Average Error: 1.0 → 0.2
Time: 4.2s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{4}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}\]
\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
↓
\frac{4}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}
double code(double v) {
	return (4.0 / (((3.0 * ((double) M_PI)) * (1.0 - (v * v))) * sqrt((2.0 - (6.0 * (v * v))))));
}
↓
double code(double v) {
	return (4.0 / fma((3.0 * sqrt(2.0)), ((double) M_PI), ((9.0 * ((pow(v, 4.0) * ((double) M_PI)) / sqrt(2.0))) - fma(9.0, ((pow(v, 2.0) * ((double) M_PI)) / sqrt(2.0)), fma(13.5, ((pow(v, 4.0) * ((double) M_PI)) / pow(sqrt(2.0), 3.0)), (3.0 * (sqrt(2.0) * (pow(v, 2.0) * ((double) M_PI)))))))));
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.0

    \[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]

  2. Taylor expanded around 0 1.2

    \[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \left(\sqrt{2} \cdot \pi\right) + 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}}\right) - \left(9 \cdot \frac{{v}^{2} \cdot \pi}{\sqrt{2}} + \left(13.5 \cdot \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}} + 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)}}\]

  3. Simplified0.2

    \[\leadsto \frac{4}{\color{blue}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}}\]

  4. Final simplification0.2

    \[\leadsto \frac{4}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}\]

Reproduce

herbie shell --seed 2020058 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))