Average Error: 0.0 → 0.0
Time: 6.0s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}\right)\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}\right)\right)
double f(double x) {
        double r68764 = 1.0;
        double r68765 = x;
        double r68766 = r68764 / r68765;
        double r68767 = r68765 * r68765;
        double r68768 = r68764 - r68767;
        double r68769 = sqrt(r68768);
        double r68770 = r68769 / r68765;
        double r68771 = r68766 + r68770;
        double r68772 = log(r68771);
        return r68772;
}
↓
double f(double x) {
        double r68773 = 1.0;
        double r68774 = x;
        double r68775 = sqrt(r68774);
        double r68776 = r68773 / r68775;
        double r68777 = 1.0;
        double r68778 = r68777 / r68775;
        double r68779 = r68774 * r68774;
        double r68780 = r68777 - r68779;
        double r68781 = sqrt(r68780);
        double r68782 = r68781 / r68775;
        double r68783 = r68778 + r68782;
        double r68784 = r68776 * r68783;
        double r68785 = log(r68784);
        return r68785;
}

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-sqr-sqrt0.0

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

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

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

  5. Applied times-frac0.0

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

  6. Applied add-sqr-sqrt0.0

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

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

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

  8. Applied times-frac0.0

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

  9. Applied distribute-lft-out0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))