Average Error: 0.5 → 0.9
Time: 5.8s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(4 \cdot {v}^{2} - 1\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(4 \cdot {v}^{2} - 1\right)
(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v) :precision binary64 (acos (- (* 4.0 (pow v 2.0)) 1.0)))
double code(double v) {
	return acos((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0));
}
↓
double code(double v) {
	return acos((4.0 * pow(v, 2.0)) - 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. Taylor expanded around 0 0.9

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

  3. Final simplification0.9

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

Reproduce

herbie shell --seed 2021024 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))