Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.2

Time: 13.8s

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 r1227400 = 1.0;
        double r1227401 = x;
        double r1227402 = r1227400 / r1227401;
        double r1227403 = r1227401 * r1227401;
        double r1227404 = r1227400 - r1227403;
        double r1227405 = sqrt(r1227404);
        double r1227406 = r1227405 / r1227401;
        double r1227407 = r1227402 + r1227406;
        double r1227408 = log(r1227407);
        return r1227408;
}

↓

double f(double x) {
        double r1227409 = 1.0;
        double r1227410 = x;
        double r1227411 = sqrt(r1227410);
        double r1227412 = r1227409 / r1227411;
        double r1227413 = r1227410 * r1227410;
        double r1227414 = r1227409 - r1227413;
        double r1227415 = sqrt(r1227414);
        double r1227416 = r1227415 / r1227411;
        double r1227417 = r1227412 + r1227416;
        double r1227418 = log(r1227417);
        double r1227419 = log(r1227412);
        double r1227420 = r1227418 + r1227419;
        return r1227420;
}

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

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

4. Applied *-un-lft-identity 0.0

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

5. Applied times-frac 0.0

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

6. Applied add-sqr-sqrt 0.0

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

7. Applied *-un-lft-identity 0.0

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

8. Applied times-frac 0.0

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

9. Applied distribute-lft-out 0.0

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

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 2019151 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))