Average Error: 0.0 → 0.1
Time: 6.0m
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)
double f(double x) {
        double r22636938 = 1.0;
        double r22636939 = x;
        double r22636940 = r22636938 / r22636939;
        double r22636941 = r22636939 * r22636939;
        double r22636942 = r22636938 - r22636941;
        double r22636943 = sqrt(r22636942);
        double r22636944 = r22636943 / r22636939;
        double r22636945 = r22636940 + r22636944;
        double r22636946 = log(r22636945);
        return r22636946;
}
↓
double f(double x) {
        double r22636947 = 1.0;
        double r22636948 = x;
        double r22636949 = r22636947 / r22636948;
        double r22636950 = r22636948 * r22636948;
        double r22636951 = r22636947 - r22636950;
        double r22636952 = sqrt(r22636951);
        double r22636953 = r22636952 / r22636948;
        double r22636954 = r22636949 + r22636953;
        double r22636955 = cbrt(r22636954);
        double r22636956 = r22636955 * r22636955;
        double r22636957 = r22636956 * r22636955;
        double r22636958 = log(r22636957);
        return r22636958;
}

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-cube-cbrt0.1

    \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)}\]

  4. Final simplification0.1

    \[\leadsto \log \left(\left(\sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}} \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right) \cdot \sqrt[3]{\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}}\right)\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))