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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0508372426442374:\\ \;\;\;\;\log \left(\sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \left(\sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)\right)\\ \mathbf{elif}\;x \le 0.9670372196853471:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Original	52.6
Target	45.1
Herbie	0.2

Derivation

Split input into 3 regimes
if x < -1.0508372426442374
1. Initial program 61.7
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
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
  \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)}\]
4. Using strategy rm
5. Applied add-cube-cbrt0.2
  \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right) \cdot \sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)}\]
if -1.0508372426442374 < x < 0.9670372196853471
1. Initial program 58.8
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around 0 0.1
  \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]
if 0.9670372196853471 < x
1. Initial program 30.7
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
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
  \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\right)\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0508372426442374:\\ \;\;\;\;\log \left(\sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \left(\sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}} \cdot \sqrt[3]{\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}}\right)\right)\\ \mathbf{elif}\;x \le 0.9670372196853471:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - \frac{1}{6} \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Error

Try it out

Target

Derivation

`if x < -1.0508372426442374`

`if -1.0508372426442374 < x < 0.9670372196853471`

`if 0.9670372196853471 < x`

Runtime

In
Out