Average Error: 31.3 → 0.2
Time: 36.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)
double f(double x) {
        double r2586127 = x;
        double r2586128 = r2586127 * r2586127;
        double r2586129 = 1.0;
        double r2586130 = r2586128 - r2586129;
        double r2586131 = sqrt(r2586130);
        double r2586132 = r2586127 + r2586131;
        double r2586133 = log(r2586132);
        return r2586133;
}
↓
double f(double x) {
        double r2586134 = 2.0;
        double r2586135 = x;
        double r2586136 = -0.125;
        double r2586137 = r2586135 * r2586135;
        double r2586138 = r2586137 * r2586135;
        double r2586139 = r2586136 / r2586138;
        double r2586140 = fma(r2586134, r2586135, r2586139);
        double r2586141 = 0.5;
        double r2586142 = r2586141 / r2586135;
        double r2586143 = r2586140 - r2586142;
        double r2586144 = log(r2586143);
        return r2586144;
}

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

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))