Hyperbolic arc-cosine

Average Error: 31.7 → 0.1

Time: 3.9s

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 r88059 = x;
        double r88060 = r88059 * r88059;
        double r88061 = 1.0;
        double r88062 = r88060 - r88061;
        double r88063 = sqrt(r88062);
        double r88064 = r88059 + r88063;
        double r88065 = log(r88064);
        return r88065;
}

↓

double f(double x) {
        double r88066 = x;
        double r88067 = 1.0;
        double r88068 = sqrt(r88067);
        double r88069 = r88066 - r88068;
        double r88070 = sqrt(r88069);
        double r88071 = r88066 + r88068;
        double r88072 = sqrt(r88071);
        double r88073 = fma(r88070, r88072, r88066);
        double r88074 = log(r88073);
        return r88074;
}

Error

Bits error versus x

Derivation

1. Initial program 31.7

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

3. Applied add-sqr-sqrt 31.7

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

4. Applied difference-of-squares 31.7

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

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 add-log-exp 0.1

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

8. Simplified 0.1

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

9. 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 2020049 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))