Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.2

Time: 13.7s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r69442 = 1.0;
        double r69443 = x;
        double r69444 = r69442 / r69443;
        double r69445 = r69443 * r69443;
        double r69446 = r69442 - r69445;
        double r69447 = sqrt(r69446);
        double r69448 = r69447 / r69443;
        double r69449 = r69444 + r69448;
        double r69450 = log(r69449);
        return r69450;
}

↓

double f(double x) {
        double r69451 = x;
        double r69452 = sqrt(r69451);
        double r69453 = log(r69452);
        double r69454 = -r69453;
        double r69455 = 1.0;
        double r69456 = r69455 / r69452;
        double r69457 = r69451 * r69451;
        double r69458 = r69455 - r69457;
        double r69459 = sqrt(r69458);
        double r69460 = r69459 / r69452;
        double r69461 = r69456 + r69460;
        double r69462 = log(r69461);
        double r69463 = r69454 + r69462;
        return r69463;
}

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. Final simplification 0.2

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

Reproduce

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