Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.1

Time: 13.6s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2791731 = 1.0;
        double r2791732 = x;
        double r2791733 = r2791731 / r2791732;
        double r2791734 = r2791732 * r2791732;
        double r2791735 = r2791731 - r2791734;
        double r2791736 = sqrt(r2791735);
        double r2791737 = r2791736 / r2791732;
        double r2791738 = r2791733 + r2791737;
        double r2791739 = log(r2791738);
        return r2791739;
}

↓

double f(double x) {
        double r2791740 = 1.0;
        double r2791741 = x;
        double r2791742 = sqrt(r2791741);
        double r2791743 = r2791740 / r2791742;
        double r2791744 = r2791741 * r2791741;
        double r2791745 = r2791740 - r2791744;
        double r2791746 = sqrt(r2791745);
        double r2791747 = r2791746 / r2791742;
        double r2791748 = r2791743 + r2791747;
        double r2791749 = r2791748 * r2791743;
        double r2791750 = log(r2791749);
        return r2791750;
}

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. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))