Average Error: 0.5 → 0.5
Time: 5.5s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(\sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}}\right)}^{3} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
\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(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}} \cdot \sqrt[3]{\sqrt{\sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}}}\right)}^{3} \cdot \sqrt{\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 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) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0)))))))))))) * ((double) cbrt(((double) sqrt(((double) sqrt(((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0)))))))))))))), 3.0)) * ((double) sqrt(((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 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 0.5

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

  3. Applied add-sqr-sqrt1.5

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

  5. Applied add-sqr-sqrt1.5

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

  6. Applied sqrt-prod0.5

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

  7. Applied associate-*r*1.5

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

  8. Simplified1.5

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

  10. Applied add-cube-cbrt1.5

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

  11. Applied unpow-prod-down2.0

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

  12. Applied associate-*l*2.0

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020148 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))