Average Error: 31.2 → 0.2
Time: 46.2s
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 r1213203 = x;
        double r1213204 = r1213203 * r1213203;
        double r1213205 = 1.0;
        double r1213206 = r1213204 - r1213205;
        double r1213207 = sqrt(r1213206);
        double r1213208 = r1213203 + r1213207;
        double r1213209 = log(r1213208);
        return r1213209;
}
↓
double f(double x) {
        double r1213210 = x;
        double r1213211 = 1.0;
        double r1213212 = r1213211 / r1213210;
        double r1213213 = -0.5;
        double r1213214 = r1213210 * r1213210;
        double r1213215 = r1213212 / r1213214;
        double r1213216 = -0.125;
        double r1213217 = fma(r1213215, r1213216, r1213210);
        double r1213218 = fma(r1213212, r1213213, r1213217);
        double r1213219 = r1213210 + r1213218;
        double r1213220 = log(r1213219);
        return r1213220;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.2

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

  2. Simplified31.2

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

  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 2019149 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))