Average Error: 0.0 → 0.2
Time: 29.6s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{\sqrt{x}} + \frac{\sqrt{1 - x \cdot x}}{\sqrt{x}}\right) + \log \left(\frac{1}{\sqrt{x}}\right)
double f(double x) {
        double r2773028 = 1.0;
        double r2773029 = x;
        double r2773030 = r2773028 / r2773029;
        double r2773031 = r2773029 * r2773029;
        double r2773032 = r2773028 - r2773031;
        double r2773033 = sqrt(r2773032);
        double r2773034 = r2773033 / r2773029;
        double r2773035 = r2773030 + r2773034;
        double r2773036 = log(r2773035);
        return r2773036;
}
↓
double f(double x) {
        double r2773037 = 1.0;
        double r2773038 = x;
        double r2773039 = sqrt(r2773038);
        double r2773040 = r2773037 / r2773039;
        double r2773041 = r2773038 * r2773038;
        double r2773042 = r2773037 - r2773041;
        double r2773043 = sqrt(r2773042);
        double r2773044 = r2773043 / r2773039;
        double r2773045 = r2773040 + r2773044;
        double r2773046 = log(r2773045);
        double r2773047 = log(r2773040);
        double r2773048 = r2773046 + r2773047;
        return r2773048;
}

Error

Bits error versus x

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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

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

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

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

  5. Applied times-frac0.0

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

  6. Applied add-sqr-sqrt0.0

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

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

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

  8. Applied times-frac0.0

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

  9. Applied distribute-lft-out0.0

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

  10. Applied log-prod0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))