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

↓

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

\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 f(double v) {
        double r229604 = 4.0;
        double r229605 = 3.0;
        double r229606 = atan2(1.0, 0.0);
        double r229607 = r229605 * r229606;
        double r229608 = 1.0;
        double r229609 = v;
        double r229610 = r229609 * r229609;
        double r229611 = r229608 - r229610;
        double r229612 = r229607 * r229611;
        double r229613 = 2.0;
        double r229614 = 6.0;
        double r229615 = r229614 * r229610;
        double r229616 = r229613 - r229615;
        double r229617 = sqrt(r229616);
        double r229618 = r229612 * r229617;
        double r229619 = r229604 / r229618;
        return r229619;
}

↓

double f(double v) {
        double r229620 = 4.0;
        double r229621 = 3.0;
        double r229622 = 2.0;
        double r229623 = sqrt(r229622);
        double r229624 = r229621 * r229623;
        double r229625 = atan2(1.0, 0.0);
        double r229626 = 9.0;
        double r229627 = v;
        double r229628 = 4.0;
        double r229629 = pow(r229627, r229628);
        double r229630 = r229629 * r229625;
        double r229631 = r229630 / r229623;
        double r229632 = r229626 * r229631;
        double r229633 = 2.0;
        double r229634 = pow(r229627, r229633);
        double r229635 = r229634 * r229625;
        double r229636 = r229635 / r229623;
        double r229637 = 13.5;
        double r229638 = 3.0;
        double r229639 = pow(r229623, r229638);
        double r229640 = r229630 / r229639;
        double r229641 = r229623 * r229635;
        double r229642 = r229621 * r229641;
        double r229643 = fma(r229637, r229640, r229642);
        double r229644 = fma(r229626, r229636, r229643);
        double r229645 = r229632 - r229644;
        double r229646 = fma(r229624, r229625, r229645);
        double r229647 = r229620 / r229646;
        return r229647;
}

Derivation

Initial program 1.0
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
Taylor expanded around 0 1.2
\[\leadsto \frac{4}{\color{blue}{\left(3 \cdot \left(\sqrt{2} \cdot \pi\right) + 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}}\right) - \left(9 \cdot \frac{{v}^{2} \cdot \pi}{\sqrt{2}} + \left(13.5 \cdot \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}} + 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)}}\]
Simplified0.2
\[\leadsto \frac{4}{\color{blue}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}}\]
Final simplification0.2
\[\leadsto \frac{4}{\mathsf{fma}\left(3 \cdot \sqrt{2}, \pi, 9 \cdot \frac{{v}^{4} \cdot \pi}{\sqrt{2}} - \mathsf{fma}\left(9, \frac{{v}^{2} \cdot \pi}{\sqrt{2}}, \mathsf{fma}\left(13.5, \frac{{v}^{4} \cdot \pi}{{\left(\sqrt{2}\right)}^{3}}, 3 \cdot \left(\sqrt{2} \cdot \left({v}^{2} \cdot \pi\right)\right)\right)\right)\right)}\]

Error

Derivation

Reproduce