Average Error: 0.1 → 0.1
Time: 10.4s
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 r5147980 = 1.0;
        double r5147981 = x;
        double r5147982 = r5147980 / r5147981;
        double r5147983 = r5147981 * r5147981;
        double r5147984 = r5147980 - r5147983;
        double r5147985 = sqrt(r5147984);
        double r5147986 = r5147985 / r5147981;
        double r5147987 = r5147982 + r5147986;
        double r5147988 = log(r5147987);
        return r5147988;
}
↓
double f(double x) {
        double r5147989 = 1.0;
        double r5147990 = x;
        double r5147991 = r5147989 / r5147990;
        double r5147992 = r5147990 * r5147990;
        double r5147993 = r5147989 - r5147992;
        double r5147994 = sqrt(r5147993);
        double r5147995 = r5147994 / r5147990;
        double r5147996 = r5147991 + r5147995;
        double r5147997 = sqrt(r5147996);
        double r5147998 = log(r5147997);
        double r5147999 = r5147998 + r5147998;
        return r5147999;
}

Error

Bits error versus x

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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 2019174 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))