Average Error: 0.1 → 0.2
Time: 10.9s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\left(-\log x\right) + \log \left(1 + \sqrt{1 - x \cdot x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\left(-\log x\right) + \log \left(1 + \sqrt{1 - x \cdot x}\right)
double f(double x) {
        double r52000 = 1.0;
        double r52001 = x;
        double r52002 = r52000 / r52001;
        double r52003 = r52001 * r52001;
        double r52004 = r52000 - r52003;
        double r52005 = sqrt(r52004);
        double r52006 = r52005 / r52001;
        double r52007 = r52002 + r52006;
        double r52008 = log(r52007);
        return r52008;
}
↓
double f(double x) {
        double r52009 = x;
        double r52010 = log(r52009);
        double r52011 = -r52010;
        double r52012 = 1.0;
        double r52013 = r52009 * r52009;
        double r52014 = r52012 - r52013;
        double r52015 = sqrt(r52014);
        double r52016 = r52012 + r52015;
        double r52017 = log(r52016);
        double r52018 = r52011 + r52017;
        return r52018;
}

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. Applied div-inv0.1

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

  5. Applied distribute-rgt-out0.1

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

  6. Applied log-prod0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))