Average Error: 31.8 → 0.1
Time: 2.9s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + 1} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x + 1} \cdot \sqrt{x - 1}\right)
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log (+ x (* (sqrt (+ x 1.0)) (sqrt (- x 1.0))))))
double code(double x) {
	return log(x + sqrt((x * x) - 1.0));
}
↓
double code(double x) {
	return log(x + (sqrt(x + 1.0) * sqrt(x - 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 31.8

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

  3. Applied add-sqr-sqrt_binary6431.8

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

  4. Applied difference-of-squares_binary6431.8

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

  5. Applied sqrt-prod_binary640.1

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

  6. Simplified0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020224 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))