}}{x} + \left(x + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\right)\\ \end{array}\]

Error

Target

Derivation

`if x < -1.0666762070372178`

`if -1.0666762070372178 < x < 0.9575491063907275`

`if 0.9575491063907275 < x`

Reproduce