Hyperbolic arc-(co)secant

Average Error: 0.0 → 0.0

Time: 12.1s

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 r3273785 = 1.0;
        double r3273786 = x;
        double r3273787 = r3273785 / r3273786;
        double r3273788 = r3273786 * r3273786;
        double r3273789 = r3273785 - r3273788;
        double r3273790 = sqrt(r3273789);
        double r3273791 = r3273790 / r3273786;
        double r3273792 = r3273787 + r3273791;
        double r3273793 = log(r3273792);
        return r3273793;
}

↓

double f(double x) {
        double r3273794 = 1.0;
        double r3273795 = x;
        double r3273796 = sqrt(r3273795);
        double r3273797 = r3273794 / r3273796;
        double r3273798 = r3273795 * r3273795;
        double r3273799 = r3273794 - r3273798;
        double r3273800 = sqrt(r3273799);
        double r3273801 = r3273800 / r3273796;
        double r3273802 = r3273797 + r3273801;
        double r3273803 = 1.0;
        double r3273804 = r3273803 / r3273796;
        double r3273805 = r3273802 * r3273804;
        double r3273806 = log(r3273805);
        return r3273806;
}

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. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019169 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))