Hyperbolic arc-(co)secant

Average Error: 0.1 → 0.2

Time: 4.7s

Precision: 64

Math

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

↓

\[\log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]

C

double f(double x) {
        double r48010 = 1.0;
        double r48011 = x;
        double r48012 = r48010 / r48011;
        double r48013 = r48011 * r48011;
        double r48014 = r48010 - r48013;
        double r48015 = sqrt(r48014);
        double r48016 = r48015 / r48011;
        double r48017 = r48012 + r48016;
        double r48018 = log(r48017);
        return r48018;
}

↓

double f(double x) {
        double r48019 = 1.0;
        double r48020 = x;
        double r48021 = cbrt(r48020);
        double r48022 = r48021 * r48021;
        double r48023 = r48019 / r48022;
        double r48024 = log(r48023);
        double r48025 = 1.0;
        double r48026 = r48025 / r48021;
        double r48027 = r48020 * r48020;
        double r48028 = r48025 - r48027;
        double r48029 = sqrt(r48028);
        double r48030 = r48029 / r48021;
        double r48031 = r48026 + r48030;
        double r48032 = log(r48031);
        double r48033 = r48024 + r48032;
        return r48033;
}

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-cube-cbrt 0.1

   \[\leadsto \log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\right)\]

4. Applied *-un-lft-identity 0.1

   \[\leadsto \log \left(\frac{1}{x} + \frac{\color{blue}{1 \cdot \sqrt{1 - x \cdot x}}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}\right)\]

5. Applied times-frac 0.1

   \[\leadsto \log \left(\frac{1}{x} + \color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}}\right)\]

6. Applied add-cube-cbrt 0.1

   \[\leadsto \log \left(\frac{1}{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}} + \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]

7. Applied *-un-lft-identity 0.1

   \[\leadsto \log \left(\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}} + \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]

8. Applied times-frac 0.1

   \[\leadsto \log \left(\color{blue}{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{1}{\sqrt[3]{x}}} + \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]

9. Applied distribute-lft-out 0.1

   \[\leadsto \log \color{blue}{\left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\right)}\]

10. Applied log-prod 0.2

    \[\leadsto \color{blue}{\log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)}\]

11. Final simplification 0.2

    \[\leadsto \log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))