Average Error: 0.1 → 0.1
Time: 12.1s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\left|\sqrt[3]{1 - x \cdot x}\right|}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\left|\sqrt[3]{1 - x \cdot x}\right|}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt[3]{1 - x \cdot x}}}{\sqrt{x}}\right)
double f(double x) {
        double r57722 = 1.0;
        double r57723 = x;
        double r57724 = r57722 / r57723;
        double r57725 = r57723 * r57723;
        double r57726 = r57722 - r57725;
        double r57727 = sqrt(r57726);
        double r57728 = r57727 / r57723;
        double r57729 = r57724 + r57728;
        double r57730 = log(r57729);
        return r57730;
}
↓
double f(double x) {
        double r57731 = 1.0;
        double r57732 = x;
        double r57733 = r57731 / r57732;
        double r57734 = r57732 * r57732;
        double r57735 = r57731 - r57734;
        double r57736 = cbrt(r57735);
        double r57737 = fabs(r57736);
        double r57738 = sqrt(r57732);
        double r57739 = r57737 / r57738;
        double r57740 = sqrt(r57736);
        double r57741 = r57740 / r57738;
        double r57742 = r57739 * r57741;
        double r57743 = r57733 + r57742;
        double r57744 = log(r57743);
        return r57744;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-sqr-sqrt0.1

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

  4. Applied add-cube-cbrt0.1

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

  5. Applied sqrt-prod0.1

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))