Hyperbolic arc-cosine

Average Error: 32.5 → 0.1

Time: 18.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r91483 = x;
        double r91484 = r91483 * r91483;
        double r91485 = 1.0;
        double r91486 = r91484 - r91485;
        double r91487 = sqrt(r91486);
        double r91488 = r91483 + r91487;
        double r91489 = log(r91488);
        return r91489;
}

↓

double f(double x) {
        double r91490 = x;
        double r91491 = 1.0;
        double r91492 = sqrt(r91491);
        double r91493 = r91490 - r91492;
        double r91494 = sqrt(r91493);
        double r91495 = r91490 + r91492;
        double r91496 = sqrt(r91495);
        double r91497 = fma(r91494, r91496, r91490);
        double r91498 = log(r91497);
        return r91498;
}

Error

Bits error versus x

Derivation

1. Initial program 32.5

   \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
2. Using strategy rm

3. Applied add-sqr-sqrt 32.5

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

4. Applied difference-of-squares 32.5

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

5. Applied sqrt-prod 0.1

   \[\leadsto \log \left(x + \color{blue}{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)\]
6. Using strategy rm

7. Applied pow1 0.1

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

8. Applied log-pow 0.1

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

9. Simplified 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))