Average Error: 1.0 → 0.0
Time: 10.4s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}\]
\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{\frac{\frac{4}{3}}{\pi - \left(v \cdot v\right) \cdot \pi}}{\sqrt{\mathsf{fma}\left(-6, v \cdot v, 2\right)}}
double f(double v) {
        double r4088681 = 4.0;
        double r4088682 = 3.0;
        double r4088683 = atan2(1.0, 0.0);
        double r4088684 = r4088682 * r4088683;
        double r4088685 = 1.0;
        double r4088686 = v;
        double r4088687 = r4088686 * r4088686;
        double r4088688 = r4088685 - r4088687;
        double r4088689 = r4088684 * r4088688;
        double r4088690 = 2.0;
        double r4088691 = 6.0;
        double r4088692 = r4088691 * r4088687;
        double r4088693 = r4088690 - r4088692;
        double r4088694 = sqrt(r4088693);
        double r4088695 = r4088689 * r4088694;
        double r4088696 = r4088681 / r4088695;
        return r4088696;
}
↓
double f(double v) {
        double r4088697 = 1.3333333333333333;
        double r4088698 = atan2(1.0, 0.0);
        double r4088699 = v;
        double r4088700 = r4088699 * r4088699;
        double r4088701 = r4088700 * r4088698;
        double r4088702 = r4088698 - r4088701;
        double r4088703 = r4088697 / r4088702;
        double r4088704 = -6.0;
        double r4088705 = 2.0;
        double r4088706 = fma(r4088704, r4088700, r4088705);
        double r4088707 = sqrt(r4088706);
        double r4088708 = r4088703 / r4088707;
        return r4088708;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))