Average Error: 31.5 → 0.1
Time: 9.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt{\sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r72655 = x;
        double r72656 = r72655 * r72655;
        double r72657 = 1.0;
        double r72658 = r72656 - r72657;
        double r72659 = sqrt(r72658);
        double r72660 = r72655 + r72659;
        double r72661 = log(r72660);
        return r72661;
}
↓
double f(double x) {
        double r72662 = x;
        double r72663 = 1.0;
        double r72664 = sqrt(r72663);
        double r72665 = r72662 + r72664;
        double r72666 = sqrt(r72665);
        double r72667 = r72662 - r72664;
        double r72668 = sqrt(r72667);
        double r72669 = r72666 * r72668;
        double r72670 = sqrt(r72669);
        double r72671 = r72670 * r72670;
        double r72672 = r72662 + r72671;
        double r72673 = log(r72672);
        return r72673;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.5

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

  3. Applied add-sqr-sqrt31.5

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

  4. Applied difference-of-squares31.5

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))