Average Error: 0.0 → 0.3
Time: 3.0s
Precision: binary64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]
↓
\[\sqrt{2} \cdot \mathsf{fma}\left(v \cdot v, -0.625, 0.25\right) \]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{2} \cdot \mathsf{fma}\left(v \cdot v, -0.625, 0.25\right)

(FPCore (v)
 :precision binary64
 (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))
↓
(FPCore (v) :precision binary64 (* (sqrt 2.0) (fma (* v v) -0.625 0.25)))
double code(double v) {
	return ((sqrt(2.0) / 4.0) * sqrt(1.0 - (3.0 * (v * v)))) * (1.0 - (v * v));
}
↓
double code(double v) {
	return sqrt(2.0) * fma((v * v), -0.625, 0.25);
}

Error

Bits error versus v

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right) \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{2} \cdot \left(\sqrt{\mathsf{fma}\left(v, v \cdot -3, 1\right)} \cdot \left(-0.25 \cdot \mathsf{fma}\left(v, v, -1\right)\right)\right)} \]

  3. Taylor expanded in v around 0 0.3

    \[\leadsto \color{blue}{0.25 \cdot \sqrt{2} - 0.625 \cdot \left({v}^{2} \cdot \sqrt{2}\right)} \]

  4. Simplified0.3

    \[\leadsto \color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(v \cdot v, -0.625, 0.25\right)} \]

  5. Final simplification0.3

    \[\leadsto \sqrt{2} \cdot \mathsf{fma}\left(v \cdot v, -0.625, 0.25\right) \]

Reproduce

herbie shell --seed 2021275 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))