Average Error: 0.0 → 0.0
Time: 16.2s
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 r3259352 = 1.0;
        double r3259353 = x;
        double r3259354 = r3259352 / r3259353;
        double r3259355 = r3259353 * r3259353;
        double r3259356 = r3259352 - r3259355;
        double r3259357 = sqrt(r3259356);
        double r3259358 = r3259357 / r3259353;
        double r3259359 = r3259354 + r3259358;
        double r3259360 = log(r3259359);
        return r3259360;
}
↓
double f(double x) {
        double r3259361 = 1.0;
        double r3259362 = x;
        double r3259363 = r3259361 / r3259362;
        double r3259364 = r3259362 * r3259362;
        double r3259365 = r3259361 - r3259364;
        double r3259366 = sqrt(r3259365);
        double r3259367 = r3259366 / r3259362;
        double r3259368 = r3259363 + r3259367;
        double r3259369 = sqrt(r3259368);
        double r3259370 = log(r3259369);
        double r3259371 = r3259370 + r3259370;
        return r3259371;
}

Error

Bits error versus x

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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 2019179 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))