Average Error: 0.6 → 0.7
Time: 5.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)
double code(double v) {
	return ((double) acos(((double) (((double) (1.0 - ((double) (5.0 * ((double) (v * v)))))) / ((double) (((double) (v * v)) - 1.0))))));
}
↓
double code(double v) {
	return ((double) acos(((double) fma(4.0, ((double) fma(v, v, ((double) pow(v, 4.0)))), ((double) -(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. Taylor expanded around 0 0.7

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

  3. Simplified0.7

    \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)}\]

  4. Final simplification0.7

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))