Hyperbolic arc-cosine

Average Error: 32.1 → 0.2

Time: 5.9s

Precision: binary64

Math

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

↓

\[\log \left(2 \cdot x + \left(\frac{-0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\]

Error

Bits error versus x

Derivation

1. Initial program 32.1

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

2. Taylor expanded around inf 0.2

   \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\]

3. Simplified 0.2

   \[\leadsto \log \color{blue}{\left(2 \cdot x + \left(\frac{-0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)}\]

4. Final simplification 0.2

   \[\leadsto \log \left(2 \cdot x + \left(\frac{-0.5}{x} - \frac{0.125}{{x}^{3}}\right)\right)\]

Reproduce

herbie shell --seed 2020157 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))