Average Error: 47.0 → 21.0
Time: 8.9s
Precision: 64
Internal precision: 2432
\[\log \left(x + \sqrt{{x}^2 - 1}\right)\]
⬇
\[\log \left(\left(\left(x + x\right) - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{{x}^3}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 47.0

    \[\log \left(x + \sqrt{{x}^2 - 1}\right)\]

  2. Applied taylor 21.0

    \[\leadsto \log \left(x + \left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)\right)\]

  3. Taylor expanded around inf 21.0

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

  4. Applied simplify 21.0

    \[\leadsto \color{blue}{\log \left(\left(\left(x + x\right) - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\]

  5. Applied simplify 21.0

    \[\leadsto \log \color{blue}{\left(\left(\left(x + x\right) - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{{x}^3}\right)}\]
  6. Removed slow pow expressions

Runtime

Time bar (total: 8.9s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064524629 4159152179 2999149171 575749698 4006532819 692958815)'
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (sqr x) 1)))))