Average Error: 0.5 → 0.5
Time: 5.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt{\cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}\right)} \cdot \sqrt{\sqrt[3]{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt{\cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}\right)} \cdot \sqrt{\sqrt[3]{{\left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)\right)}^{3}}}
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) sqrt(((double) acos(((double) cbrt(((double) pow(((double) (((double) (1.0 - ((double) (5.0 * ((double) pow(v, 2.0)))))) / ((double) (((double) pow(v, 2.0)) - 1.0)))), 3.0)))))))) * ((double) sqrt(((double) cbrt(((double) pow(((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) pow(v, 2.0)))))) / ((double) (((double) pow(v, 2.0)) - 1.0)))))), 3.0))))))));
}

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-cbrt-cube0.5

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

  4. Applied add-cbrt-cube0.6

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

  5. Applied cbrt-undiv0.5

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

  6. Simplified0.5

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

  8. Applied add-sqr-sqrt1.5

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

  10. Applied add-cbrt-cube0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020130 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))