Average Error: 31.9 → 0.3
Time: 2.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \mathsf{fma}\left(0.5, \frac{1}{x}, 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \mathsf{fma}\left(0.5, \frac{1}{x}, 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)
double code(double x) {
	return log((x + sqrt(((x * x) - 1.0))));
}
↓
double code(double x) {
	return log((x + (x - fma(0.5, (1.0 / x), (0.125 * (1.0 / pow(x, 3.0)))))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  2. Taylor expanded around inf 0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\]

  3. Simplified0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - \mathsf{fma}\left(0.5, \frac{1}{x}, 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\]

  4. Final simplification0.3

    \[\leadsto \log \left(x + \left(x - \mathsf{fma}\left(0.5, \frac{1}{x}, 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))