Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.3

Time: 27.8s

Precision: 64

Math

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

↓

\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{3} - {v}^{6}\right) \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

C

double f(double v, double t) {
        double r184741 = 1.0;
        double r184742 = 5.0;
        double r184743 = v;
        double r184744 = r184743 * r184743;
        double r184745 = r184742 * r184744;
        double r184746 = r184741 - r184745;
        double r184747 = atan2(1.0, 0.0);
        double r184748 = t;
        double r184749 = r184747 * r184748;
        double r184750 = 2.0;
        double r184751 = 3.0;
        double r184752 = r184751 * r184744;
        double r184753 = r184741 - r184752;
        double r184754 = r184750 * r184753;
        double r184755 = sqrt(r184754);
        double r184756 = r184749 * r184755;
        double r184757 = r184741 - r184744;
        double r184758 = r184756 * r184757;
        double r184759 = r184746 / r184758;
        return r184759;
}

↓

double f(double v, double t) {
        double r184760 = 1.0;
        double r184761 = 5.0;
        double r184762 = v;
        double r184763 = r184762 * r184762;
        double r184764 = r184761 * r184763;
        double r184765 = r184760 - r184764;
        double r184766 = atan2(1.0, 0.0);
        double r184767 = r184765 / r184766;
        double r184768 = 3.0;
        double r184769 = pow(r184760, r184768);
        double r184770 = 6.0;
        double r184771 = pow(r184762, r184770);
        double r184772 = r184769 - r184771;
        double r184773 = t;
        double r184774 = 2.0;
        double r184775 = 3.0;
        double r184776 = r184775 * r184763;
        double r184777 = pow(r184776, r184768);
        double r184778 = r184769 - r184777;
        double r184779 = r184774 * r184778;
        double r184780 = sqrt(r184779);
        double r184781 = r184773 * r184780;
        double r184782 = r184772 * r184781;
        double r184783 = r184767 / r184782;
        double r184784 = r184760 * r184760;
        double r184785 = r184776 * r184776;
        double r184786 = r184760 * r184776;
        double r184787 = r184785 + r184786;
        double r184788 = r184784 + r184787;
        double r184789 = sqrt(r184788);
        double r184790 = r184763 * r184763;
        double r184791 = r184760 * r184763;
        double r184792 = r184790 + r184791;
        double r184793 = r184784 + r184792;
        double r184794 = r184789 * r184793;
        double r184795 = r184783 * r184794;
        return r184795;
}

Error

Bits error versus v

Bits error versus t

Derivation

1. Initial program 0.4

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
2. Using strategy rm

3. Applied associate-*l* 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right)} \cdot \left(1 - v \cdot v\right)}\]
4. Using strategy rm

5. Applied flip3-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right)\right) \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}}\]

6. Applied flip3-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \color{blue}{\frac{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right)\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

7. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{\color{blue}{\frac{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right)\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

8. Applied sqrt-div 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \color{blue}{\frac{\sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right)\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

9. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \color{blue}{\frac{t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

10. Applied associate-*r/ 0.4

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

11. Applied frac-times 0.4

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}}}\]

12. Applied associate-/r/ 0.4

    \[\leadsto \color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\pi \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)}\]

13. Simplified 0.3

    \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{3} - {v}^{6}\right) \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)}} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

14. Final simplification 0.3

    \[\leadsto \frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{3} - {v}^{6}\right) \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019303 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))