Average Error: 31.7 → 0.0
Time: 14.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x} + \sqrt{\sqrt{1}}}\right) \cdot \sqrt{\sqrt{x} - \sqrt{\sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x} + \sqrt{\sqrt{1}}}\right) \cdot \sqrt{\sqrt{x} - \sqrt{\sqrt{1}}}\right)
double f(double x) {
        double r75215 = x;
        double r75216 = r75215 * r75215;
        double r75217 = 1.0;
        double r75218 = r75216 - r75217;
        double r75219 = sqrt(r75218);
        double r75220 = r75215 + r75219;
        double r75221 = log(r75220);
        return r75221;
}
↓
double f(double x) {
        double r75222 = x;
        double r75223 = 1.0;
        double r75224 = sqrt(r75223);
        double r75225 = r75222 + r75224;
        double r75226 = sqrt(r75225);
        double r75227 = sqrt(r75222);
        double r75228 = sqrt(r75224);
        double r75229 = r75227 + r75228;
        double r75230 = sqrt(r75229);
        double r75231 = r75226 * r75230;
        double r75232 = r75227 - r75228;
        double r75233 = sqrt(r75232);
        double r75234 = r75231 * r75233;
        double r75235 = r75222 + r75234;
        double r75236 = log(r75235);
        return r75236;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.7

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

  3. Applied add-sqr-sqrt31.7

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

  4. Applied difference-of-squares31.7

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

  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

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

  8. Applied sqrt-prod0.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Applied difference-of-squares0.0

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

  11. Applied sqrt-prod0.0

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

  12. Applied associate-*r*0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))