Average Error: 31.8 → 0.3
Time: 9.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right)\right)
double f(double x) {
        double r972148 = x;
        double r972149 = r972148 * r972148;
        double r972150 = 1.0;
        double r972151 = r972149 - r972150;
        double r972152 = sqrt(r972151);
        double r972153 = r972148 + r972152;
        double r972154 = log(r972153);
        return r972154;
}
↓
double f(double x) {
        double r972155 = -0.125;
        double r972156 = x;
        double r972157 = r972156 * r972156;
        double r972158 = r972157 * r972156;
        double r972159 = r972155 / r972158;
        double r972160 = 2.0;
        double r972161 = -0.5;
        double r972162 = r972161 / r972156;
        double r972163 = fma(r972160, r972156, r972162);
        double r972164 = r972159 + r972163;
        double r972165 = log(r972164);
        return r972165;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.8

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

  2. Simplified31.8

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

  3. Taylor expanded around inf 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))