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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0712784506357302:\\ \;\;\;\;\log \left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{elif}\;x \le 0.008931035360035398:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\sqrt{1^2 + x^2}^*} \cdot \sqrt{\sqrt{1^2 + x^2}^*} + x\right)\\ \end{array}\]

Original	52.6
Target	44.9
Herbie	0.1

Derivation

Split input into 3 regimes
if x < -1.0712784506357302
1. Initial program 61.8
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Initial simplification61.0
  \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
3. Taylor expanded around -inf 0.2
  \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]
4. Simplified0.2
  \[\leadsto \log \color{blue}{\left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)}\]
if -1.0712784506357302 < x < 0.008931035360035398
1. Initial program 58.8
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Initial simplification58.8
  \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
3. 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 0.008931035360035398 < x
1. Initial program 31.3
  \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
2. Initial simplification0.0
  \[\leadsto \log \left(x + \sqrt{1^2 + x^2}^*\right)\]
3. Using strategy rm
4. Applied add-sqr-sqrt0.0
  \[\leadsto \log \left(x + \color{blue}{\sqrt{\sqrt{1^2 + x^2}^*} \cdot \sqrt{\sqrt{1^2 + x^2}^*}}\right)\]
Recombined 3 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0712784506357302:\\ \;\;\;\;\log \left((\left((\left(\frac{\frac{1}{8}}{x}\right) \cdot \left(\frac{1}{x}\right) + \left(-\frac{1}{2}\right))_*\right) \cdot \left(\frac{1}{x}\right) + \left(\frac{-\frac{1}{16}}{{x}^{5}}\right))_*\right)\\ \mathbf{elif}\;x \le 0.008931035360035398:\\ \;\;\;\;\left({x}^{5} \cdot \frac{3}{40} + x\right) - {x}^{3} \cdot \frac{1}{6}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\sqrt{1^2 + x^2}^*} \cdot \sqrt{\sqrt{1^2 + x^2}^*} + x\right)\\ \end{array}\]

Error

Target

Derivation

`if x < -1.0712784506357302`

`if -1.0712784506357302 < x < 0.008931035360035398`

`if 0.008931035360035398 < x`

Runtime