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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.02484362312584243:\\ \;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right)\\ \mathbf{elif}\;x \le 0.89339731849518578:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.02484362312584243:\\
\;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right)\\

\mathbf{elif}\;x \le 0.89339731849518578:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}

double f(double x) {
        double r224685 = x;
        double r224686 = r224685 * r224685;
        double r224687 = 1.0;
        double r224688 = r224686 + r224687;
        double r224689 = sqrt(r224688);
        double r224690 = r224685 + r224689;
        double r224691 = log(r224690);
        return r224691;
}

↓

double f(double x) {
        double r224692 = x;
        double r224693 = -1.0248436231258424;
        bool r224694 = r224692 <= r224693;
        double r224695 = 0.125;
        double r224696 = 3.0;
        double r224697 = pow(r224692, r224696);
        double r224698 = r224695 / r224697;
        double r224699 = 0.5;
        double r224700 = r224699 / r224692;
        double r224701 = 0.0625;
        double r224702 = -r224701;
        double r224703 = 5.0;
        double r224704 = pow(r224692, r224703);
        double r224705 = r224702 / r224704;
        double r224706 = r224700 - r224705;
        double r224707 = r224698 - r224706;
        double r224708 = sqrt(r224707);
        double r224709 = log(r224708);
        double r224710 = r224709 + r224709;
        double r224711 = 0.8933973184951858;
        bool r224712 = r224692 <= r224711;
        double r224713 = 1.0;
        double r224714 = sqrt(r224713);
        double r224715 = log(r224714);
        double r224716 = r224692 / r224714;
        double r224717 = r224715 + r224716;
        double r224718 = 0.16666666666666666;
        double r224719 = pow(r224714, r224696);
        double r224720 = r224697 / r224719;
        double r224721 = r224718 * r224720;
        double r224722 = r224717 - r224721;
        double r224723 = r224692 + r224700;
        double r224724 = r224723 - r224698;
        double r224725 = r224692 + r224724;
        double r224726 = log(r224725);
        double r224727 = r224712 ? r224722 : r224726;
        double r224728 = r224694 ? r224710 : r224727;
        return r224728;
}

Original	53.1
Target	45.1
Herbie	0.3

Derivation

Split input into 3 regimes
if x < -1.0248436231258424
1. Initial program 63.0
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around -inf 0.2
  \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]
3. Simplified0.2
  \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)}\]
4. Using strategy rm
5. Applied add-sqr-sqrt0.2
  \[\leadsto \log \color{blue}{\left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)} \cdot \sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right)}\]
6. Applied log-prod0.2
  \[\leadsto \color{blue}{\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right)}\]
if -1.0248436231258424 < x < 0.8933973184951858
1. Initial program 58.5
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around 0 0.3
  \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]
if 0.8933973184951858 < x
1. Initial program 32.0
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around inf 0.2
  \[\leadsto \log \left(x + \color{blue}{\left(\left(x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\]
3. Simplified0.2
  \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)}\right)\]
Recombined 3 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02484362312584243:\\ \;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)}\right)\\ \mathbf{elif}\;x \le 0.89339731849518578:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Error

Try it out

Target

Derivation

`if x < -1.0248436231258424`

`if -1.0248436231258424 < x < 0.8933973184951858`

`if 0.8933973184951858 < x`

Reproduce

In
Out