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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0457624597562116:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 1.0059846727535393:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{4}}{x \cdot x} - \frac{\frac{3}{32}}{{x}^{4}}\right) + \sqrt[3]{\left(\left(\log 2 + \log x\right) \cdot \left(\log 2 + \log x\right)\right) \cdot \left(\log 2 + \log x\right)}\\ \end{array}\]

Original	52.1
Target	44.4
Herbie	0.3

Derivation

Split input into 3 regimes
if x < -1.0457624597562116
1. Initial program 61.8
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around -inf 0.1
  \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]
3. Simplified0.1
  \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)}\]
if -1.0457624597562116 < x < 1.0059846727535393
1. Initial program 58.6
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around 0 0.2
  \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]
if 1.0059846727535393 < x
1. Initial program 30.1
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around inf 0.3
  \[\leadsto \color{blue}{\left(\log 2 + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right) - \left(\log \left(\frac{1}{x}\right) + \frac{3}{32} \cdot \frac{1}{{x}^{4}}\right)}\]
3. Simplified0.3
  \[\leadsto \color{blue}{\left(\log 2 + \log x\right) + \left(\frac{\frac{1}{4}}{x \cdot x} - \frac{\frac{3}{32}}{{x}^{4}}\right)}\]
4. Using strategy rm
5. Applied add-cbrt-cube0.7
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\log 2 + \log x\right) \cdot \left(\log 2 + \log x\right)\right) \cdot \left(\log 2 + \log x\right)}} + \left(\frac{\frac{1}{4}}{x \cdot x} - \frac{\frac{3}{32}}{{x}^{4}}\right)\]
Recombined 3 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0457624597562116:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 1.0059846727535393:\\ \;\;\;\;\left(x + \frac{3}{40} \cdot {x}^{5}\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{4}}{x \cdot x} - \frac{\frac{3}{32}}{{x}^{4}}\right) + \sqrt[3]{\left(\left(\log 2 + \log x\right) \cdot \left(\log 2 + \log x\right)\right) \cdot \left(\log 2 + \log x\right)}\\ \end{array}\]

Error

Try it out

Target

Derivation

`if x < -1.0457624597562116`

`if -1.0457624597562116 < x < 1.0059846727535393`

`if 1.0059846727535393 < x`

Runtime

In
Out