Average Error: 0.0 → 0.0
Time: 5.2s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \left(\frac{\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}}{\sqrt{x}} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}}\right) \cdot \frac{\sqrt[3]{\sqrt{1 - x \cdot x}}}{\sqrt[3]{\sqrt{x}}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \left(\frac{\sqrt[3]{\sqrt{1 - x \cdot x}} \cdot \sqrt[3]{\sqrt{1 - x \cdot x}}}{\sqrt{x}} \cdot \frac{\sqrt[3]{\sqrt{1}}}{\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}}\right) \cdot \frac{\sqrt[3]{\sqrt{1 - x \cdot x}}}{\sqrt[3]{\sqrt{x}}}\right)
double f(double x) {
        double r100424 = 1.0;
        double r100425 = x;
        double r100426 = r100424 / r100425;
        double r100427 = r100425 * r100425;
        double r100428 = r100424 - r100427;
        double r100429 = sqrt(r100428);
        double r100430 = r100429 / r100425;
        double r100431 = r100426 + r100430;
        double r100432 = log(r100431);
        return r100432;
}
↓
double f(double x) {
        double r100433 = 1.0;
        double r100434 = x;
        double r100435 = r100433 / r100434;
        double r100436 = r100434 * r100434;
        double r100437 = r100433 - r100436;
        double r100438 = sqrt(r100437);
        double r100439 = cbrt(r100438);
        double r100440 = r100439 * r100439;
        double r100441 = sqrt(r100434);
        double r100442 = r100440 / r100441;
        double r100443 = 1.0;
        double r100444 = sqrt(r100443);
        double r100445 = cbrt(r100444);
        double r100446 = cbrt(r100441);
        double r100447 = r100446 * r100446;
        double r100448 = r100445 / r100447;
        double r100449 = r100442 * r100448;
        double r100450 = r100439 / r100446;
        double r100451 = r100449 * r100450;
        double r100452 = r100435 + r100451;
        double r100453 = log(r100452);
        return r100453;
}

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-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 add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  7. Applied add-cube-cbrt0.0

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

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

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

  9. Applied sqrt-prod0.0

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

  10. Applied cbrt-prod0.0

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

  11. Applied times-frac0.0

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

  12. Applied associate-*r*0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))