Average Error: 30.8 → 0.2
Time: 24.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)
double f(double x) {
        double r1937701 = x;
        double r1937702 = r1937701 * r1937701;
        double r1937703 = 1.0;
        double r1937704 = r1937702 - r1937703;
        double r1937705 = sqrt(r1937704);
        double r1937706 = r1937701 + r1937705;
        double r1937707 = log(r1937706);
        return r1937707;
}
↓
double f(double x) {
        double r1937708 = 2.0;
        double r1937709 = x;
        double r1937710 = -0.5;
        double r1937711 = r1937710 / r1937709;
        double r1937712 = fma(r1937708, r1937709, r1937711);
        double r1937713 = 0.125;
        double r1937714 = r1937713 / r1937709;
        double r1937715 = r1937709 * r1937709;
        double r1937716 = r1937714 / r1937715;
        double r1937717 = r1937712 - r1937716;
        double r1937718 = log(r1937717);
        return r1937718;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.8

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

  2. Simplified30.8

    \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)}\]

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))