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

↓

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

\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 r179624 = x;
        double r179625 = r179624 * r179624;
        double r179626 = 1.0;
        double r179627 = r179625 + r179626;
        double r179628 = sqrt(r179627);
        double r179629 = r179624 + r179628;
        double r179630 = log(r179629);
        return r179630;
}

↓

double f(double x) {
        double r179631 = x;
        double r179632 = -0.9990009314811332;
        bool r179633 = r179631 <= r179632;
        double r179634 = 0.125;
        double r179635 = 3.0;
        double r179636 = pow(r179631, r179635);
        double r179637 = r179634 / r179636;
        double r179638 = 0.5;
        double r179639 = r179638 / r179631;
        double r179640 = 0.0625;
        double r179641 = 5.0;
        double r179642 = pow(r179631, r179641);
        double r179643 = r179640 / r179642;
        double r179644 = r179639 + r179643;
        double r179645 = r179637 - r179644;
        double r179646 = sqrt(r179645);
        double r179647 = log(r179646);
        double r179648 = r179647 + r179647;
        double r179649 = 0.8927294336574271;
        bool r179650 = r179631 <= r179649;
        double r179651 = 1.0;
        double r179652 = sqrt(r179651);
        double r179653 = log(r179652);
        double r179654 = r179631 / r179652;
        double r179655 = r179653 + r179654;
        double r179656 = 0.16666666666666666;
        double r179657 = pow(r179652, r179635);
        double r179658 = r179636 / r179657;
        double r179659 = r179656 * r179658;
        double r179660 = r179655 - r179659;
        double r179661 = r179631 - r179637;
        double r179662 = r179639 + r179661;
        double r179663 = r179631 + r179662;
        double r179664 = log(r179663);
        double r179665 = r179650 ? r179660 : r179664;
        double r179666 = r179633 ? r179648 : r179665;
        return r179666;
}

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
Out