Average Error: 0.0 → 0.0
Time: 12.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{\frac{x}{\sqrt{1 - x \cdot x}}} + \frac{1}{x}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{\frac{x}{\sqrt{1 - x \cdot x}}} + \frac{1}{x}\right)
double f(double x) {
        double r1229027 = 1.0;
        double r1229028 = x;
        double r1229029 = r1229027 / r1229028;
        double r1229030 = r1229028 * r1229028;
        double r1229031 = r1229027 - r1229030;
        double r1229032 = sqrt(r1229031);
        double r1229033 = r1229032 / r1229028;
        double r1229034 = r1229029 + r1229033;
        double r1229035 = log(r1229034);
        return r1229035;
}
↓
double f(double x) {
        double r1229036 = 1.0;
        double r1229037 = x;
        double r1229038 = r1229037 * r1229037;
        double r1229039 = r1229036 - r1229038;
        double r1229040 = sqrt(r1229039);
        double r1229041 = r1229037 / r1229040;
        double r1229042 = r1229036 / r1229041;
        double r1229043 = r1229036 / r1229037;
        double r1229044 = r1229042 + r1229043;
        double r1229045 = log(r1229044);
        return r1229045;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.0

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

  4. Applied associate-/l*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))