Average Error: 0.6 → 0.6
Time: 11.3s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 - \sqrt[3]{125 \cdot {v}^{6}}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 - \sqrt[3]{125 \cdot {v}^{6}}}{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 (* 125.0 (pow v 6.0)))) (- (* 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(125.0 * pow(v, 6.0))) / ((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-cbrt-cube_binary64_18190.6

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

  4. Simplified0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - \sqrt[3]{\color{blue}{125 \cdot {v}^{6}}}}{v \cdot v - 1}\right)\]

  5. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - \sqrt[3]{125 \cdot {v}^{6}}}{v \cdot v - 1}\right)\]

Reproduce

herbie shell --seed 2021007 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))