Average Error: 31.4 → 0.2
Time: 16.5s
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 r2865223 = x;
        double r2865224 = r2865223 * r2865223;
        double r2865225 = 1.0;
        double r2865226 = r2865224 - r2865225;
        double r2865227 = sqrt(r2865226);
        double r2865228 = r2865223 + r2865227;
        double r2865229 = log(r2865228);
        return r2865229;
}
↓
double f(double x) {
        double r2865230 = 2.0;
        double r2865231 = x;
        double r2865232 = -0.125;
        double r2865233 = r2865231 * r2865231;
        double r2865234 = r2865233 * r2865231;
        double r2865235 = r2865232 / r2865234;
        double r2865236 = 0.5;
        double r2865237 = r2865236 / r2865231;
        double r2865238 = r2865235 - r2865237;
        double r2865239 = fma(r2865230, r2865231, r2865238);
        double r2865240 = log(r2865239);
        return r2865240;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.4

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

  2. Simplified31.4

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

  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

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))