Average Error: 32.0 → 0.2
Time: 4.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(x - \left(\frac{0.125}{{x}^{3}} + \frac{0.5}{x}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(x - \left(\frac{0.125}{{x}^{3}} + \frac{0.5}{x}\right)\right)\right)
double f(double x) {
        double r46054 = x;
        double r46055 = r46054 * r46054;
        double r46056 = 1.0;
        double r46057 = r46055 - r46056;
        double r46058 = sqrt(r46057);
        double r46059 = r46054 + r46058;
        double r46060 = log(r46059);
        return r46060;
}
↓
double f(double x) {
        double r46061 = x;
        double r46062 = 0.125;
        double r46063 = 3.0;
        double r46064 = pow(r46061, r46063);
        double r46065 = r46062 / r46064;
        double r46066 = 0.5;
        double r46067 = r46066 / r46061;
        double r46068 = r46065 + r46067;
        double r46069 = r46061 - r46068;
        double r46070 = r46061 + r46069;
        double r46071 = log(r46070);
        return r46071;
}

Error

Bits error versus x

Try it out

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 0.2

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)\right)}\right)\]

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))