Average Error: 0.6 → 0.6
Time: 5.9s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \left(\sqrt[3]{5} \cdot \sqrt[3]{5}\right) \cdot \left(\sqrt[3]{5} \cdot \left(v \cdot v\right)\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - \left(\sqrt[3]{5} \cdot \sqrt[3]{5}\right) \cdot \left(\sqrt[3]{5} \cdot \left(v \cdot v\right)\right)}{v \cdot v - 1}\right)
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v)
 :precision binary64
 (acos
  (/
   (- 1.0 (* (* (cbrt 5.0) (cbrt 5.0)) (* (cbrt 5.0) (* v v))))
   (- (* v v) 1.0))))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return acos((1.0 - ((cbrt(5.0) * cbrt(5.0)) * (cbrt(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 0.6

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

  3. Applied add-cube-cbrt_binary64_21310.6

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

  4. Applied associate-*l*_binary64_20420.6

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

  5. Simplified0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2020288 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))