Average Error: 31.1 → 0.2
Time: 19.1s
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 r2533076 = x;
        double r2533077 = r2533076 * r2533076;
        double r2533078 = 1.0;
        double r2533079 = r2533077 - r2533078;
        double r2533080 = sqrt(r2533079);
        double r2533081 = r2533076 + r2533080;
        double r2533082 = log(r2533081);
        return r2533082;
}
↓
double f(double x) {
        double r2533083 = -0.125;
        double r2533084 = x;
        double r2533085 = r2533084 * r2533084;
        double r2533086 = r2533085 * r2533084;
        double r2533087 = r2533083 / r2533086;
        double r2533088 = 2.0;
        double r2533089 = -0.5;
        double r2533090 = r2533089 / r2533084;
        double r2533091 = fma(r2533088, r2533084, r2533090);
        double r2533092 = r2533087 + r2533091;
        double r2533093 = log(r2533092);
        return r2533093;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.1

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

  2. Simplified31.1

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

  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

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

  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 2019158 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))