Average Error: 0.0 → 0.0
Time: 9.9s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\sqrt{1 - x \cdot x}}{\sqrt[3]{x}}\right)
double f(double x) {
        double r55102 = 1.0;
        double r55103 = x;
        double r55104 = r55102 / r55103;
        double r55105 = r55103 * r55103;
        double r55106 = r55102 - r55105;
        double r55107 = sqrt(r55106);
        double r55108 = r55107 / r55103;
        double r55109 = r55104 + r55108;
        double r55110 = log(r55109);
        return r55110;
}
↓
double f(double x) {
        double r55111 = 1.0;
        double r55112 = x;
        double r55113 = r55111 / r55112;
        double r55114 = 1.0;
        double r55115 = cbrt(r55112);
        double r55116 = r55115 * r55115;
        double r55117 = r55114 / r55116;
        double r55118 = r55112 * r55112;
        double r55119 = r55111 - r55118;
        double r55120 = sqrt(r55119);
        double r55121 = r55120 / r55115;
        double r55122 = r55117 * r55121;
        double r55123 = r55113 + r55122;
        double r55124 = log(r55123);
        return r55124;
}

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

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

  5. Applied sqrt-prod0.0

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

  6. Applied times-frac0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))