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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.9982267368695869613759441563161090016365:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.9479026978334682551619039259094279259443:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + \frac{0.25}{x \cdot x}\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.9982267368695869613759441563161090016365:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\

\mathbf{elif}\;x \le 0.9479026978334682551619039259094279259443:\\
\;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\left(\log 2 + \frac{0.25}{x \cdot x}\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)\\

\end{array}

double f(double x) {
        double r101830 = x;
        double r101831 = r101830 * r101830;
        double r101832 = 1.0;
        double r101833 = r101831 + r101832;
        double r101834 = sqrt(r101833);
        double r101835 = r101830 + r101834;
        double r101836 = log(r101835);
        return r101836;
}

↓

double f(double x) {
        double r101837 = x;
        double r101838 = -0.998226736869587;
        bool r101839 = r101837 <= r101838;
        double r101840 = 0.125;
        double r101841 = 3.0;
        double r101842 = pow(r101837, r101841);
        double r101843 = r101840 / r101842;
        double r101844 = 0.0625;
        double r101845 = 5.0;
        double r101846 = pow(r101837, r101845);
        double r101847 = r101844 / r101846;
        double r101848 = r101843 - r101847;
        double r101849 = 0.5;
        double r101850 = r101849 / r101837;
        double r101851 = r101848 - r101850;
        double r101852 = log(r101851);
        double r101853 = 0.9479026978334683;
        bool r101854 = r101837 <= r101853;
        double r101855 = 1.0;
        double r101856 = sqrt(r101855);
        double r101857 = log(r101856);
        double r101858 = -0.16666666666666666;
        double r101859 = r101837 * r101837;
        double r101860 = r101859 / r101855;
        double r101861 = r101858 * r101860;
        double r101862 = 1.0;
        double r101863 = r101861 + r101862;
        double r101864 = r101837 / r101856;
        double r101865 = r101863 * r101864;
        double r101866 = r101857 + r101865;
        double r101867 = 2.0;
        double r101868 = log(r101867);
        double r101869 = 0.25;
        double r101870 = r101869 / r101859;
        double r101871 = r101868 + r101870;
        double r101872 = 0.09375;
        double r101873 = 4.0;
        double r101874 = pow(r101837, r101873);
        double r101875 = r101872 / r101874;
        double r101876 = log(r101837);
        double r101877 = r101875 - r101876;
        double r101878 = r101871 - r101877;
        double r101879 = r101854 ? r101866 : r101878;
        double r101880 = r101839 ? r101852 : r101879;
        return r101880;
}

Original	53.0
Target	45.4
Herbie	0.3

Derivation

Split input into 3 regimes
if x < -0.998226736869587
1. Initial program 62.6
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around -inf 0.3
  \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]
3. Simplified0.3
  \[\leadsto \log \color{blue}{\left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)}\]
if -0.998226736869587 < x < 0.9479026978334683
1. Initial program 58.7
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around 0 0.3
  \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]
3. Simplified0.3
  \[\leadsto \color{blue}{\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}}\]
if 0.9479026978334683 < x
1. Initial program 31.9
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around inf 0.3
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{1}{{x}^{2}} + \log 2\right) - \left(\log \left(\frac{1}{x}\right) + 0.09375 \cdot \frac{1}{{x}^{4}}\right)}\]
3. Simplified0.3
  \[\leadsto \color{blue}{\left(\log 2 + \frac{0.25}{x \cdot x}\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)}\]
Recombined 3 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9982267368695869613759441563161090016365:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.9479026978334682551619039259094279259443:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + \frac{0.25}{x \cdot x}\right) - \left(\frac{0.09375}{{x}^{4}} - \log x\right)\\ \end{array}\]

Error

Try it out

Target

Derivation

`if x < -0.998226736869587`

`if -0.998226736869587 < x < 0.9479026978334683`

`if 0.9479026978334683 < x`

Reproduce

In
Out