Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.0

Time: 10.2s

Precision: 64

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

Math

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

↓

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

C

double f(double x) {
        double r89666 = 1.0;
        double r89667 = x;
        double r89668 = r89666 / r89667;
        double r89669 = r89667 * r89667;
        double r89670 = r89666 - r89669;
        double r89671 = sqrt(r89670);
        double r89672 = r89671 / r89667;
        double r89673 = r89668 + r89672;
        double r89674 = log(r89673);
        return r89674;
}

↓

double f(double x) {
        double r89675 = 1.0;
        double r89676 = x;
        double r89677 = r89675 / r89676;
        double r89678 = r89676 * r89676;
        double r89679 = r89675 - r89678;
        double r89680 = sqrt(r89679);
        double r89681 = r89680 / r89676;
        double r89682 = r89677 + r89681;
        double r89683 = sqrt(r89682);
        double r89684 = 1.0;
        double r89685 = sqrt(r89676);
        double r89686 = r89684 / r89685;
        double r89687 = sqrt(r89686);
        double r89688 = r89683 * r89687;
        double r89689 = r89675 / r89685;
        double r89690 = r89680 / r89685;
        double r89691 = r89689 + r89690;
        double r89692 = sqrt(r89691);
        double r89693 = r89688 * r89692;
        double r89694 = log(r89693);
        return r89694;
}

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

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

6. Applied *-un-lft-identity 0.0

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

7. Applied times-frac 0.0

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

8. Applied add-sqr-sqrt 0.0

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

9. Applied *-un-lft-identity 0.0

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

10. Applied times-frac 0.0

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

11. Applied distribute-lft-out 0.0

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

12. Applied sqrt-prod 0.0

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

13. Applied associate-*r* 0.0

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

14. Final simplification 0.0

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

Reproduce

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