Average Error: 0.4 → 0.1
Time: 23.8s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{\frac{\frac{1 - v \cdot \left(v \cdot 5\right)}{\pi}}{\sqrt{\left(1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot 27\right)\right) \cdot 2}}}{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot t} \cdot \sqrt{\left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) + 1}\right) \cdot \left(1 + v \cdot v\right)\]
\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)}
↓
\left(\frac{\frac{\frac{1 - v \cdot \left(v \cdot 5\right)}{\pi}}{\sqrt{\left(1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot 27\right)\right) \cdot 2}}}{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot t} \cdot \sqrt{\left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) + 1}\right) \cdot \left(1 + v \cdot v\right)
double f(double v, double t) {
        double r6753564 = 1.0;
        double r6753565 = 5.0;
        double r6753566 = v;
        double r6753567 = r6753566 * r6753566;
        double r6753568 = r6753565 * r6753567;
        double r6753569 = r6753564 - r6753568;
        double r6753570 = atan2(1.0, 0.0);
        double r6753571 = t;
        double r6753572 = r6753570 * r6753571;
        double r6753573 = 2.0;
        double r6753574 = 3.0;
        double r6753575 = r6753574 * r6753567;
        double r6753576 = r6753564 - r6753575;
        double r6753577 = r6753573 * r6753576;
        double r6753578 = sqrt(r6753577);
        double r6753579 = r6753572 * r6753578;
        double r6753580 = r6753564 - r6753567;
        double r6753581 = r6753579 * r6753580;
        double r6753582 = r6753569 / r6753581;
        return r6753582;
}
↓
double f(double v, double t) {
        double r6753583 = 1.0;
        double r6753584 = v;
        double r6753585 = 5.0;
        double r6753586 = r6753584 * r6753585;
        double r6753587 = r6753584 * r6753586;
        double r6753588 = r6753583 - r6753587;
        double r6753589 = atan2(1.0, 0.0);
        double r6753590 = r6753588 / r6753589;
        double r6753591 = r6753584 * r6753584;
        double r6753592 = r6753584 * r6753591;
        double r6753593 = 27.0;
        double r6753594 = r6753592 * r6753593;
        double r6753595 = r6753592 * r6753594;
        double r6753596 = r6753583 - r6753595;
        double r6753597 = 2.0;
        double r6753598 = r6753596 * r6753597;
        double r6753599 = sqrt(r6753598);
        double r6753600 = r6753590 / r6753599;
        double r6753601 = r6753591 * r6753591;
        double r6753602 = r6753583 - r6753601;
        double r6753603 = t;
        double r6753604 = r6753602 * r6753603;
        double r6753605 = r6753600 / r6753604;
        double r6753606 = 3.0;
        double r6753607 = r6753606 * r6753591;
        double r6753608 = r6753607 * r6753607;
        double r6753609 = r6753607 + r6753608;
        double r6753610 = r6753609 + r6753583;
        double r6753611 = sqrt(r6753610);
        double r6753612 = r6753605 * r6753611;
        double r6753613 = r6753583 + r6753591;
        double r6753614 = r6753612 * r6753613;
        return r6753614;
}

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 0.4

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

  3. Applied associate-*l*0.4

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

  5. Applied flip--0.4

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

  6. Applied associate-*r/0.4

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

  7. Applied associate-/r/0.4

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

  8. Simplified0.3

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

  10. Applied flip3--0.3

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

  11. Applied associate-*r/0.3

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

  12. Applied sqrt-div0.3

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

  13. Applied associate-*l/0.3

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

  14. Applied associate-*l/0.3

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

  15. Applied associate-/r/0.3

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

  16. Simplified0.1

    \[\leadsto \left(\color{blue}{\frac{\frac{\frac{1 - \left(5 \cdot v\right) \cdot v}{\pi}}{\sqrt{2 \cdot \left(1 - \left(27 \cdot \left(v \cdot \left(v \cdot v\right)\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)\right)}}}{t \cdot \left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}} \cdot \sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}\right) \cdot \left(1 + v \cdot v\right)\]

  17. Final simplification0.1

    \[\leadsto \left(\frac{\frac{\frac{1 - v \cdot \left(v \cdot 5\right)}{\pi}}{\sqrt{\left(1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot \left(v \cdot v\right)\right) \cdot 27\right)\right) \cdot 2}}}{\left(1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot t} \cdot \sqrt{\left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right) + 1}\right) \cdot \left(1 + v \cdot v\right)\]

Reproduce

herbie shell --seed 2019168 
(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)))))