Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.2

Time: 14.3s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r1400184 = 1.0;
        double r1400185 = x;
        double r1400186 = r1400184 / r1400185;
        double r1400187 = r1400185 * r1400185;
        double r1400188 = r1400184 - r1400187;
        double r1400189 = sqrt(r1400188);
        double r1400190 = r1400189 / r1400185;
        double r1400191 = r1400186 + r1400190;
        double r1400192 = log(r1400191);
        return r1400192;
}

↓

double f(double x) {
        double r1400193 = 1.0;
        double r1400194 = x;
        double r1400195 = r1400194 * r1400194;
        double r1400196 = r1400193 - r1400195;
        double r1400197 = sqrt(r1400196);
        double r1400198 = r1400193 + r1400197;
        double r1400199 = log(r1400198);
        double r1400200 = log(r1400194);
        double r1400201 = r1400199 - r1400200;
        return r1400201;
}

Error

Bits error versus x

Derivation

1. Initial program 0.0

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

3. Applied div-inv 0.0

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

4. Applied distribute-rgt1-in 0.0

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

6. Applied un-div-inv 0.0

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

7. Applied log-div 0.2

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

8. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))