Average Error: 0.5 → 0.8
Time: 6.6s
Precision: binary64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right) \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(4, v \cdot v, -1\right)\right) \]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\mathsf{fma}\left(4, v \cdot v, -1\right)\right)

(FPCore (v)
 :precision binary64
 (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))
↓
(FPCore (v) :precision binary64 (acos (fma 4.0 (* 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(fma(4.0, (v * v), -1.0));
}

Error

Bits error versus v

Derivation

  1. Initial program 0.5

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

  2. Simplified0.5

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

  3. Taylor expanded in v around 0 0.8

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

  4. Simplified0.8

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

  5. Final simplification0.8

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

Reproduce

herbie shell --seed 2021313 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))