Average Error: 31.8 → 0.3
Time: 10.5s
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 r1211303 = x;
        double r1211304 = r1211303 * r1211303;
        double r1211305 = 1.0;
        double r1211306 = r1211304 - r1211305;
        double r1211307 = sqrt(r1211306);
        double r1211308 = r1211303 + r1211307;
        double r1211309 = log(r1211308);
        return r1211309;
}
↓
double f(double x) {
        double r1211310 = x;
        double r1211311 = 0.5;
        double r1211312 = r1211311 / r1211310;
        double r1211313 = 0.125;
        double r1211314 = r1211313 / r1211310;
        double r1211315 = r1211310 * r1211310;
        double r1211316 = r1211314 / r1211315;
        double r1211317 = r1211312 + r1211316;
        double r1211318 = r1211310 - r1211317;
        double r1211319 = r1211310 + r1211318;
        double r1211320 = log(r1211319);
        return r1211320;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.8

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

  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 2019155 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))