Average Error: 31.1 → 0.4
Time: 1.3m
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\left(\log 2 - \frac{\frac{\frac{1}{4}}{x}}{x}\right) + \left(\log x - \frac{\frac{3}{32}}{{x}^{4}}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 31.1

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

  2. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\frac{1}{4} \cdot \frac{1}{{x}^{2}} + \left(\frac{3}{32} \cdot \frac{1}{{x}^{4}} + \log \left(\frac{1}{x}\right)\right)\right)}\]

  3. Applied simplify0.4

    \[\leadsto \color{blue}{\left(\log 2 - \frac{\frac{\frac{1}{4}}{x}}{x}\right) + \left(\log x - \frac{\frac{3}{32}}{{x}^{4}}\right)}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))