Average Error: 0.0 → 0.0
Time: 11.1s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) + \log \left(\sqrt{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\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 r857364 = 1.0;
        double r857365 = x;
        double r857366 = r857364 / r857365;
        double r857367 = r857365 * r857365;
        double r857368 = r857364 - r857367;
        double r857369 = sqrt(r857368);
        double r857370 = r857369 / r857365;
        double r857371 = r857366 + r857370;
        double r857372 = log(r857371);
        return r857372;
}
↓
double f(double x) {
        double r857373 = 1.0;
        double r857374 = x;
        double r857375 = r857373 / r857374;
        double r857376 = r857374 * r857374;
        double r857377 = r857373 - r857376;
        double r857378 = sqrt(r857377);
        double r857379 = r857378 / r857374;
        double r857380 = r857375 + r857379;
        double r857381 = sqrt(r857380);
        double r857382 = log(r857381);
        double r857383 = r857382 + r857382;
        return r857383;
}

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. Applied log-prod0.0

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

  5. Final simplification0.0

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

Reproduce

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