Average Error: 1.0 → 0.0
Time: 3.3s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}\right)} \cdot \left(\left(1 + v \cdot v\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)\]
\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{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}\right)} \cdot \left(\left(1 + v \cdot v\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)
double f(double v) {
        double r282587 = 4.0;
        double r282588 = 3.0;
        double r282589 = atan2(1.0, 0.0);
        double r282590 = r282588 * r282589;
        double r282591 = 1.0;
        double r282592 = v;
        double r282593 = r282592 * r282592;
        double r282594 = r282591 - r282593;
        double r282595 = r282590 * r282594;
        double r282596 = 2.0;
        double r282597 = 6.0;
        double r282598 = r282597 * r282593;
        double r282599 = r282596 - r282598;
        double r282600 = sqrt(r282599);
        double r282601 = r282595 * r282600;
        double r282602 = r282587 / r282601;
        return r282602;
}
↓
double f(double v) {
        double r282603 = 4.0;
        double r282604 = 3.0;
        double r282605 = atan2(1.0, 0.0);
        double r282606 = r282604 * r282605;
        double r282607 = r282603 / r282606;
        double r282608 = v;
        double r282609 = 1.0;
        double r282610 = fma(r282608, r282608, r282609);
        double r282611 = r282608 * r282608;
        double r282612 = r282609 - r282611;
        double r282613 = 2.0;
        double r282614 = 3.0;
        double r282615 = pow(r282613, r282614);
        double r282616 = 6.0;
        double r282617 = r282616 * r282611;
        double r282618 = pow(r282617, r282614);
        double r282619 = r282615 - r282618;
        double r282620 = sqrt(r282619);
        double r282621 = r282612 * r282620;
        double r282622 = r282610 * r282621;
        double r282623 = r282607 / r282622;
        double r282624 = r282609 + r282611;
        double r282625 = r282613 * r282613;
        double r282626 = r282617 * r282617;
        double r282627 = r282613 * r282617;
        double r282628 = r282626 + r282627;
        double r282629 = r282625 + r282628;
        double r282630 = sqrt(r282629);
        double r282631 = r282624 * r282630;
        double r282632 = r282623 * r282631;
        return r282632;
}

Error

Bits error versus v

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. Using strategy rm

  3. Applied flip3--1.0

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

  4. Applied sqrt-div1.0

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

  5. Applied flip--1.0

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

  6. Applied associate-*r/1.0

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

  7. Applied frac-times1.0

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

  8. Applied associate-/r/1.0

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

  9. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}\right)}} \cdot \left(\left(1 + v \cdot v\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)\]

  10. Final simplification0.0

    \[\leadsto \frac{\frac{4}{3 \cdot \pi}}{\mathsf{fma}\left(v, v, 1\right) \cdot \left(\left(1 - v \cdot v\right) \cdot \sqrt{{2}^{3} - {\left(6 \cdot \left(v \cdot v\right)\right)}^{3}}\right)} \cdot \left(\left(1 + v \cdot v\right) \cdot \sqrt{2 \cdot 2 + \left(\left(6 \cdot \left(v \cdot v\right)\right) \cdot \left(6 \cdot \left(v \cdot v\right)\right) + 2 \cdot \left(6 \cdot \left(v \cdot v\right)\right)\right)}\right)\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))