Average Error: 0.1 → 0.1
Time: 10.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \sqrt{1 - x \cdot x} \cdot \frac{1}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \sqrt{1 - x \cdot x} \cdot \frac{1}{x}\right)
double f(double x) {
        double r51693 = 1.0;
        double r51694 = x;
        double r51695 = r51693 / r51694;
        double r51696 = r51694 * r51694;
        double r51697 = r51693 - r51696;
        double r51698 = sqrt(r51697);
        double r51699 = r51698 / r51694;
        double r51700 = r51695 + r51699;
        double r51701 = log(r51700);
        return r51701;
}
↓
double f(double x) {
        double r51702 = 1.0;
        double r51703 = x;
        double r51704 = r51702 / r51703;
        double r51705 = r51703 * r51703;
        double r51706 = r51702 - r51705;
        double r51707 = sqrt(r51706);
        double r51708 = 1.0;
        double r51709 = r51708 / r51703;
        double r51710 = r51707 * r51709;
        double r51711 = r51704 + r51710;
        double r51712 = log(r51711);
        return r51712;
}

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 div-inv0.1

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

  4. Final simplification0.1

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

Reproduce

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