Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.2

Time: 14.2s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r97397 = 1.0;
        double r97398 = x;
        double r97399 = r97397 / r97398;
        double r97400 = r97398 * r97398;
        double r97401 = r97397 - r97400;
        double r97402 = sqrt(r97401);
        double r97403 = r97402 / r97398;
        double r97404 = r97399 + r97403;
        double r97405 = log(r97404);
        return r97405;
}

↓

double f(double x) {
        double r97406 = x;
        double r97407 = sqrt(r97406);
        double r97408 = log(r97407);
        double r97409 = -r97408;
        double r97410 = 1.0;
        double r97411 = r97406 * r97406;
        double r97412 = r97410 - r97411;
        double r97413 = sqrt(r97412);
        double r97414 = r97413 / r97407;
        double r97415 = r97410 / r97407;
        double r97416 = r97414 + r97415;
        double r97417 = log(r97416);
        double r97418 = r97409 + r97417;
        return r97418;
}

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

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

4. Applied *-un-lft-identity 0.1

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

5. Applied times-frac 0.1

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

6. Applied add-sqr-sqrt 0.1

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

7. Applied *-un-lft-identity 0.1

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

8. Applied times-frac 0.1

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

9. Applied distribute-lft-out 0.1

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

10. Applied log-prod 0.2

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

11. Simplified 0.2

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

12. Simplified 0.2

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

13. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))