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

Average Error: 0.5 → 0.2

Time: 21.9s

Precision: 64

Math

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

↓

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

C

double f(double v, double t) {
        double r163601 = 1.0;
        double r163602 = 5.0;
        double r163603 = v;
        double r163604 = r163603 * r163603;
        double r163605 = r163602 * r163604;
        double r163606 = r163601 - r163605;
        double r163607 = atan2(1.0, 0.0);
        double r163608 = t;
        double r163609 = r163607 * r163608;
        double r163610 = 2.0;
        double r163611 = 3.0;
        double r163612 = r163611 * r163604;
        double r163613 = r163601 - r163612;
        double r163614 = r163610 * r163613;
        double r163615 = sqrt(r163614);
        double r163616 = r163609 * r163615;
        double r163617 = r163601 - r163604;
        double r163618 = r163616 * r163617;
        double r163619 = r163606 / r163618;
        return r163619;
}

↓

double f(double v, double t) {
        double r163620 = 1.0;
        double r163621 = 3.0;
        double r163622 = pow(r163620, r163621);
        double r163623 = 5.0;
        double r163624 = v;
        double r163625 = r163624 * r163624;
        double r163626 = r163623 * r163625;
        double r163627 = pow(r163626, r163621);
        double r163628 = r163622 - r163627;
        double r163629 = sqrt(r163628);
        double r163630 = r163620 - r163626;
        double r163631 = sqrt(r163630);
        double r163632 = atan2(1.0, 0.0);
        double r163633 = r163631 / r163632;
        double r163634 = r163620 * r163620;
        double r163635 = 4.0;
        double r163636 = pow(r163624, r163635);
        double r163637 = 3.0;
        double r163638 = r163637 * r163637;
        double r163639 = r163636 * r163638;
        double r163640 = r163634 - r163639;
        double r163641 = 2.0;
        double r163642 = r163640 * r163641;
        double r163643 = sqrt(r163642);
        double r163644 = r163633 / r163643;
        double r163645 = r163629 * r163644;
        double r163646 = t;
        double r163647 = r163626 + r163620;
        double r163648 = r163626 * r163647;
        double r163649 = r163634 + r163648;
        double r163650 = sqrt(r163649);
        double r163651 = r163646 * r163650;
        double r163652 = r163645 / r163651;
        double r163653 = pow(r163625, r163621);
        double r163654 = r163622 - r163653;
        double r163655 = r163652 / r163654;
        double r163656 = r163637 * r163625;
        double r163657 = r163620 + r163656;
        double r163658 = sqrt(r163657);
        double r163659 = r163625 * r163625;
        double r163660 = r163620 * r163625;
        double r163661 = r163659 + r163660;
        double r163662 = r163634 + r163661;
        double r163663 = r163658 * r163662;
        double r163664 = r163655 * r163663;
        return r163664;
}

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 flip3-- 0.5

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

4. Applied flip-- 0.5

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

5. Applied associate-*r/ 0.5

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

6. Applied sqrt-div 0.5

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

7. Applied associate-*r/ 0.5

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

8. Applied frac-times 0.5

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

9. Applied associate-/r/ 0.5

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

10. Simplified 0.4

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

12. Applied *-un-lft-identity 0.4

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

13. Applied add-sqr-sqrt 0.5

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

14. Applied times-frac 0.4

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

15. Applied times-frac 0.4

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

16. Simplified 0.4

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

17. Simplified 0.4

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

19. Applied associate-*l/ 0.2

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

21. Applied flip3-- 0.2

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

22. Applied sqrt-div 0.2

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

23. Applied associate-*l/ 0.2

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

24. Applied associate-/l/ 0.2

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

25. Simplified 0.2

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

26. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019212 
(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)))))