Average Error: 1.0 → 0.0
Time: 4.6s
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 r217726 = 4.0;
        double r217727 = 3.0;
        double r217728 = atan2(1.0, 0.0);
        double r217729 = r217727 * r217728;
        double r217730 = 1.0;
        double r217731 = v;
        double r217732 = r217731 * r217731;
        double r217733 = r217730 - r217732;
        double r217734 = r217729 * r217733;
        double r217735 = 2.0;
        double r217736 = 6.0;
        double r217737 = r217736 * r217732;
        double r217738 = r217735 - r217737;
        double r217739 = sqrt(r217738);
        double r217740 = r217734 * r217739;
        double r217741 = r217726 / r217740;
        return r217741;
}
↓
double f(double v) {
        double r217742 = 4.0;
        double r217743 = 3.0;
        double r217744 = atan2(1.0, 0.0);
        double r217745 = r217743 * r217744;
        double r217746 = r217742 / r217745;
        double r217747 = v;
        double r217748 = 1.0;
        double r217749 = fma(r217747, r217747, r217748);
        double r217750 = r217747 * r217747;
        double r217751 = r217748 - r217750;
        double r217752 = 2.0;
        double r217753 = 3.0;
        double r217754 = pow(r217752, r217753);
        double r217755 = 6.0;
        double r217756 = r217755 * r217750;
        double r217757 = pow(r217756, r217753);
        double r217758 = r217754 - r217757;
        double r217759 = sqrt(r217758);
        double r217760 = r217751 * r217759;
        double r217761 = r217749 * r217760;
        double r217762 = r217746 / r217761;
        double r217763 = r217748 + r217750;
        double r217764 = r217752 * r217752;
        double r217765 = r217756 * r217756;
        double r217766 = r217752 * r217756;
        double r217767 = r217765 + r217766;
        double r217768 = r217764 + r217767;
        double r217769 = sqrt(r217768);
        double r217770 = r217763 * r217769;
        double r217771 = r217762 * r217770;
        return r217771;
}

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 2019347 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))