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

Average Error: 0.4 → 0.3

Time: 33.5s

Precision: 64

Math

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

↓

\[\left(\left(\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}}{{1}^{6} - {\left({v}^{4}\right)}^{3}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi \cdot \sqrt{\left({1}^{4} - {v}^{8} \cdot {3}^{4}\right) \cdot 2}}\right) \cdot \left(\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left({v}^{4} \cdot {v}^{4} + \left(1 \cdot 1\right) \cdot {v}^{4}\right)\right)\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

C

double f(double v, double t) {
        double r186465 = 1.0;
        double r186466 = 5.0;
        double r186467 = v;
        double r186468 = r186467 * r186467;
        double r186469 = r186466 * r186468;
        double r186470 = r186465 - r186469;
        double r186471 = atan2(1.0, 0.0);
        double r186472 = t;
        double r186473 = r186471 * r186472;
        double r186474 = 2.0;
        double r186475 = 3.0;
        double r186476 = r186475 * r186468;
        double r186477 = r186465 - r186476;
        double r186478 = r186474 * r186477;
        double r186479 = sqrt(r186478);
        double r186480 = r186473 * r186479;
        double r186481 = r186465 - r186468;
        double r186482 = r186480 * r186481;
        double r186483 = r186470 / r186482;
        return r186483;
}

↓

double f(double v, double t) {
        double r186484 = 1.0;
        double r186485 = 5.0;
        double r186486 = v;
        double r186487 = r186486 * r186486;
        double r186488 = r186485 * r186487;
        double r186489 = r186484 - r186488;
        double r186490 = sqrt(r186489);
        double r186491 = t;
        double r186492 = r186490 / r186491;
        double r186493 = 6.0;
        double r186494 = pow(r186484, r186493);
        double r186495 = 4.0;
        double r186496 = pow(r186486, r186495);
        double r186497 = 3.0;
        double r186498 = pow(r186496, r186497);
        double r186499 = r186494 - r186498;
        double r186500 = r186492 / r186499;
        double r186501 = atan2(1.0, 0.0);
        double r186502 = pow(r186484, r186495);
        double r186503 = 8.0;
        double r186504 = pow(r186486, r186503);
        double r186505 = 3.0;
        double r186506 = pow(r186505, r186495);
        double r186507 = r186504 * r186506;
        double r186508 = r186502 - r186507;
        double r186509 = 2.0;
        double r186510 = r186508 * r186509;
        double r186511 = sqrt(r186510);
        double r186512 = r186501 * r186511;
        double r186513 = r186490 / r186512;
        double r186514 = r186500 * r186513;
        double r186515 = r186484 * r186484;
        double r186516 = r186505 * r186505;
        double r186517 = r186516 * r186496;
        double r186518 = r186515 + r186517;
        double r186519 = sqrt(r186518);
        double r186520 = r186515 * r186515;
        double r186521 = r186496 * r186496;
        double r186522 = r186515 * r186496;
        double r186523 = r186521 + r186522;
        double r186524 = r186520 + r186523;
        double r186525 = r186519 * r186524;
        double r186526 = r186514 * r186525;
        double r186527 = r186505 * r186487;
        double r186528 = r186484 + r186527;
        double r186529 = sqrt(r186528);
        double r186530 = r186484 + r186487;
        double r186531 = r186529 * r186530;
        double r186532 = r186526 * r186531;
        return r186532;
}

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 flip-- 0.4

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

4. Applied flip-- 0.4

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

5. Applied associate-*r/ 0.4

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

6. Applied sqrt-div 0.4

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

7. Applied associate-*r/ 0.4

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

8. Applied frac-times 0.4

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

9. Applied associate-/r/ 0.4

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

10. Simplified 0.4

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

12. Applied flip3-- 0.4

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

13. Applied flip-- 0.4

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

14. Applied associate-*r/ 0.4

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

15. Applied sqrt-div 0.4

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

16. Applied associate-*r/ 0.4

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

17. Applied frac-times 0.4

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

18. Applied associate-/r/ 0.4

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

19. Simplified 0.4

    \[\leadsto \left(\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{t \cdot \pi}}{\left({1}^{6} - {\left({v}^{4}\right)}^{3}\right) \cdot \sqrt{\left(1 \cdot {1}^{3} - {v}^{8} \cdot \left({3}^{3} \cdot 3\right)\right) \cdot 2}}} \cdot \left(\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left({v}^{4} \cdot {v}^{4} + \left(1 \cdot 1\right) \cdot {v}^{4}\right)\right)\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]
20. Using strategy rm

21. Applied add-sqr-sqrt 0.4

    \[\leadsto \left(\frac{\frac{\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}{t \cdot \pi}}{\left({1}^{6} - {\left({v}^{4}\right)}^{3}\right) \cdot \sqrt{\left(1 \cdot {1}^{3} - {v}^{8} \cdot \left({3}^{3} \cdot 3\right)\right) \cdot 2}} \cdot \left(\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left({v}^{4} \cdot {v}^{4} + \left(1 \cdot 1\right) \cdot {v}^{4}\right)\right)\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

22. Applied times-frac 0.4

    \[\leadsto \left(\frac{\color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}}{\left({1}^{6} - {\left({v}^{4}\right)}^{3}\right) \cdot \sqrt{\left(1 \cdot {1}^{3} - {v}^{8} \cdot \left({3}^{3} \cdot 3\right)\right) \cdot 2}} \cdot \left(\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left({v}^{4} \cdot {v}^{4} + \left(1 \cdot 1\right) \cdot {v}^{4}\right)\right)\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

23. Applied times-frac 0.3

    \[\leadsto \left(\color{blue}{\left(\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}}{{1}^{6} - {\left({v}^{4}\right)}^{3}} \cdot \frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{\sqrt{\left(1 \cdot {1}^{3} - {v}^{8} \cdot \left({3}^{3} \cdot 3\right)\right) \cdot 2}}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left({v}^{4} \cdot {v}^{4} + \left(1 \cdot 1\right) \cdot {v}^{4}\right)\right)\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

24. Simplified 0.3

    \[\leadsto \left(\left(\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}}{{1}^{6} - {\left({v}^{4}\right)}^{3}} \cdot \color{blue}{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi \cdot \sqrt{\left({1}^{4} - {v}^{8} \cdot {3}^{4}\right) \cdot 2}}}\right) \cdot \left(\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left({v}^{4} \cdot {v}^{4} + \left(1 \cdot 1\right) \cdot {v}^{4}\right)\right)\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

25. Final simplification 0.3

    \[\leadsto \left(\left(\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{t}}{{1}^{6} - {\left({v}^{4}\right)}^{3}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi \cdot \sqrt{\left({1}^{4} - {v}^{8} \cdot {3}^{4}\right) \cdot 2}}\right) \cdot \left(\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}} \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left({v}^{4} \cdot {v}^{4} + \left(1 \cdot 1\right) \cdot {v}^{4}\right)\right)\right)\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

Reproduce

herbie shell --seed 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)))))