Average Error: 0.5 → 0.1
Time: 34.4s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right) \cdot \left(\left(\sqrt{{v}^{4} \cdot \left(3 \cdot 3\right) + 1 \cdot 1} \cdot \left(1 \cdot 1 + {v}^{4}\right)\right) \cdot \frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{{1}^{3} \cdot 1 - {v}^{8}}}{\pi}}{\sqrt{\left({1}^{3} \cdot 1 - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right) \cdot 2}}}{t}\right)\]
\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)}
↓
\left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right) \cdot \left(\left(\sqrt{{v}^{4} \cdot \left(3 \cdot 3\right) + 1 \cdot 1} \cdot \left(1 \cdot 1 + {v}^{4}\right)\right) \cdot \frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{{1}^{3} \cdot 1 - {v}^{8}}}{\pi}}{\sqrt{\left({1}^{3} \cdot 1 - \left(3 \cdot {3}^{3}\right) \cdot {v}^{8}\right) \cdot 2}}}{t}\right)
double f(double v, double t) {
        double r183415 = 1.0;
        double r183416 = 5.0;
        double r183417 = v;
        double r183418 = r183417 * r183417;
        double r183419 = r183416 * r183418;
        double r183420 = r183415 - r183419;
        double r183421 = atan2(1.0, 0.0);
        double r183422 = t;
        double r183423 = r183421 * r183422;
        double r183424 = 2.0;
        double r183425 = 3.0;
        double r183426 = r183425 * r183418;
        double r183427 = r183415 - r183426;
        double r183428 = r183424 * r183427;
        double r183429 = sqrt(r183428);
        double r183430 = r183423 * r183429;
        double r183431 = r183415 - r183418;
        double r183432 = r183430 * r183431;
        double r183433 = r183420 / r183432;
        return r183433;
}
↓
double f(double v, double t) {
        double r183434 = 1.0;
        double r183435 = 3.0;
        double r183436 = v;
        double r183437 = r183436 * r183436;
        double r183438 = r183435 * r183437;
        double r183439 = r183434 + r183438;
        double r183440 = sqrt(r183439);
        double r183441 = r183434 + r183437;
        double r183442 = r183440 * r183441;
        double r183443 = 4.0;
        double r183444 = pow(r183436, r183443);
        double r183445 = r183435 * r183435;
        double r183446 = r183444 * r183445;
        double r183447 = r183434 * r183434;
        double r183448 = r183446 + r183447;
        double r183449 = sqrt(r183448);
        double r183450 = r183447 + r183444;
        double r183451 = r183449 * r183450;
        double r183452 = 5.0;
        double r183453 = r183452 * r183437;
        double r183454 = r183434 - r183453;
        double r183455 = 3.0;
        double r183456 = pow(r183434, r183455);
        double r183457 = r183456 * r183434;
        double r183458 = 8.0;
        double r183459 = pow(r183436, r183458);
        double r183460 = r183457 - r183459;
        double r183461 = r183454 / r183460;
        double r183462 = atan2(1.0, 0.0);
        double r183463 = r183461 / r183462;
        double r183464 = pow(r183435, r183455);
        double r183465 = r183435 * r183464;
        double r183466 = r183465 * r183459;
        double r183467 = r183457 - r183466;
        double r183468 = 2.0;
        double r183469 = r183467 * r183468;
        double r183470 = sqrt(r183469);
        double r183471 = r183463 / r183470;
        double r183472 = t;
        double r183473 = r183471 / r183472;
        double r183474 = r183451 * r183473;
        double r183475 = r183442 * r183474;
        return r183475;
}

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 flip--0.5

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

  4. Applied flip--0.5

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

  5. Applied associate-*r/0.5

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

  6. Applied sqrt-div0.5

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

  7. Applied associate-*r/0.5

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

  8. Applied frac-times0.5

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

  9. Applied associate-/r/0.5

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

  10. Simplified0.5

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

  12. Applied flip--0.5

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

  13. Applied associate-*r/0.5

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

  14. Applied sqrt-div0.5

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

  15. Applied associate-*r/0.5

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

  16. Applied flip--0.5

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

  17. Applied associate-*l/0.5

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

  18. Applied frac-times0.5

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

  19. Applied associate-/r/0.5

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

  20. Simplified0.1

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

  21. Final simplification0.1

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

Reproduce

herbie shell --seed 2019235 
(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)))))