Average Error: 0.1 → 0.1
Time: 5.0s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\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}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r85342 = 1.0;
        double r85343 = x;
        double r85344 = r85342 / r85343;
        double r85345 = r85343 * r85343;
        double r85346 = r85342 - r85345;
        double r85347 = sqrt(r85346);
        double r85348 = r85347 / r85343;
        double r85349 = r85344 + r85348;
        double r85350 = log(r85349);
        return r85350;
}
↓
double f(double x) {
        double r85351 = 1.0;
        double r85352 = x;
        double r85353 = r85351 / r85352;
        double r85354 = r85352 * r85352;
        double r85355 = r85351 - r85354;
        double r85356 = sqrt(r85355);
        double r85357 = r85356 / r85352;
        double r85358 = r85353 + r85357;
        double r85359 = sqrt(r85358);
        double r85360 = log(r85359);
        double r85361 = r85360 + r85360;
        return r85361;
}

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

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

  4. Applied log-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))