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

Average Error: 0.5 → 0.5

Time: 27.6s

Precision: 64

Math

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

↓

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

C

double f(double v, double t) {
        double r194525 = 1.0;
        double r194526 = 5.0;
        double r194527 = v;
        double r194528 = r194527 * r194527;
        double r194529 = r194526 * r194528;
        double r194530 = r194525 - r194529;
        double r194531 = atan2(1.0, 0.0);
        double r194532 = t;
        double r194533 = r194531 * r194532;
        double r194534 = 2.0;
        double r194535 = 3.0;
        double r194536 = r194535 * r194528;
        double r194537 = r194525 - r194536;
        double r194538 = r194534 * r194537;
        double r194539 = sqrt(r194538);
        double r194540 = r194533 * r194539;
        double r194541 = r194525 - r194528;
        double r194542 = r194540 * r194541;
        double r194543 = r194530 / r194542;
        return r194543;
}

↓

double f(double v, double t) {
        double r194544 = 1.0;
        double r194545 = 5.0;
        double r194546 = v;
        double r194547 = r194546 * r194546;
        double r194548 = r194545 * r194547;
        double r194549 = r194544 - r194548;
        double r194550 = atan2(1.0, 0.0);
        double r194551 = t;
        double r194552 = r194550 * r194551;
        double r194553 = 2.0;
        double r194554 = 3.0;
        double r194555 = pow(r194544, r194554);
        double r194556 = 3.0;
        double r194557 = r194556 * r194547;
        double r194558 = pow(r194557, r194554);
        double r194559 = r194555 - r194558;
        double r194560 = r194553 * r194559;
        double r194561 = sqrt(r194560);
        double r194562 = cbrt(r194561);
        double r194563 = r194562 * r194562;
        double r194564 = r194552 * r194563;
        double r194565 = sqrt(r194553);
        double r194566 = cbrt(r194565);
        double r194567 = r194564 * r194566;
        double r194568 = sqrt(r194559);
        double r194569 = cbrt(r194568);
        double r194570 = r194567 * r194569;
        double r194571 = r194544 * r194544;
        double r194572 = r194557 * r194557;
        double r194573 = r194544 * r194557;
        double r194574 = r194572 + r194573;
        double r194575 = r194571 + r194574;
        double r194576 = sqrt(r194575);
        double r194577 = r194570 / r194576;
        double r194578 = r194544 - r194547;
        double r194579 = r194577 * r194578;
        double r194580 = r194549 / r194579;
        return r194580;
}

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

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

4. Applied associate-*r/ 0.5

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

5. Applied sqrt-div 0.5

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

6. Applied associate-*r/ 0.5

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

8. Applied add-cube-cbrt 0.5

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

9. Applied associate-*r* 0.5

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

11. Applied sqrt-prod 0.5

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

12. Applied cbrt-prod 0.5

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

13. Applied associate-*r* 0.5

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

14. Final simplification 0.5

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

Reproduce

herbie shell --seed 2019294 
(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)))))