Average Error: 0.1 → 0.1
Time: 10.1s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1} - x}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \frac{\sqrt{\sqrt{1} + x}}{\sqrt{x}} \cdot \frac{\sqrt{\sqrt{1} - x}}{\sqrt{x}}\right)
double f(double x) {
        double r56847 = 1.0;
        double r56848 = x;
        double r56849 = r56847 / r56848;
        double r56850 = r56848 * r56848;
        double r56851 = r56847 - r56850;
        double r56852 = sqrt(r56851);
        double r56853 = r56852 / r56848;
        double r56854 = r56849 + r56853;
        double r56855 = log(r56854);
        return r56855;
}
↓
double f(double x) {
        double r56856 = 1.0;
        double r56857 = x;
        double r56858 = r56856 / r56857;
        double r56859 = sqrt(r56856);
        double r56860 = r56859 + r56857;
        double r56861 = sqrt(r56860);
        double r56862 = sqrt(r56857);
        double r56863 = r56861 / r56862;
        double r56864 = r56859 - r56857;
        double r56865 = sqrt(r56864);
        double r56866 = r56865 / r56862;
        double r56867 = r56863 * r56866;
        double r56868 = r56858 + r56867;
        double r56869 = log(r56868);
        return r56869;
}

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

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

  4. Applied add-sqr-sqrt0.1

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

  5. Applied difference-of-squares0.1

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

  6. Applied sqrt-prod0.1

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

  7. Applied times-frac0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))