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

Average Error: 0.4 → 0.3

Time: 26.5s

Precision: 64

Math

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

↓

\[\left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(\sqrt{\left({1}^{6} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{6}\right) \cdot 2} \cdot t\right)} \cdot \left(\sqrt{{1}^{3} + {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

C

double f(double v, double t) {
        double r193511 = 1.0;
        double r193512 = 5.0;
        double r193513 = v;
        double r193514 = r193513 * r193513;
        double r193515 = r193512 * r193514;
        double r193516 = r193511 - r193515;
        double r193517 = atan2(1.0, 0.0);
        double r193518 = t;
        double r193519 = r193517 * r193518;
        double r193520 = 2.0;
        double r193521 = 3.0;
        double r193522 = r193521 * r193514;
        double r193523 = r193511 - r193522;
        double r193524 = r193520 * r193523;
        double r193525 = sqrt(r193524);
        double r193526 = r193519 * r193525;
        double r193527 = r193511 - r193514;
        double r193528 = r193526 * r193527;
        double r193529 = r193516 / r193528;
        return r193529;
}

↓

double f(double v, double t) {
        double r193530 = 1.0;
        double r193531 = 5.0;
        double r193532 = v;
        double r193533 = r193532 * r193532;
        double r193534 = r193531 * r193533;
        double r193535 = r193530 - r193534;
        double r193536 = atan2(1.0, 0.0);
        double r193537 = r193535 / r193536;
        double r193538 = 6.0;
        double r193539 = pow(r193530, r193538);
        double r193540 = 12.0;
        double r193541 = pow(r193532, r193540);
        double r193542 = r193539 - r193541;
        double r193543 = 3.0;
        double r193544 = r193543 * r193533;
        double r193545 = pow(r193544, r193538);
        double r193546 = r193539 - r193545;
        double r193547 = 2.0;
        double r193548 = r193546 * r193547;
        double r193549 = sqrt(r193548);
        double r193550 = t;
        double r193551 = r193549 * r193550;
        double r193552 = r193542 * r193551;
        double r193553 = r193537 / r193552;
        double r193554 = 3.0;
        double r193555 = pow(r193530, r193554);
        double r193556 = pow(r193544, r193554);
        double r193557 = r193555 + r193556;
        double r193558 = sqrt(r193557);
        double r193559 = pow(r193532, r193538);
        double r193560 = r193555 + r193559;
        double r193561 = r193558 * r193560;
        double r193562 = r193553 * r193561;
        double r193563 = r193530 * r193530;
        double r193564 = r193544 * r193544;
        double r193565 = r193530 * r193544;
        double r193566 = r193564 + r193565;
        double r193567 = r193563 + r193566;
        double r193568 = sqrt(r193567);
        double r193569 = r193533 * r193533;
        double r193570 = r193530 * r193533;
        double r193571 = r193569 + r193570;
        double r193572 = r193563 + r193571;
        double r193573 = r193568 * r193572;
        double r193574 = r193562 * r193573;
        return r193574;
}

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

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

15. Applied flip-- 0.3

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

16. Applied flip-- 0.3

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

17. Applied associate-*r/ 0.3

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

18. Applied sqrt-div 0.3

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

19. Applied associate-*r/ 0.3

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

20. Applied frac-times 0.3

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

21. Applied associate-/r/ 0.3

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

22. Simplified 0.3

    \[\leadsto \left(\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(\sqrt{\left({1}^{6} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{6}\right) \cdot 2} \cdot t\right)}} \cdot \left(\sqrt{{1}^{3} + {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

23. Final simplification 0.3

    \[\leadsto \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(\sqrt{\left({1}^{6} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{6}\right) \cdot 2} \cdot t\right)} \cdot \left(\sqrt{{1}^{3} + {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(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)))))