Hyperbolic arc-cosine

Average Error: 32.4 → 0.1

Time: 5.0s

Precision: 64

Math

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

↓

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

C

double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}

↓

double code(double x) {
	return log((x + ((sqrt((x + sqrt(1.0))) * sqrt(sqrt((x - sqrt(1.0))))) * sqrt(sqrt((x - sqrt(1.0)))))));
}

Error

Bits error versus x

Derivation

1. Initial program 32.4

   \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
2. Using strategy rm

3. Applied add-sqr-sqrt 32.4

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

4. Applied difference-of-squares 32.4

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

5. Applied sqrt-prod 0.1

   \[\leadsto \log \left(x + \color{blue}{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)\]
6. Using strategy rm

7. Applied add-sqr-sqrt 0.1

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

8. Applied sqrt-prod 0.1

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

9. Applied associate-*r* 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))