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 code(double v) {
	return ((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0))))));
}
↓
double code(double v) {
	return ((double) (((double) pow(((double) (((double) cbrt(((double) sqrt(((double) sqrt(((double) acos(((double) (((double) (4.0 * ((double) (((double) pow(v, 2.0)) + ((double) pow(v, 4.0)))))) - 1.0)))))))))) * ((double) cbrt(((double) sqrt(((double) sqrt(((double) acos(((double) (((double) (4.0 * ((double) (((double) pow(v, 2.0)) + ((double) pow(v, 4.0)))))) - 1.0)))))))))))), 3.0)) * ((double) sqrt(((double) acos(((double) (((double) (4.0 * ((double) (((double) pow(v, 2.0)) + ((double) pow(v, 4.0)))))) - 1.0))))))));
}

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-sqrt1.8

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

  7. Applied add-sqr-sqrt1.8

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

  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*1.8

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

  10. Simplified1.8

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

  12. Applied add-cube-cbrt1.8

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

  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 2020122 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))