Average Error: 31.1 → 0.2
Time: 20.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right) + \log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} + \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right) + \log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} + \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)
double f(double x) {
        double r2255505 = x;
        double r2255506 = r2255505 * r2255505;
        double r2255507 = 1.0;
        double r2255508 = r2255506 - r2255507;
        double r2255509 = sqrt(r2255508);
        double r2255510 = r2255505 + r2255509;
        double r2255511 = log(r2255510);
        return r2255511;
}
↓
double f(double x) {
        double r2255512 = x;
        double r2255513 = sqrt(r2255512);
        double r2255514 = 0.5;
        double r2255515 = sqrt(r2255514);
        double r2255516 = r2255513 / r2255515;
        double r2255517 = 0.125;
        double r2255518 = r2255517 / r2255512;
        double r2255519 = r2255512 * r2255512;
        double r2255520 = r2255518 / r2255519;
        double r2255521 = r2255514 / r2255512;
        double r2255522 = r2255520 + r2255521;
        double r2255523 = sqrt(r2255522);
        double r2255524 = r2255516 - r2255523;
        double r2255525 = log(r2255524);
        double r2255526 = r2255516 + r2255523;
        double r2255527 = log(r2255526);
        double r2255528 = r2255525 + r2255527;
        return r2255528;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.1

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

  2. Taylor expanded around inf 0.3

    \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

  3. Simplified0.3

    \[\leadsto \log \color{blue}{\left(\frac{x}{\frac{1}{2}} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}\right)\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt0.3

    \[\leadsto \log \left(\frac{x}{\frac{1}{2}} - \color{blue}{\sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}} \cdot \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}}\right)\]

  6. Applied add-sqr-sqrt0.3

    \[\leadsto \log \left(\frac{x}{\color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}} \cdot \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)\]

  7. Applied add-sqr-sqrt0.3

    \[\leadsto \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}} \cdot \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)\]

  8. Applied times-frac0.3

    \[\leadsto \log \left(\color{blue}{\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} \cdot \frac{\sqrt{x}}{\sqrt{\frac{1}{2}}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}} \cdot \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)\]

  9. Applied difference-of-squares0.3

    \[\leadsto \log \color{blue}{\left(\left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} + \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right) \cdot \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)\right)}\]

  10. Applied log-prod0.2

    \[\leadsto \color{blue}{\log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} + \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right) + \log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)}\]

  11. Final simplification0.2

    \[\leadsto \log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} - \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right) + \log \left(\frac{\sqrt{x}}{\sqrt{\frac{1}{2}}} + \sqrt{\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} + \frac{\frac{1}{2}}{x}}\right)\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))