Average Error: 1.0 → 0.0
Time: 19.5s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\sqrt[3]{\frac{\frac{\frac{64}{27}}{\left(\pi - \left(v \cdot v\right) \cdot \pi\right) \cdot \left(\pi - \left(v \cdot v\right) \cdot \pi\right)}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\sqrt[3]{\frac{\frac{\frac{64}{27}}{\left(\pi - \left(v \cdot v\right) \cdot \pi\right) \cdot \left(\pi - \left(v \cdot v\right) \cdot \pi\right)}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r4683601 = 4.0;
        double r4683602 = 3.0;
        double r4683603 = atan2(1.0, 0.0);
        double r4683604 = r4683602 * r4683603;
        double r4683605 = 1.0;
        double r4683606 = v;
        double r4683607 = r4683606 * r4683606;
        double r4683608 = r4683605 - r4683607;
        double r4683609 = r4683604 * r4683608;
        double r4683610 = 2.0;
        double r4683611 = 6.0;
        double r4683612 = r4683611 * r4683607;
        double r4683613 = r4683610 - r4683612;
        double r4683614 = sqrt(r4683613);
        double r4683615 = r4683609 * r4683614;
        double r4683616 = r4683601 / r4683615;
        return r4683616;
}
↓
double f(double v) {
        double r4683617 = 2.3703703703703702;
        double r4683618 = atan2(1.0, 0.0);
        double r4683619 = v;
        double r4683620 = r4683619 * r4683619;
        double r4683621 = r4683620 * r4683618;
        double r4683622 = r4683618 - r4683621;
        double r4683623 = r4683622 * r4683622;
        double r4683624 = r4683617 / r4683623;
        double r4683625 = r4683624 / r4683622;
        double r4683626 = cbrt(r4683625);
        double r4683627 = -6.0;
        double r4683628 = 2.0;
        double r4683629 = fma(r4683627, r4683620, r4683628);
        double r4683630 = sqrt(r4683629);
        double r4683631 = r4683626 / r4683630;
        return r4683631;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{4}{3}}{\pi - \pi \cdot \left(v \cdot v\right)}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube1.0

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

  5. Applied add-cbrt-cube0.0

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

  6. Applied cbrt-undiv0.0

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

  7. Simplified0.0

    \[\leadsto \frac{\sqrt[3]{\color{blue}{\frac{\frac{\frac{64}{27}}{\left(\pi - \pi \cdot \left(v \cdot v\right)\right) \cdot \left(\pi - \pi \cdot \left(v \cdot v\right)\right)}}{\pi - \pi \cdot \left(v \cdot v\right)}}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]

  8. Final simplification0.0

    \[\leadsto \frac{\sqrt[3]{\frac{\frac{\frac{64}{27}}{\left(\pi - \left(v \cdot v\right) \cdot \pi\right) \cdot \left(\pi - \left(v \cdot v\right) \cdot \pi\right)}}{\pi - \left(v \cdot v\right) \cdot \pi}}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))