Average Error: 0.1 → 0.1
Time: 10.7s
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 r2915033 = 1.0;
        double r2915034 = x;
        double r2915035 = r2915033 / r2915034;
        double r2915036 = r2915034 * r2915034;
        double r2915037 = r2915033 - r2915036;
        double r2915038 = sqrt(r2915037);
        double r2915039 = r2915038 / r2915034;
        double r2915040 = r2915035 + r2915039;
        double r2915041 = log(r2915040);
        return r2915041;
}
↓
double f(double x) {
        double r2915042 = 1.0;
        double r2915043 = x;
        double r2915044 = r2915042 / r2915043;
        double r2915045 = r2915043 * r2915043;
        double r2915046 = r2915042 - r2915045;
        double r2915047 = sqrt(r2915046);
        double r2915048 = r2915047 / r2915043;
        double r2915049 = r2915044 + r2915048;
        double r2915050 = sqrt(r2915049);
        double r2915051 = log(r2915050);
        double r2915052 = r2915051 + r2915051;
        return r2915052;
}

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))))