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

Average Error: 0.5 → 0.1

Time: 20.9s

Precision: 64

Math

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

↓

\[\left(\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\frac{\frac{\frac{\frac{1}{1 \cdot 1 - {v}^{4}}}{\pi}}{\sqrt{\left({1}^{3} \cdot 1 - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right) \cdot 2}}}{t} \cdot \sqrt{1 \cdot 1 + {v}^{4} \cdot \left(3 \cdot 3\right)}\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

C

double f(double v, double t) {
        double r190667 = 1.0;
        double r190668 = 5.0;
        double r190669 = v;
        double r190670 = r190669 * r190669;
        double r190671 = r190668 * r190670;
        double r190672 = r190667 - r190671;
        double r190673 = atan2(1.0, 0.0);
        double r190674 = t;
        double r190675 = r190673 * r190674;
        double r190676 = 2.0;
        double r190677 = 3.0;
        double r190678 = r190677 * r190670;
        double r190679 = r190667 - r190678;
        double r190680 = r190676 * r190679;
        double r190681 = sqrt(r190680);
        double r190682 = r190675 * r190681;
        double r190683 = r190667 - r190670;
        double r190684 = r190682 * r190683;
        double r190685 = r190672 / r190684;
        return r190685;
}

↓

double f(double v, double t) {
        double r190686 = 1.0;
        double r190687 = 5.0;
        double r190688 = v;
        double r190689 = r190688 * r190688;
        double r190690 = r190687 * r190689;
        double r190691 = r190686 - r190690;
        double r190692 = 1.0;
        double r190693 = r190686 * r190686;
        double r190694 = 4.0;
        double r190695 = pow(r190688, r190694);
        double r190696 = r190693 - r190695;
        double r190697 = r190692 / r190696;
        double r190698 = atan2(1.0, 0.0);
        double r190699 = r190697 / r190698;
        double r190700 = 3.0;
        double r190701 = pow(r190686, r190700);
        double r190702 = r190701 * r190686;
        double r190703 = 3.0;
        double r190704 = pow(r190703, r190700);
        double r190705 = r190703 * r190704;
        double r190706 = 8.0;
        double r190707 = pow(r190688, r190706);
        double r190708 = r190705 * r190707;
        double r190709 = r190702 - r190708;
        double r190710 = 2.0;
        double r190711 = r190709 * r190710;
        double r190712 = sqrt(r190711);
        double r190713 = r190699 / r190712;
        double r190714 = t;
        double r190715 = r190713 / r190714;
        double r190716 = r190703 * r190703;
        double r190717 = r190695 * r190716;
        double r190718 = r190693 + r190717;
        double r190719 = sqrt(r190718);
        double r190720 = r190715 * r190719;
        double r190721 = r190691 * r190720;
        double r190722 = r190703 * r190689;
        double r190723 = r190686 + r190722;
        double r190724 = sqrt(r190723);
        double r190725 = r190686 + r190689;
        double r190726 = r190724 * r190725;
        double r190727 = r190721 * r190726;
        return r190727;
}

Error

Bits error versus v

Bits error versus t

Derivation

1. Initial program 0.5

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

3. Applied flip-- 0.5

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

4. Applied flip-- 0.5

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

5. Applied associate-*r/ 0.5

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

6. Applied sqrt-div 0.5

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

7. Applied associate-*r/ 0.5

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

8. Applied frac-times 0.5

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

9. Applied associate-/r/ 0.5

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

10. Simplified 0.5

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

12. Applied div-inv 0.5

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

13. Simplified 0.3

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

15. Applied flip-- 0.3

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

16. Applied associate-*l/ 0.3

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

17. Applied sqrt-div 0.3

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

18. Applied associate-*r/ 0.3

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

19. Applied associate-/r/ 0.3

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

20. Simplified 0.1

    \[\leadsto \left(\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\color{blue}{\frac{\frac{\frac{\frac{1}{1 \cdot 1 - {v}^{4}}}{\pi}}{\sqrt{\left({1}^{3} \cdot 1 - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right) \cdot 2}}}{t}} \cdot \sqrt{1 \cdot 1 + {v}^{4} \cdot \left(3 \cdot 3\right)}\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

21. Final simplification 0.1

    \[\leadsto \left(\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\frac{\frac{\frac{\frac{1}{1 \cdot 1 - {v}^{4}}}{\pi}}{\sqrt{\left({1}^{3} \cdot 1 - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right) \cdot 2}}}{t} \cdot \sqrt{1 \cdot 1 + {v}^{4} \cdot \left(3 \cdot 3\right)}\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

Reproduce

herbie shell --seed 2019208 
(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)))))