Average Error: 0.1 → 0.1
Time: 10.5s
Precision: 64
\[\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)\]
↓
\[\log \left(\frac{1}{x} + \left(\frac{\sqrt{\sqrt{1} + x}}{\sqrt{x}} \cdot \sqrt{\sqrt{\sqrt{1}} + \sqrt{x}}\right) \cdot \frac{\sqrt{\sqrt{\sqrt{1}} - \sqrt{x}}}{\sqrt{x}}\right)\]
\log \left(\frac{1}{x} + \frac{\sqrt{1 - x \cdot x}}{x}\right)
↓
\log \left(\frac{1}{x} + \left(\frac{\sqrt{\sqrt{1} + x}}{\sqrt{x}} \cdot \sqrt{\sqrt{\sqrt{1}} + \sqrt{x}}\right) \cdot \frac{\sqrt{\sqrt{\sqrt{1}} - \sqrt{x}}}{\sqrt{x}}\right)
double f(double x) {
        double r98329 = 1.0;
        double r98330 = x;
        double r98331 = r98329 / r98330;
        double r98332 = r98330 * r98330;
        double r98333 = r98329 - r98332;
        double r98334 = sqrt(r98333);
        double r98335 = r98334 / r98330;
        double r98336 = r98331 + r98335;
        double r98337 = log(r98336);
        return r98337;
}
↓
double f(double x) {
        double r98338 = 1.0;
        double r98339 = x;
        double r98340 = r98338 / r98339;
        double r98341 = sqrt(r98338);
        double r98342 = r98341 + r98339;
        double r98343 = sqrt(r98342);
        double r98344 = sqrt(r98339);
        double r98345 = r98343 / r98344;
        double r98346 = sqrt(r98341);
        double r98347 = r98346 + r98344;
        double r98348 = sqrt(r98347);
        double r98349 = r98345 * r98348;
        double r98350 = r98346 - r98344;
        double r98351 = sqrt(r98350);
        double r98352 = r98351 / r98344;
        double r98353 = r98349 * r98352;
        double r98354 = r98340 + r98353;
        double r98355 = log(r98354);
        return r98355;
}

Error

Bits error versus x

Try it out

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. Using strategy rm

  9. Applied *-un-lft-identity0.1

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

  10. Applied sqrt-prod0.1

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

  11. Applied add-sqr-sqrt0.1

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

  12. Applied add-sqr-sqrt0.1

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

  13. Applied sqrt-prod0.1

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

  14. Applied difference-of-squares0.1

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

  15. Applied sqrt-prod0.1

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

  16. Applied times-frac0.1

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

  17. Applied associate-*r*0.1

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

  18. Simplified0.1

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

  19. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1 x) (/ (sqrt (- 1 (* x x))) x))))