Average Error: 31.7 → 0.1
Time: 5.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x} + \sqrt{\sqrt{1}}}\right) \cdot \sqrt{\sqrt{x} - \sqrt{\sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x} + \sqrt{\sqrt{1}}}\right) \cdot \sqrt{\sqrt{x} - \sqrt{\sqrt{1}}}\right)
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(sqrt(1.0))))) * sqrt((sqrt(x) - sqrt(sqrt(1.0)))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied add-sqr-sqrt31.7

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

  4. Applied difference-of-squares31.7

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied add-sqr-sqrt0.1

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

  10. Applied difference-of-squares0.1

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

  11. Applied sqrt-prod0.1

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

  12. Applied associate-*r*0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))