Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.1

Time: 10.4s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2309337 = 1.0;
        double r2309338 = x;
        double r2309339 = r2309337 / r2309338;
        double r2309340 = r2309338 * r2309338;
        double r2309341 = r2309337 - r2309340;
        double r2309342 = sqrt(r2309341);
        double r2309343 = r2309342 / r2309338;
        double r2309344 = r2309339 + r2309343;
        double r2309345 = log(r2309344);
        return r2309345;
}

↓

double f(double x) {
        double r2309346 = 1.0;
        double r2309347 = x;
        double r2309348 = r2309346 / r2309347;
        double r2309349 = sqrt(r2309348);
        double r2309350 = r2309347 * r2309347;
        double r2309351 = r2309346 - r2309350;
        double r2309352 = sqrt(r2309351);
        double r2309353 = r2309352 / r2309347;
        double r2309354 = fma(r2309349, r2309349, r2309353);
        double r2309355 = log(r2309354);
        return r2309355;
}

Error

Bits error versus x

Derivation

1. Initial program 0.1

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

3. Applied add-sqr-sqrt 0.1

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

4. Applied fma-def 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))