Average Error: 30.6 → 0.3
Time: 10.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\right)
double f(double x) {
        double r1166661 = x;
        double r1166662 = r1166661 * r1166661;
        double r1166663 = 1.0;
        double r1166664 = r1166662 - r1166663;
        double r1166665 = sqrt(r1166664);
        double r1166666 = r1166661 + r1166665;
        double r1166667 = log(r1166666);
        return r1166667;
}
↓
double f(double x) {
        double r1166668 = x;
        double r1166669 = 0.5;
        double r1166670 = r1166669 / r1166668;
        double r1166671 = 0.125;
        double r1166672 = r1166671 / r1166668;
        double r1166673 = r1166668 * r1166668;
        double r1166674 = r1166672 / r1166673;
        double r1166675 = r1166670 + r1166674;
        double r1166676 = r1166668 - r1166675;
        double r1166677 = r1166668 + r1166676;
        double r1166678 = log(r1166677);
        return r1166678;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.6

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

  2. Taylor expanded around inf 0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

  3. Simplified0.3

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}\right)\right)}\right)\]

  4. Final simplification0.3

    \[\leadsto \log \left(x + \left(x - \left(\frac{\frac{1}{2}}{x} + \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))