Average Error: 31.3 → 0.1
Time: 4.0s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)
double code(double x) {
	return ((double) log(((double) (x + ((double) sqrt(((double) (((double) (x * x)) - 1.0))))))));
}
↓
double code(double x) {
	return ((double) log(((double) (x + ((double) (((double) (((double) cbrt(((double) (((double) sqrt(((double) (x + ((double) sqrt(1.0)))))) * ((double) sqrt(((double) (x - ((double) sqrt(1.0)))))))))) * ((double) cbrt(((double) (((double) sqrt(((double) (x + ((double) sqrt(1.0)))))) * ((double) sqrt(((double) (x - ((double) sqrt(1.0)))))))))))) * ((double) cbrt(((double) (((double) sqrt(((double) (x + ((double) sqrt(1.0)))))) * ((double) sqrt(((double) (x - ((double) sqrt(1.0))))))))))))))));
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  3. Applied add-sqr-sqrt31.3

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

  4. Applied difference-of-squares31.3

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

  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-cube-cbrt0.1

    \[\leadsto \log \left(x + \color{blue}{\left(\sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}}\right)\]

  8. Final simplification0.1

    \[\leadsto \log \left(x + \left(\sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right) \cdot \sqrt[3]{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)\]

Reproduce

herbie shell --seed 2020161 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))