Average Error: 30.7 → 0.3
Time: 17.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\]
\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} - \frac{\frac{1}{2}}{x}\right)\right)
double f(double x) {
        double r2431030 = x;
        double r2431031 = r2431030 * r2431030;
        double r2431032 = 1.0;
        double r2431033 = r2431031 - r2431032;
        double r2431034 = sqrt(r2431033);
        double r2431035 = r2431030 + r2431034;
        double r2431036 = log(r2431035);
        return r2431036;
}
↓
double f(double x) {
        double r2431037 = 2.0;
        double r2431038 = x;
        double r2431039 = -0.125;
        double r2431040 = r2431038 * r2431038;
        double r2431041 = r2431040 * r2431038;
        double r2431042 = r2431039 / r2431041;
        double r2431043 = 0.5;
        double r2431044 = r2431043 / r2431038;
        double r2431045 = r2431042 - r2431044;
        double r2431046 = fma(r2431037, r2431038, r2431045);
        double r2431047 = log(r2431046);
        return r2431047;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.7

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

  2. Simplified30.7

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

  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

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))