Average Error: 0.0 → 0.3
Time: 3.5s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) + \log \left(\frac{1}{\sqrt[3]{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]
\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)
double f(double x) {
        double r70240 = 1.0;
        double r70241 = x;
        double r70242 = r70240 / r70241;
        double r70243 = r70241 * r70241;
        double r70244 = r70240 - r70243;
        double r70245 = sqrt(r70244);
        double r70246 = r70245 / r70241;
        double r70247 = r70242 + r70246;
        double r70248 = log(r70247);
        return r70248;
}
↓
double f(double x) {
        double r70249 = 1.0;
        double r70250 = x;
        double r70251 = cbrt(r70250);
        double r70252 = r70251 * r70251;
        double r70253 = r70249 / r70252;
        double r70254 = log(r70253);
        double r70255 = 1.0;
        double r70256 = r70255 / r70251;
        double r70257 = r70250 * r70250;
        double r70258 = r70255 - r70257;
        double r70259 = sqrt(r70258);
        double r70260 = r70259 / r70251;
        double r70261 = r70256 + r70260;
        double r70262 = log(r70261);
        double r70263 = r70254 + r70262;
        return r70263;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-cube-cbrt0.0

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

  4. Applied *-un-lft-identity0.0

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

  5. Applied times-frac0.0

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

  6. Applied add-cube-cbrt0.1

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

  7. Applied *-un-lft-identity0.1

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

  8. Applied times-frac0.1

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

  9. Applied distribute-lft-out0.1

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

  10. Applied log-prod0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))