Average Error: 31.6 → 0.3
Time: 3.1s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right) + \frac{-0.125}{{x}^{3}}\right) \]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right) + \frac{-0.125}{{x}^{3}}\right)

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log (+ (fma x 2.0 (/ -0.5 x)) (/ -0.125 (pow x 3.0)))))
double code(double x) {
	return log(x + sqrt((x * x) - 1.0));
}
↓
double code(double x) {
	return log(fma(x, 2.0, (-0.5 / x)) + (-0.125 / pow(x, 3.0)));
}

Error

Bits error versus x

Derivation

  1. Initial program 31.6

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

  2. Simplified31.6

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

  3. Taylor expanded in x around inf 0.3

    \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)} \]

  4. Simplified0.3

    \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right) + \frac{-0.125}{{x}^{3}}\right)} \]

  5. Final simplification0.3

    \[\leadsto \log \left(\mathsf{fma}\left(x, 2, \frac{-0.5}{x}\right) + \frac{-0.125}{{x}^{3}}\right) \]

Reproduce

herbie shell --seed 2022097 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))