Average Error: 46.3 → 21.0
Time: 24.0s
Precision: 64
Internal precision: 128
\[\log \left(x + \sqrt{{x}^2 - 1}\right)\]
⬇
\[\log \left(\left(\left(x + x\right) - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{{x}^3}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 46.3

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

  2. Applied taylor 21.0

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

  3. Taylor expanded around inf 21.0

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

  4. Applied simplify 21.0

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

Runtime

Time bar (total: 24.0s) Debug log

Please include this information when filing a bug report:

herbie --seed '#(2488491742 537576963 2292106893 1780561712 129042182 1227067085)'
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (sqr x) 1)))))