Average Error: 0.0 → 0.0
Time: 10.1s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r1258860 = 1.0;
        double r1258861 = x;
        double r1258862 = r1258860 / r1258861;
        double r1258863 = r1258861 * r1258861;
        double r1258864 = r1258860 - r1258863;
        double r1258865 = sqrt(r1258864);
        double r1258866 = r1258865 / r1258861;
        double r1258867 = r1258862 + r1258866;
        double r1258868 = log(r1258867);
        return r1258868;
}
↓
double f(double x) {
        double r1258869 = 1.0;
        double r1258870 = x;
        double r1258871 = r1258869 / r1258870;
        double r1258872 = r1258870 * r1258870;
        double r1258873 = r1258869 - r1258872;
        double r1258874 = sqrt(r1258873);
        double r1258875 = r1258874 / r1258870;
        double r1258876 = r1258871 + r1258875;
        double r1258877 = sqrt(r1258876);
        double r1258878 = r1258877 * r1258877;
        double r1258879 = log(r1258878);
        return r1258879;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))