Average Error: 0.5 → 0.1
Time: 3.8m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{1 + \left(\left(v \cdot v\right) \cdot 3 + \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)} \cdot \left(\frac{\frac{\frac{1}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}}{t} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right)\]
\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)}
↓
\sqrt{1 + \left(\left(v \cdot v\right) \cdot 3 + \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)} \cdot \left(\frac{\frac{\frac{1}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}}{t} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right)
double f(double v, double t) {
        double r71587429 = 1.0;
        double r71587430 = 5.0;
        double r71587431 = v;
        double r71587432 = r71587431 * r71587431;
        double r71587433 = r71587430 * r71587432;
        double r71587434 = r71587429 - r71587433;
        double r71587435 = atan2(1.0, 0.0);
        double r71587436 = t;
        double r71587437 = r71587435 * r71587436;
        double r71587438 = 2.0;
        double r71587439 = 3.0;
        double r71587440 = r71587439 * r71587432;
        double r71587441 = r71587429 - r71587440;
        double r71587442 = r71587438 * r71587441;
        double r71587443 = sqrt(r71587442);
        double r71587444 = r71587437 * r71587443;
        double r71587445 = r71587429 - r71587432;
        double r71587446 = r71587444 * r71587445;
        double r71587447 = r71587434 / r71587446;
        return r71587447;
}
↓
double f(double v, double t) {
        double r71587448 = 1.0;
        double r71587449 = v;
        double r71587450 = r71587449 * r71587449;
        double r71587451 = 3.0;
        double r71587452 = r71587450 * r71587451;
        double r71587453 = r71587452 * r71587452;
        double r71587454 = r71587452 + r71587453;
        double r71587455 = r71587448 + r71587454;
        double r71587456 = sqrt(r71587455);
        double r71587457 = atan2(1.0, 0.0);
        double r71587458 = r71587448 / r71587457;
        double r71587459 = 2.0;
        double r71587460 = 54.0;
        double r71587461 = r71587449 * r71587450;
        double r71587462 = r71587461 * r71587461;
        double r71587463 = r71587460 * r71587462;
        double r71587464 = r71587459 - r71587463;
        double r71587465 = sqrt(r71587464);
        double r71587466 = r71587458 / r71587465;
        double r71587467 = t;
        double r71587468 = r71587466 / r71587467;
        double r71587469 = 5.0;
        double r71587470 = r71587469 * r71587450;
        double r71587471 = r71587448 - r71587470;
        double r71587472 = r71587448 - r71587450;
        double r71587473 = r71587471 / r71587472;
        double r71587474 = r71587468 * r71587473;
        double r71587475 = r71587456 * r71587474;
        return r71587475;
}

Error

Bits error versus v

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-div0.5

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

  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. Applied associate-*l/0.5

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

  8. Applied associate-/r/0.5

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

  10. Applied *-un-lft-identity0.5

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

  11. Applied times-frac0.5

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

  12. Simplified0.4

    \[\leadsto \left(\color{blue}{\frac{\frac{\frac{1}{t}}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\]
  13. Using strategy rm

  14. Applied *-un-lft-identity0.4

    \[\leadsto \left(\frac{\frac{\frac{1}{t}}{\pi}}{\color{blue}{1 \cdot \sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\]

  15. Applied div-inv0.4

    \[\leadsto \left(\frac{\color{blue}{\frac{1}{t} \cdot \frac{1}{\pi}}}{1 \cdot \sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\]

  16. Applied times-frac0.3

    \[\leadsto \left(\color{blue}{\left(\frac{\frac{1}{t}}{1} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}\right)} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\]

  17. Simplified0.3

    \[\leadsto \left(\left(\color{blue}{\frac{1}{t}} \cdot \frac{\frac{1}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}\right) \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\]
  18. Using strategy rm

  19. Applied associate-*l/0.1

    \[\leadsto \left(\color{blue}{\frac{1 \cdot \frac{\frac{1}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}}{t}} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\]

  20. Simplified0.1

    \[\leadsto \left(\frac{\color{blue}{\frac{\frac{1}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}}}{t} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right) \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\]

  21. Final simplification0.1

    \[\leadsto \sqrt{1 + \left(\left(v \cdot v\right) \cdot 3 + \left(\left(v \cdot v\right) \cdot 3\right) \cdot \left(\left(v \cdot v\right) \cdot 3\right)\right)} \cdot \left(\frac{\frac{\frac{1}{\pi}}{\sqrt{2 - 54 \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}}{t} \cdot \frac{1 - 5 \cdot \left(v \cdot v\right)}{1 - v \cdot v}\right)\]

Reproduce

herbie shell --seed 2019119 
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))