Average Error: 0.6 → 0.7
Time: 6.6s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{\cos^{-1} \left(4 \cdot \left({v}^{4} + v \cdot v\right) + -1\right)} \cdot \sqrt[3]{\cos^{-1} \left(4 \cdot \left({v}^{4} + v \cdot v\right) + -1\right)}\right)}^{3}}\right)}^{3} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\cos^{-1} \left(4 \cdot \left({v}^{4} + v \cdot v\right) + -1\right)}\right)}^{3}}\right)}^{3}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{\cos^{-1} \left(4 \cdot \left({v}^{4} + v \cdot v\right) + -1\right)} \cdot \sqrt[3]{\cos^{-1} \left(4 \cdot \left({v}^{4} + v \cdot v\right) + -1\right)}\right)}^{3}}\right)}^{3} \cdot {\left(\sqrt[3]{{\left(\sqrt[3]{\cos^{-1} \left(4 \cdot \left({v}^{4} + v \cdot v\right) + -1\right)}\right)}^{3}}\right)}^{3}}
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (cbrt
  (*
   (pow
    (cbrt
     (pow
      (*
       (cbrt (acos (+ (* 4.0 (+ (pow v 4.0) (* v v))) -1.0)))
       (cbrt (acos (+ (* 4.0 (+ (pow v 4.0) (* v v))) -1.0))))
      3.0))
    3.0)
   (pow
    (cbrt (pow (cbrt (acos (+ (* 4.0 (+ (pow v 4.0) (* v v))) -1.0))) 3.0))
    3.0))))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return cbrt(pow(cbrt(pow((cbrt(acos((4.0 * (pow(v, 4.0) + (v * v))) + -1.0)) * cbrt(acos((4.0 * (pow(v, 4.0) + (v * v))) + -1.0))), 3.0)), 3.0) * pow(cbrt(pow(cbrt(acos((4.0 * (pow(v, 4.0) + (v * v))) + -1.0)), 3.0)), 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 0.6

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

  2. Taylor expanded around 0 0.7

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

  3. Simplified0.7

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

  5. Applied add-cbrt-cube_binary64_14781.7

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

  6. Simplified1.7

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

  8. Applied add-cbrt-cube_binary64_14781.7

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

  9. Simplified1.7

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

  11. Applied add-cube-cbrt_binary64_14772.2

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

  12. Applied unpow-prod-down_binary64_15212.2

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

  13. Applied cbrt-prod_binary64_14730.7

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

  14. Applied unpow-prod-down_binary64_15210.7

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

  15. Final simplification0.7

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

Reproduce

herbie shell --seed 2021045 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))