Average Error: 32.1 → 0.4
Time: 5.1s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log 2 + \left(\left(\log x - \frac{0.25}{x \cdot x}\right) - \frac{0.09375}{{x}^{4}}\right) \]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 + \left(\left(\log x - \frac{0.25}{x \cdot x}\right) - \frac{0.09375}{{x}^{4}}\right)

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (+ (log 2.0) (- (- (log x) (/ 0.25 (* x x))) (/ 0.09375 (pow x 4.0)))))
double code(double x) {
	return log(x + sqrt((x * x) - 1.0));
}
↓
double code(double x) {
	return log(2.0) + ((log(x) - (0.25 / (x * x))) - (0.09375 / pow(x, 4.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  2. Simplified32.1

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

  3. Taylor expanded in x around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(\log \left(\frac{1}{x}\right) + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

  4. Simplified0.4

    \[\leadsto \color{blue}{\log 2 + \left(\left(\log x - \frac{0.25}{x \cdot x}\right) - \frac{0.09375}{{x}^{4}}\right)} \]

  5. Final simplification0.4

    \[\leadsto \log 2 + \left(\left(\log x - \frac{0.25}{x \cdot x}\right) - \frac{0.09375}{{x}^{4}}\right) \]

Reproduce

herbie shell --seed 2021307 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))