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

Average Error: 0.5 → 0.5

Time: 36.9s

Precision: 64

Math

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

↓

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

C

double f(double v, double t) {
        double r12414520 = 1.0;
        double r12414521 = 5.0;
        double r12414522 = v;
        double r12414523 = r12414522 * r12414522;
        double r12414524 = r12414521 * r12414523;
        double r12414525 = r12414520 - r12414524;
        double r12414526 = atan2(1.0, 0.0);
        double r12414527 = t;
        double r12414528 = r12414526 * r12414527;
        double r12414529 = 2.0;
        double r12414530 = 3.0;
        double r12414531 = r12414530 * r12414523;
        double r12414532 = r12414520 - r12414531;
        double r12414533 = r12414529 * r12414532;
        double r12414534 = sqrt(r12414533);
        double r12414535 = r12414528 * r12414534;
        double r12414536 = r12414520 - r12414523;
        double r12414537 = r12414535 * r12414536;
        double r12414538 = r12414525 / r12414537;
        return r12414538;
}

↓

double f(double v, double t) {
        double r12414539 = 1.0;
        double r12414540 = r12414539 * r12414539;
        double r12414541 = 5.0;
        double r12414542 = v;
        double r12414543 = r12414542 * r12414542;
        double r12414544 = r12414541 * r12414543;
        double r12414545 = r12414544 * r12414544;
        double r12414546 = r12414540 - r12414545;
        double r12414547 = atan2(1.0, 0.0);
        double r12414548 = t;
        double r12414549 = r12414547 * r12414548;
        double r12414550 = 2.0;
        double r12414551 = 3.0;
        double r12414552 = r12414551 * r12414543;
        double r12414553 = r12414552 * r12414552;
        double r12414554 = r12414540 - r12414553;
        double r12414555 = r12414550 * r12414554;
        double r12414556 = sqrt(r12414555);
        double r12414557 = r12414549 * r12414556;
        double r12414558 = r12414539 * r12414540;
        double r12414559 = r12414543 * r12414543;
        double r12414560 = r12414543 * r12414559;
        double r12414561 = r12414558 - r12414560;
        double r12414562 = r12414557 * r12414561;
        double r12414563 = r12414562 * r12414546;
        double r12414564 = r12414546 / r12414563;
        double r12414565 = r12414539 + r12414552;
        double r12414566 = sqrt(r12414565);
        double r12414567 = r12414539 * r12414543;
        double r12414568 = r12414559 + r12414567;
        double r12414569 = r12414540 + r12414568;
        double r12414570 = r12414566 * r12414569;
        double r12414571 = r12414539 - r12414544;
        double r12414572 = r12414570 * r12414571;
        double r12414573 = r12414564 * r12414572;
        return r12414573;
}

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

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

4. Applied associate-/l/ 0.5

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

6. Applied flip-+ 0.5

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

7. Applied flip3-- 0.5

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

8. Applied flip-- 0.5

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

9. Applied associate-*r/ 0.5

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

10. Applied sqrt-div 0.5

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

11. Applied associate-*r/ 0.5

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

12. Applied frac-times 0.5

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

13. Applied frac-times 0.5

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

14. Applied associate-/r/ 0.5

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

15. Simplified 0.5

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

16. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019173 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))