Average Error: 30.6 → 0.3

Time: 50.7s

Precision: 64

Internal Precision: 2368

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

↓

\[\log \left((2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_* - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\]

Error

Derivation

Initial program 30.6
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
Initial simplification30.6
\[\leadsto \log \left(x + \sqrt{(x \cdot x + -1)_*}\right)\]
Taylor expanded around inf 0.3
\[\leadsto \log \color{blue}{\left(2 \cdot x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]
Simplified0.3
\[\leadsto \log \color{blue}{\left((2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_* - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\]
Final simplification0.3
\[\leadsto \log \left((2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_* - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\]

	Baseline	Herbie	Oracle	Span	%
Regimes	0.3	0.3	0.0	0.3	0%

herbie shell --seed 2018296 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))