Average Error: 31.3 → 0.1
Time: 16.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt[3]{\sqrt{x - 1}} \cdot \left(\sqrt{1 + x} \cdot \left(\sqrt[3]{\sqrt{x - 1}} \cdot \sqrt[3]{\sqrt{x - 1}}\right)\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt[3]{\sqrt{x - 1}} \cdot \left(\sqrt{1 + x} \cdot \left(\sqrt[3]{\sqrt{x - 1}} \cdot \sqrt[3]{\sqrt{x - 1}}\right)\right)\right)
double f(double x) {
        double r1695381 = x;
        double r1695382 = r1695381 * r1695381;
        double r1695383 = 1.0;
        double r1695384 = r1695382 - r1695383;
        double r1695385 = sqrt(r1695384);
        double r1695386 = r1695381 + r1695385;
        double r1695387 = log(r1695386);
        return r1695387;
}
↓
double f(double x) {
        double r1695388 = x;
        double r1695389 = 1.0;
        double r1695390 = r1695388 - r1695389;
        double r1695391 = sqrt(r1695390);
        double r1695392 = cbrt(r1695391);
        double r1695393 = r1695389 + r1695388;
        double r1695394 = sqrt(r1695393);
        double r1695395 = r1695392 * r1695392;
        double r1695396 = r1695394 * r1695395;
        double r1695397 = r1695392 * r1695396;
        double r1695398 = r1695388 + r1695397;
        double r1695399 = log(r1695398);
        return r1695399;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied difference-of-sqr-131.3

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

  4. Applied sqrt-prod0.1

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

  6. Applied add-cube-cbrt0.1

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

  7. Applied associate-*r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))