Average Error: 31.4 → 0.0
Time: 27.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{\sqrt{x} - 1} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + \sqrt{x}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{\sqrt{x} - 1} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + \sqrt{x}}\right)\right)
double f(double x) {
        double r9262602 = x;
        double r9262603 = r9262602 * r9262602;
        double r9262604 = 1.0;
        double r9262605 = r9262603 - r9262604;
        double r9262606 = sqrt(r9262605);
        double r9262607 = r9262602 + r9262606;
        double r9262608 = log(r9262607);
        return r9262608;
}
↓
double f(double x) {
        double r9262609 = x;
        double r9262610 = sqrt(r9262609);
        double r9262611 = 1.0;
        double r9262612 = r9262610 - r9262611;
        double r9262613 = sqrt(r9262612);
        double r9262614 = r9262611 + r9262609;
        double r9262615 = sqrt(r9262614);
        double r9262616 = r9262611 + r9262610;
        double r9262617 = sqrt(r9262616);
        double r9262618 = r9262615 * r9262617;
        double r9262619 = r9262613 * r9262618;
        double r9262620 = r9262609 + r9262619;
        double r9262621 = log(r9262620);
        return r9262621;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.4

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

  3. Applied *-un-lft-identity31.4

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

  4. Applied difference-of-squares31.4

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

  5. Applied sqrt-prod0.0

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

  7. Applied *-un-lft-identity0.0

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

  8. Applied add-sqr-sqrt0.0

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

  9. Applied difference-of-squares0.0

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

  10. Applied sqrt-prod0.0

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

  11. Applied associate-*r*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))