Average Error: 0.5 → 0.5
Time: 5.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1}{\frac{v \cdot v - 1}{1 - 5 \cdot \left(v \cdot v\right)}}\right)
double code(double v) {
	return acos(((1.0 - (5.0 * (v * v))) / ((v * v) - 1.0)));
}
↓
double code(double v) {
	return acos((1.0 / (((v * v) - 1.0) / (1.0 - (5.0 * (v * v))))));
}

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 clear-num0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))