Average Error: 30.9 → 0.2
Time: 32.9s
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 r1763415 = x;
        double r1763416 = r1763415 * r1763415;
        double r1763417 = 1.0;
        double r1763418 = r1763416 - r1763417;
        double r1763419 = sqrt(r1763418);
        double r1763420 = r1763415 + r1763419;
        double r1763421 = log(r1763420);
        return r1763421;
}
↓
double f(double x) {
        double r1763422 = 2.0;
        double r1763423 = x;
        double r1763424 = -0.125;
        double r1763425 = r1763423 * r1763423;
        double r1763426 = r1763425 * r1763423;
        double r1763427 = r1763424 / r1763426;
        double r1763428 = 0.5;
        double r1763429 = r1763428 / r1763423;
        double r1763430 = r1763427 - r1763429;
        double r1763431 = fma(r1763422, r1763423, r1763430);
        double r1763432 = log(r1763431);
        return r1763432;
}

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

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

  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 2019143 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))