Average Error: 0.5 → 0.3
Time: 12.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(1 \cdot 1 - {v}^{4}\right) \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 + v \cdot v\right)\right)\]
\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)}
↓
\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left(1 \cdot 1 - {v}^{4}\right) \cdot \left(\left(t \cdot \sqrt{2}\right) \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right)} \cdot \left(\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)\right)
double f(double v, double t) {
        double r246587 = 1.0;
        double r246588 = 5.0;
        double r246589 = v;
        double r246590 = r246589 * r246589;
        double r246591 = r246588 * r246590;
        double r246592 = r246587 - r246591;
        double r246593 = atan2(1.0, 0.0);
        double r246594 = t;
        double r246595 = r246593 * r246594;
        double r246596 = 2.0;
        double r246597 = 3.0;
        double r246598 = r246597 * r246590;
        double r246599 = r246587 - r246598;
        double r246600 = r246596 * r246599;
        double r246601 = sqrt(r246600);
        double r246602 = r246595 * r246601;
        double r246603 = r246587 - r246590;
        double r246604 = r246602 * r246603;
        double r246605 = r246592 / r246604;
        return r246605;
}
↓
double f(double v, double t) {
        double r246606 = 1.0;
        double r246607 = 5.0;
        double r246608 = v;
        double r246609 = r246608 * r246608;
        double r246610 = r246607 * r246609;
        double r246611 = r246606 - r246610;
        double r246612 = atan2(1.0, 0.0);
        double r246613 = r246611 / r246612;
        double r246614 = r246606 * r246606;
        double r246615 = 4.0;
        double r246616 = pow(r246608, r246615);
        double r246617 = r246614 - r246616;
        double r246618 = t;
        double r246619 = 2.0;
        double r246620 = sqrt(r246619);
        double r246621 = r246618 * r246620;
        double r246622 = 3.0;
        double r246623 = pow(r246606, r246622);
        double r246624 = 3.0;
        double r246625 = r246624 * r246609;
        double r246626 = pow(r246625, r246622);
        double r246627 = r246623 - r246626;
        double r246628 = sqrt(r246627);
        double r246629 = r246621 * r246628;
        double r246630 = r246617 * r246629;
        double r246631 = r246613 / r246630;
        double r246632 = r246625 * r246625;
        double r246633 = r246606 * r246625;
        double r246634 = r246632 + r246633;
        double r246635 = r246614 + r246634;
        double r246636 = sqrt(r246635);
        double r246637 = r246606 + r246609;
        double r246638 = r246636 * r246637;
        double r246639 = r246631 * r246638;
        return r246639;
}

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 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 sqrt-prod0.4

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

  6. Applied associate-*r*0.4

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

  8. Applied flip--0.4

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

  9. Applied flip3--0.4

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

  10. Applied sqrt-div0.5

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

  11. Applied associate-*r/0.5

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

  12. Applied associate-*r/0.5

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

  13. Applied frac-times0.5

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

  14. Applied associate-/r/0.5

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 
(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)))))