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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0736690649339895:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9600099568077087:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + {x}^{5} \cdot \frac{3}{40}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)} + \left(x + \frac{\frac{1}{2}}{x}\right)\right)\right)\\ \end{array}\]

Original	52.7
Target	44.9
Herbie	0.2

Derivation

Split input into 3 regimes
if x < -1.0736690649339895
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
  \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \frac{\frac{1}{16}}{{x}^{5}}\right)\right)}\]
if -1.0736690649339895 < x < 0.9600099568077087
1. Initial program 58.5
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Taylor expanded around 0 0.2
  \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]
3. Simplified0.2
  \[\leadsto \color{blue}{\left({x}^{5} \cdot \frac{3}{40} + x \cdot \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right) + x}\]
if 0.9600099568077087 < x
1. Initial program 30.8
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
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
  \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{\frac{1}{2}}{x}\right) + \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right)}\right)\]
Recombined 3 regimes into one program.
Final simplification0.2
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0736690649339895:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9600099568077087:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right) \cdot x + {x}^{5} \cdot \frac{3}{40}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{\frac{-1}{8}}{x \cdot \left(x \cdot x\right)} + \left(x + \frac{\frac{1}{2}}{x}\right)\right)\right)\\ \end{array}\]

Error

Target

Derivation

`if x < -1.0736690649339895`

`if -1.0736690649339895 < x < 0.9600099568077087`

`if 0.9600099568077087 < x`

Reproduce