Average Error: 31.6 → 0.2
Time: 3.8s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(1 + \frac{0.5}{x} \cdot \frac{-0.5}{x}\right) - \log \left(\frac{0.5}{x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(1 + \frac{0.5}{x} \cdot \frac{-0.5}{x}\right) - \log \left(\frac{0.5}{x}\right)
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (- (log (+ 1.0 (* (/ 0.5 x) (/ -0.5 x)))) (log (/ 0.5 x))))
double code(double x) {
	return log(x + sqrt((x * x) - 1.0));
}
↓
double code(double x) {
	return log(1.0 + ((0.5 / x) * (-0.5 / x))) - log(0.5 / x);
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  2. Taylor expanded around inf 0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - 0.5 \cdot \frac{1}{x}\right)}\right)\]

  3. Simplified0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - \frac{0.5}{x}\right)}\right)\]
  4. Using strategy rm

  5. Applied flip-+_binary64_278063.2

    \[\leadsto \log \color{blue}{\left(\frac{x \cdot x - \left(x - \frac{0.5}{x}\right) \cdot \left(x - \frac{0.5}{x}\right)}{x - \left(x - \frac{0.5}{x}\right)}\right)}\]

  6. Applied log-div_binary64_289363.2

    \[\leadsto \color{blue}{\log \left(x \cdot x - \left(x - \frac{0.5}{x}\right) \cdot \left(x - \frac{0.5}{x}\right)\right) - \log \left(x - \left(x - \frac{0.5}{x}\right)\right)}\]

  7. Simplified63.2

    \[\leadsto \color{blue}{\log \left(1 + \frac{0.5}{x} \cdot \frac{-0.5}{x}\right)} - \log \left(x - \left(x - \frac{0.5}{x}\right)\right)\]

  8. Simplified0.2

    \[\leadsto \log \left(1 + \frac{0.5}{x} \cdot \frac{-0.5}{x}\right) - \color{blue}{\log \left(\frac{0.5}{x}\right)}\]

  9. Final simplification0.2

    \[\leadsto \log \left(1 + \frac{0.5}{x} \cdot \frac{-0.5}{x}\right) - \log \left(\frac{0.5}{x}\right)\]

Reproduce

herbie shell --seed 2021019 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))