Average Error: 1.0 → 0.0
Time: 23.1s
Precision: 64
\[\frac{4}{\left(\left(3 \cdot \pi\right) \cdot \left(1 - v \cdot v\right)\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
↓
\[\frac{\left({\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right)\right)}^{3} + {1}^{6}\right) \cdot \frac{\frac{4}{{1}^{3} - {v}^{6}}}{3 \cdot \pi}}{\mathsf{fma}\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right), \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v, v, 1\right), -1 \cdot 1\right), {1}^{4}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}\]
\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{\left({\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right)\right)}^{3} + {1}^{6}\right) \cdot \frac{\frac{4}{{1}^{3} - {v}^{6}}}{3 \cdot \pi}}{\mathsf{fma}\left(\mathsf{fma}\left(1, v \cdot v, {v}^{4}\right), \mathsf{fma}\left(v \cdot v, \mathsf{fma}\left(v, v, 1\right), -1 \cdot 1\right), {1}^{4}\right) \cdot \sqrt{2 - 6 \cdot \left(v \cdot v\right)}}
double f(double v) {
        double r236761 = 4.0;
        double r236762 = 3.0;
        double r236763 = atan2(1.0, 0.0);
        double r236764 = r236762 * r236763;
        double r236765 = 1.0;
        double r236766 = v;
        double r236767 = r236766 * r236766;
        double r236768 = r236765 - r236767;
        double r236769 = r236764 * r236768;
        double r236770 = 2.0;
        double r236771 = 6.0;
        double r236772 = r236771 * r236767;
        double r236773 = r236770 - r236772;
        double r236774 = sqrt(r236773);
        double r236775 = r236769 * r236774;
        double r236776 = r236761 / r236775;
        return r236776;
}
↓
double f(double v) {
        double r236777 = 1.0;
        double r236778 = v;
        double r236779 = r236778 * r236778;
        double r236780 = 4.0;
        double r236781 = pow(r236778, r236780);
        double r236782 = fma(r236777, r236779, r236781);
        double r236783 = 3.0;
        double r236784 = pow(r236782, r236783);
        double r236785 = 6.0;
        double r236786 = pow(r236777, r236785);
        double r236787 = r236784 + r236786;
        double r236788 = 4.0;
        double r236789 = pow(r236777, r236783);
        double r236790 = pow(r236778, r236785);
        double r236791 = r236789 - r236790;
        double r236792 = r236788 / r236791;
        double r236793 = 3.0;
        double r236794 = atan2(1.0, 0.0);
        double r236795 = r236793 * r236794;
        double r236796 = r236792 / r236795;
        double r236797 = r236787 * r236796;
        double r236798 = fma(r236778, r236778, r236777);
        double r236799 = r236777 * r236777;
        double r236800 = -r236799;
        double r236801 = fma(r236779, r236798, r236800);
        double r236802 = pow(r236777, r236780);
        double r236803 = fma(r236782, r236801, r236802);
        double r236804 = 2.0;
        double r236805 = 6.0;
        double r236806 = r236805 * r236779;
        double r236807 = r236804 - r236806;
        double r236808 = sqrt(r236807);
        double r236809 = r236803 * r236808;
        double r236810 = r236797 / r236809;
        return r236810;
}

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

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

  4. Applied associate-*r/1.0

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

  5. Applied associate-*l/1.0

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

  6. Applied associate-/r/1.0

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

  7. Simplified0.0

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

  9. Applied flip3-+0.0

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

  10. Applied frac-times0.0

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

  11. Simplified0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Equation (22+)"
  :precision binary64
  (/ 4 (* (* (* 3 PI) (- 1 (* v v))) (sqrt (- 2 (* 6 (* v v)))))))