Average Error: 0.5 → 0.5
Time: 15.0s
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 r232635 = 1.0;
        double r232636 = 5.0;
        double r232637 = v;
        double r232638 = r232637 * r232637;
        double r232639 = r232636 * r232638;
        double r232640 = r232635 - r232639;
        double r232641 = r232638 - r232635;
        double r232642 = r232640 / r232641;
        double r232643 = acos(r232642);
        return r232643;
}
↓
double f(double v) {
        double r232644 = 1.0;
        double r232645 = 5.0;
        double r232646 = v;
        double r232647 = 2.0;
        double r232648 = pow(r232646, r232647);
        double r232649 = r232645 * r232648;
        double r232650 = r232644 - r232649;
        double r232651 = r232648 - r232644;
        double r232652 = r232650 / r232651;
        double r232653 = acos(r232652);
        double r232654 = sqrt(r232653);
        double r232655 = r232654 * r232654;
        double r232656 = 3.0;
        double r232657 = pow(r232655, r232656);
        double r232658 = cbrt(r232657);
        return r232658;
}

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-cube0.5

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

  4. Applied add-cbrt-cube0.5

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

  5. Applied cbrt-unprod0.5

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

  6. Applied add-cbrt-cube0.5

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

  7. Applied cbrt-unprod0.5

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

  8. Simplified0.5

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

  10. Applied add-cbrt-cube1.5

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

  11. Simplified1.5

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

  13. Applied add-sqr-sqrt0.5

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

  14. 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 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))