Average Error: 31.7 → 0.1
Time: 5.7s
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 r81505 = x;
        double r81506 = r81505 * r81505;
        double r81507 = 1.0;
        double r81508 = r81506 - r81507;
        double r81509 = sqrt(r81508);
        double r81510 = r81505 + r81509;
        double r81511 = log(r81510);
        return r81511;
}
↓
double f(double x) {
        double r81512 = x;
        double r81513 = 1.0;
        double r81514 = sqrt(r81513);
        double r81515 = r81512 + r81514;
        double r81516 = sqrt(r81515);
        double r81517 = sqrt(r81512);
        double r81518 = sqrt(r81514);
        double r81519 = r81517 + r81518;
        double r81520 = sqrt(r81519);
        double r81521 = r81516 * r81520;
        double r81522 = r81517 - r81518;
        double r81523 = sqrt(r81522);
        double r81524 = r81521 * r81523;
        double r81525 = r81512 + r81524;
        double r81526 = log(r81525);
        return r81526;
}

Error

Bits error versus x

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Results

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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-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied add-sqr-sqrt0.1

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

  10. Applied difference-of-squares0.1

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

  11. Applied sqrt-prod0.1

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

  12. Applied associate-*r*0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020049 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))