Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.2

Time: 21.1s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r2728808 = 1.0;
        double r2728809 = x;
        double r2728810 = r2728808 / r2728809;
        double r2728811 = r2728809 * r2728809;
        double r2728812 = r2728808 - r2728811;
        double r2728813 = sqrt(r2728812);
        double r2728814 = r2728813 / r2728809;
        double r2728815 = r2728810 + r2728814;
        double r2728816 = log(r2728815);
        return r2728816;
}

↓

double f(double x) {
        double r2728817 = 1.0;
        double r2728818 = x;
        double r2728819 = sqrt(r2728818);
        double r2728820 = r2728817 / r2728819;
        double r2728821 = r2728818 * r2728818;
        double r2728822 = r2728817 - r2728821;
        double r2728823 = sqrt(r2728822);
        double r2728824 = r2728823 / r2728819;
        double r2728825 = r2728820 + r2728824;
        double r2728826 = log(r2728825);
        double r2728827 = log(r2728820);
        double r2728828 = r2728826 + r2728827;
        return r2728828;
}

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. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019146 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))