Average Error: 30.9 → 0.2
Time: 7.8s
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 r1200248 = x;
        double r1200249 = r1200248 * r1200248;
        double r1200250 = 1.0;
        double r1200251 = r1200249 - r1200250;
        double r1200252 = sqrt(r1200251);
        double r1200253 = r1200248 + r1200252;
        double r1200254 = log(r1200253);
        return r1200254;
}
↓
double f(double x) {
        double r1200255 = -0.125;
        double r1200256 = x;
        double r1200257 = r1200256 * r1200256;
        double r1200258 = r1200257 * r1200256;
        double r1200259 = r1200255 / r1200258;
        double r1200260 = 2.0;
        double r1200261 = -0.5;
        double r1200262 = r1200261 / r1200256;
        double r1200263 = fma(r1200260, r1200256, r1200262);
        double r1200264 = r1200259 + r1200263;
        double r1200265 = log(r1200264);
        return r1200265;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.9

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

  2. Simplified30.9

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))