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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.99900093148113323:\\ \;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)}\right)\\ \mathbf{elif}\;x \le 0.892729433657427074:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.99900093148113323:\\
\;\;\;\;\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 0.892729433657427074:\\
\;\;\;\;\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(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\

\end{array}

double f(double x) {
        double r135795 = x;
        double r135796 = r135795 * r135795;
        double r135797 = 1.0;
        double r135798 = r135796 + r135797;
        double r135799 = sqrt(r135798);
        double r135800 = r135795 + r135799;
        double r135801 = log(r135800);
        return r135801;
}

↓

double f(double x) {
        double r135802 = x;
        double r135803 = -0.9990009314811332;
        bool r135804 = r135802 <= r135803;
        double r135805 = 0.125;
        double r135806 = 3.0;
        double r135807 = pow(r135802, r135806);
        double r135808 = r135805 / r135807;
        double r135809 = 0.5;
        double r135810 = r135809 / r135802;
        double r135811 = 0.0625;
        double r135812 = 5.0;
        double r135813 = pow(r135802, r135812);
        double r135814 = r135811 / r135813;
        double r135815 = r135810 + r135814;
        double r135816 = r135808 - r135815;
        double r135817 = sqrt(r135816);
        double r135818 = log(r135817);
        double r135819 = r135818 + r135818;
        double r135820 = 0.8927294336574271;
        bool r135821 = r135802 <= r135820;
        double r135822 = 1.0;
        double r135823 = sqrt(r135822);
        double r135824 = log(r135823);
        double r135825 = r135802 / r135823;
        double r135826 = r135824 + r135825;
        double r135827 = 0.16666666666666666;
        double r135828 = pow(r135823, r135806);
        double r135829 = r135807 / r135828;
        double r135830 = r135827 * r135829;
        double r135831 = r135826 - r135830;
        double r135832 = r135802 - r135808;
        double r135833 = r135810 + r135832;
        double r135834 = r135802 + r135833;
        double r135835 = log(r135834);
        double r135836 = r135821 ? r135831 : r135835;
        double r135837 = r135804 ? r135819 : r135836;
        return r135837;
}

Original	53.2
Target	45.1
Herbie	0.2

Derivation

Split input into 3 regimes
if x < -0.9990009314811332
1. Initial program 63.1
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around -inf 0.1
  \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]
3. Simplified0.1
  \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)}\]
4. Using strategy rm
5. Applied add-sqr-sqrt0.1
  \[\leadsto \log \color{blue}{\left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)} \cdot \sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)}\right)}\]
6. Applied log-prod0.1
  \[\leadsto \color{blue}{\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)}\right)}\]
if -0.9990009314811332 < x < 0.8927294336574271
1. Initial program 58.7
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around 0 0.3
  \[\leadsto \color{blue}{\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}}\]
if 0.8927294336574271 < x
1. Initial program 31.4
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around inf 0.3
  \[\leadsto \log \left(x + \color{blue}{\left(\left(x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\]
3. Simplified0.3
  \[\leadsto \log \left(x + \color{blue}{\left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)}\right)\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.99900093148113323:\\ \;\;\;\;\log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)}\right) + \log \left(\sqrt{\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)}\right)\\ \mathbf{elif}\;x \le 0.892729433657427074:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

Error

Try it out

Target

Derivation

`if x < -0.9990009314811332`

`if -0.9990009314811332 < x < 0.8927294336574271`

`if 0.8927294336574271 < x`

Reproduce

In
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