Average Error: 31.3 → 0.2
Time: 44.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \mathsf{fma}\left(\frac{1}{x}, \frac{-1}{2}, \mathsf{fma}\left(\frac{\frac{1}{x}}{x \cdot x}, \frac{-1}{8}, x\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \mathsf{fma}\left(\frac{1}{x}, \frac{-1}{2}, \mathsf{fma}\left(\frac{\frac{1}{x}}{x \cdot x}, \frac{-1}{8}, x\right)\right)\right)
double f(double x) {
        double r1269027 = x;
        double r1269028 = r1269027 * r1269027;
        double r1269029 = 1.0;
        double r1269030 = r1269028 - r1269029;
        double r1269031 = sqrt(r1269030);
        double r1269032 = r1269027 + r1269031;
        double r1269033 = log(r1269032);
        return r1269033;
}
↓
double f(double x) {
        double r1269034 = x;
        double r1269035 = 1.0;
        double r1269036 = r1269035 / r1269034;
        double r1269037 = -0.5;
        double r1269038 = r1269034 * r1269034;
        double r1269039 = r1269036 / r1269038;
        double r1269040 = -0.125;
        double r1269041 = fma(r1269039, r1269040, r1269034);
        double r1269042 = fma(r1269036, r1269037, r1269041);
        double r1269043 = r1269034 + r1269042;
        double r1269044 = log(r1269043);
        return r1269044;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.3

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

  2. Simplified31.3

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

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))