Average Error: 32.1 → 0.1
Time: 10.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)
double f(double x) {
        double r58219 = x;
        double r58220 = r58219 * r58219;
        double r58221 = 1.0;
        double r58222 = r58220 - r58221;
        double r58223 = sqrt(r58222);
        double r58224 = r58219 + r58223;
        double r58225 = log(r58224);
        return r58225;
}
↓
double f(double x) {
        double r58226 = x;
        double r58227 = 1.0;
        double r58228 = sqrt(r58227);
        double r58229 = r58226 + r58228;
        double r58230 = sqrt(r58229);
        double r58231 = sqrt(r58230);
        double r58232 = r58226 - r58228;
        double r58233 = sqrt(r58232);
        double r58234 = r58231 * r58233;
        double r58235 = r58231 * r58234;
        double r58236 = r58226 + r58235;
        double r58237 = log(r58236);
        return r58237;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  3. Applied add-sqr-sqrt32.1

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

  4. Applied difference-of-squares32.1

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied sqrt-prod0.1

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

  9. Applied associate-*l*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))