Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.0

Time: 10.5s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x) {
        double r114396 = 1.0;
        double r114397 = x;
        double r114398 = r114396 / r114397;
        double r114399 = r114397 * r114397;
        double r114400 = r114396 - r114399;
        double r114401 = sqrt(r114400);
        double r114402 = r114401 / r114397;
        double r114403 = r114398 + r114402;
        double r114404 = log(r114403);
        return r114404;
}

↓

double f(double x) {
        double r114405 = 1.0;
        double r114406 = x;
        double r114407 = sqrt(r114406);
        double r114408 = r114405 / r114407;
        double r114409 = sqrt(r114408);
        double r114410 = 1.0;
        double r114411 = r114410 / r114407;
        double r114412 = r114406 * r114406;
        double r114413 = r114410 - r114412;
        double r114414 = sqrt(r114413);
        double r114415 = r114414 / r114407;
        double r114416 = r114411 + r114415;
        double r114417 = sqrt(r114416);
        double r114418 = r114410 / r114406;
        double r114419 = r114414 / r114406;
        double r114420 = r114418 + r114419;
        double r114421 = sqrt(r114420);
        double r114422 = r114417 * r114421;
        double r114423 = r114409 * r114422;
        double r114424 = log(r114423);
        return r114424;
}

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 add-sqr-sqrt 0.0

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

5. Applied add-sqr-sqrt 0.0

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

6. Applied *-un-lft-identity 0.0

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

7. Applied times-frac 0.0

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

8. Applied add-sqr-sqrt 0.0

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

9. Applied *-un-lft-identity 0.0

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

10. Applied times-frac 0.0

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

11. Applied distribute-lft-out 0.0

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

12. Applied sqrt-prod 0.0

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

13. Applied associate-*l* 0.0

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

14. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))