Average Error: 32.0 → 0.3
Time: 2.7s
Precision: binary64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \left(\frac{0.5}{x} + \frac{0.125}{{x}^{3}}\right)\right)\right)
(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))
↓
(FPCore (x)
 :precision binary64
 (log (+ x (- x (+ (/ 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(x + (x - ((0.5 / x) + (0.125 / pow(x, 3.0)))));
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2020253 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))