Average Error: 1.0 → 0.0
Time: 7.1s
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 r300615 = 4.0;
        double r300616 = 3.0;
        double r300617 = atan2(1.0, 0.0);
        double r300618 = r300616 * r300617;
        double r300619 = 1.0;
        double r300620 = v;
        double r300621 = r300620 * r300620;
        double r300622 = r300619 - r300621;
        double r300623 = r300618 * r300622;
        double r300624 = 2.0;
        double r300625 = 6.0;
        double r300626 = r300625 * r300621;
        double r300627 = r300624 - r300626;
        double r300628 = sqrt(r300627);
        double r300629 = r300623 * r300628;
        double r300630 = r300615 / r300629;
        return r300630;
}
↓
double f(double v) {
        double r300631 = 4.0;
        double r300632 = 3.0;
        double r300633 = atan2(1.0, 0.0);
        double r300634 = r300632 * r300633;
        double r300635 = r300631 / r300634;
        double r300636 = v;
        double r300637 = 1.0;
        double r300638 = fma(r300636, r300636, r300637);
        double r300639 = r300636 * r300636;
        double r300640 = r300637 - r300639;
        double r300641 = 2.0;
        double r300642 = 3.0;
        double r300643 = pow(r300641, r300642);
        double r300644 = 6.0;
        double r300645 = r300644 * r300639;
        double r300646 = pow(r300645, r300642);
        double r300647 = r300643 - r300646;
        double r300648 = sqrt(r300647);
        double r300649 = r300640 * r300648;
        double r300650 = r300638 * r300649;
        double r300651 = r300635 / r300650;
        double r300652 = r300637 + r300639;
        double r300653 = r300641 * r300641;
        double r300654 = r300645 * r300645;
        double r300655 = r300641 * r300645;
        double r300656 = r300654 + r300655;
        double r300657 = r300653 + r300656;
        double r300658 = sqrt(r300657);
        double r300659 = r300652 * r300658;
        double r300660 = r300651 * r300659;
        return r300660;
}

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 2020057 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))