Average Error: 0.4 → 0.1
Time: 21.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\sqrt{1 + \left(3 \cdot \left(v \cdot v\right) + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\sqrt{2 \cdot \left(1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)\right)}}}{t}}{1 - v \cdot v}\]
\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(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)} \cdot \frac{\frac{\frac{\frac{1 - \left(v \cdot v\right) \cdot 5}{\pi}}{\sqrt{2 \cdot \left(1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)\right)}}}{t}}{1 - v \cdot v}
double f(double v, double t) {
        double r3965685 = 1.0;
        double r3965686 = 5.0;
        double r3965687 = v;
        double r3965688 = r3965687 * r3965687;
        double r3965689 = r3965686 * r3965688;
        double r3965690 = r3965685 - r3965689;
        double r3965691 = atan2(1.0, 0.0);
        double r3965692 = t;
        double r3965693 = r3965691 * r3965692;
        double r3965694 = 2.0;
        double r3965695 = 3.0;
        double r3965696 = r3965695 * r3965688;
        double r3965697 = r3965685 - r3965696;
        double r3965698 = r3965694 * r3965697;
        double r3965699 = sqrt(r3965698);
        double r3965700 = r3965693 * r3965699;
        double r3965701 = r3965685 - r3965688;
        double r3965702 = r3965700 * r3965701;
        double r3965703 = r3965690 / r3965702;
        return r3965703;
}
↓
double f(double v, double t) {
        double r3965704 = 1.0;
        double r3965705 = 3.0;
        double r3965706 = v;
        double r3965707 = r3965706 * r3965706;
        double r3965708 = r3965705 * r3965707;
        double r3965709 = r3965708 * r3965708;
        double r3965710 = r3965708 + r3965709;
        double r3965711 = r3965704 + r3965710;
        double r3965712 = sqrt(r3965711);
        double r3965713 = 5.0;
        double r3965714 = r3965707 * r3965713;
        double r3965715 = r3965704 - r3965714;
        double r3965716 = atan2(1.0, 0.0);
        double r3965717 = r3965715 / r3965716;
        double r3965718 = 2.0;
        double r3965719 = r3965708 * r3965709;
        double r3965720 = r3965704 - r3965719;
        double r3965721 = r3965718 * r3965720;
        double r3965722 = sqrt(r3965721);
        double r3965723 = r3965717 / r3965722;
        double r3965724 = t;
        double r3965725 = r3965723 / r3965724;
        double r3965726 = r3965704 - r3965707;
        double r3965727 = r3965725 / r3965726;
        double r3965728 = r3965712 * r3965727;
        return r3965728;
}

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 flip3--0.4

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

  4. Applied associate-*r/0.4

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

  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. Simplified0.5

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

  11. Applied associate-/r*0.3

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

  13. Applied *-un-lft-identity0.3

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

  14. Applied add-sqr-sqrt0.3

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

  15. Applied times-frac0.3

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

  16. Applied times-frac0.3

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

  18. Applied associate-*l/0.1

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

  19. Simplified0.1

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

  20. Final simplification0.1

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

Reproduce

herbie shell --seed 2019153 
(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)))))