Average Error: 32.1 → 0.4
Time: 5.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log 2 + \left(\left(\log x - \frac{\frac{0.25}{x}}{x}\right) - \frac{0.09375}{{x}^{4}}\right)
double f(double x) {
        double r91990 = x;
        double r91991 = r91990 * r91990;
        double r91992 = 1.0;
        double r91993 = r91991 - r91992;
        double r91994 = sqrt(r91993);
        double r91995 = r91990 + r91994;
        double r91996 = log(r91995);
        return r91996;
}
↓
double f(double x) {
        double r91997 = 2.0;
        double r91998 = log(r91997);
        double r91999 = x;
        double r92000 = log(r91999);
        double r92001 = 0.25;
        double r92002 = r92001 / r91999;
        double r92003 = r92002 / r91999;
        double r92004 = r92000 - r92003;
        double r92005 = 0.09375;
        double r92006 = 4.0;
        double r92007 = pow(r91999, r92006);
        double r92008 = r92005 / r92007;
        double r92009 = r92004 - r92008;
        double r92010 = r91998 + r92009;
        return r92010;
}

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. Taylor expanded around inf 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))