Average Error: 0.0 → 0.0
Time: 11.7s
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 r1379073 = 1.0;
        double r1379074 = x;
        double r1379075 = r1379073 / r1379074;
        double r1379076 = r1379074 * r1379074;
        double r1379077 = r1379073 - r1379076;
        double r1379078 = sqrt(r1379077);
        double r1379079 = r1379078 / r1379074;
        double r1379080 = r1379075 + r1379079;
        double r1379081 = log(r1379080);
        return r1379081;
}
↓
double f(double x) {
        double r1379082 = 1.0;
        double r1379083 = x;
        double r1379084 = r1379082 / r1379083;
        double r1379085 = r1379083 * r1379083;
        double r1379086 = r1379082 - r1379085;
        double r1379087 = sqrt(r1379086);
        double r1379088 = r1379087 / r1379083;
        double r1379089 = r1379084 + r1379088;
        double r1379090 = sqrt(r1379089);
        double r1379091 = log(r1379090);
        double r1379092 = r1379091 + r1379091;
        return r1379092;
}

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 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))