Average Error: 0.5 → 0.5
Time: 15.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt[3]{{\left(\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right)}^{3}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt[3]{{\left(\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right)}^{3}}
double f(double v) {
        double r267620 = 1.0;
        double r267621 = 5.0;
        double r267622 = v;
        double r267623 = r267622 * r267622;
        double r267624 = r267621 * r267623;
        double r267625 = r267620 - r267624;
        double r267626 = r267623 - r267620;
        double r267627 = r267625 / r267626;
        double r267628 = acos(r267627);
        return r267628;
}
↓
double f(double v) {
        double r267629 = 1.0;
        double r267630 = 5.0;
        double r267631 = v;
        double r267632 = 2.0;
        double r267633 = pow(r267631, r267632);
        double r267634 = r267630 * r267633;
        double r267635 = r267629 - r267634;
        double r267636 = r267633 - r267629;
        double r267637 = r267635 / r267636;
        double r267638 = acos(r267637);
        double r267639 = sqrt(r267638);
        double r267640 = r267639 * r267639;
        double r267641 = 3.0;
        double r267642 = pow(r267640, r267641);
        double r267643 = cbrt(r267642);
        return r267643;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube1.5

    \[\leadsto \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right) \cdot \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}\]

  4. Simplified1.5

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.5

    \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right)}}^{3}}\]

  7. Final simplification0.5

    \[\leadsto \sqrt[3]{{\left(\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}\right)}^{3}}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))