Average Error: 32.6 → 0.5
Time: 5.4s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log 2 - \left(\log \left(\frac{1}{x}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right) \]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 - \left(\log \left(\frac{1}{x}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (- (log 2.0) (+ (log (/ 1.0 x)) (* 0.25 (/ 1.0 (pow x 2.0))))))
double code(double x) {
	return log(x + sqrt((x * x) - 1.0));
}
↓
double code(double x) {
	return log(2.0) - (log(1.0 / x) + (0.25 * (1.0 / pow(x, 2.0))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  2. Taylor expanded around inf 0.5

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)} \]

  3. Final simplification0.5

    \[\leadsto \log 2 - \left(\log \left(\frac{1}{x}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right) \]

Reproduce

herbie shell --seed 2021197 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))