Average Error: 0.5 → 0.6
Time: 5.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\left(\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\left(\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt[3]{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return acos(((cbrt((1.0 - (5.0 * (v * v)))) * cbrt((1.0 - (5.0 * (v * v))))) * (cbrt((1.0 - (5.0 * (v * v)))) / ((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 *-un-lft-identity0.5

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

  4. Applied add-cube-cbrt0.6

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

  5. Applied times-frac0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020060 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))