Average Error: 32.1 → 0.3
Time: 6.0s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
↓
\[\log 2 - \left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(\log \left(\frac{1}{x}\right) + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 - \left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(\log \left(\frac{1}{x}\right) + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right)

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (-
  (log 2.0)
  (+
   (* 0.09375 (/ 1.0 (pow x 4.0)))
   (+
    (log (/ 1.0 x))
    (+
     (* 0.052083333333333336 (/ 1.0 (pow x 6.0)))
     (* 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) - ((0.09375 * (1.0 / pow(x, 4.0))) + (log(1.0 / x) + ((0.052083333333333336 * (1.0 / pow(x, 6.0))) + (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 32.1

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

  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 0.3

    \[\leadsto \color{blue}{\log 2 - \left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(\log \left(\frac{1}{x}\right) + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

  4. Final simplification0.3

    \[\leadsto \log 2 - \left(0.09375 \cdot \frac{1}{{x}^{4}} + \left(\log \left(\frac{1}{x}\right) + \left(0.052083333333333336 \cdot \frac{1}{{x}^{6}} + 0.25 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

Reproduce

herbie shell --seed 2021344 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))