Hyperbolic arc-cosine

Average Error: 31.3 → 0.0

Time: 33.3s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2302016 = x;
        double r2302017 = r2302016 * r2302016;
        double r2302018 = 1.0;
        double r2302019 = r2302017 - r2302018;
        double r2302020 = sqrt(r2302019);
        double r2302021 = r2302016 + r2302020;
        double r2302022 = log(r2302021);
        return r2302022;
}

↓

double f(double x) {
        double r2302023 = x;
        double r2302024 = sqrt(r2302023);
        double r2302025 = 1.0;
        double r2302026 = r2302023 - r2302025;
        double r2302027 = sqrt(r2302026);
        double r2302028 = r2302023 + r2302025;
        double r2302029 = sqrt(r2302028);
        double r2302030 = r2302027 * r2302029;
        double r2302031 = fma(r2302024, r2302024, r2302030);
        double r2302032 = log(r2302031);
        return r2302032;
}

Error

Bits error versus x

Derivation

1. Initial program 31.3

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

3. Applied difference-of-sqr-1 31.3

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

4. Applied sqrt-prod 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied fma-def 0.0

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

8. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))