Average Error: 0.6 → 0.8
Time: 4.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3} \cdot \sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3} \cdot \sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}
double f(double v) {
        double r321576 = 1.0;
        double r321577 = 5.0;
        double r321578 = v;
        double r321579 = r321578 * r321578;
        double r321580 = r321577 * r321579;
        double r321581 = r321576 - r321580;
        double r321582 = r321579 - r321576;
        double r321583 = r321581 / r321582;
        double r321584 = acos(r321583);
        return r321584;
}
↓
double f(double v) {
        double r321585 = 4.0;
        double r321586 = v;
        double r321587 = 2.0;
        double r321588 = pow(r321586, r321587);
        double r321589 = 4.0;
        double r321590 = pow(r321586, r321589);
        double r321591 = r321588 + r321590;
        double r321592 = r321585 * r321591;
        double r321593 = 1.0;
        double r321594 = r321592 - r321593;
        double r321595 = acos(r321594);
        double r321596 = sqrt(r321595);
        double r321597 = sqrt(r321596);
        double r321598 = cbrt(r321597);
        double r321599 = r321598 * r321598;
        double r321600 = 3.0;
        double r321601 = pow(r321599, r321600);
        double r321602 = r321601 * r321596;
        return r321602;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.6

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

  2. Taylor expanded around 0 0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\]

  3. Simplified0.8

    \[\leadsto \cos^{-1} \color{blue}{\left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\]
  4. Using strategy rm

  5. Applied add-sqr-sqrt1.7

    \[\leadsto \color{blue}{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)} \cdot \sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\]
  6. Using strategy rm

  7. Applied add-sqr-sqrt1.7

    \[\leadsto \sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)} \cdot \sqrt{\color{blue}{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)} \cdot \sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\]

  8. Applied sqrt-prod0.8

    \[\leadsto \sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)} \cdot \color{blue}{\left(\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}} \cdot \sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\right)}\]

  9. Applied associate-*r*1.7

    \[\leadsto \color{blue}{\left(\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)} \cdot \sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\right) \cdot \sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\]

  10. Simplified1.7

    \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\right)}^{3}} \cdot \sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\]
  11. Using strategy rm

  12. Applied add-cube-cbrt1.7

    \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}}^{3} \cdot \sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\]

  13. Applied unpow-prod-down2.3

    \[\leadsto \color{blue}{\left({\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3}\right)} \cdot \sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\]

  14. Applied associate-*l*2.3

    \[\leadsto \color{blue}{{\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3} \cdot \left({\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3} \cdot \sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\right)}\]

  15. Simplified0.8

    \[\leadsto {\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3} \cdot \color{blue}{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}\]

  16. Final simplification0.8

    \[\leadsto {\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}}}\right)}^{3} \cdot \sqrt{\cos^{-1} \left(4 \cdot \left({v}^{2} + {v}^{4}\right) - 1\right)}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))