Average Error: 0.4 → 0.4
Time: 1.6m
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1\right) \cdot \left(\frac{\frac{1}{1 - \left(\left(v \cdot v\right) \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot v\right)}}{\left(\pi \cdot \sqrt{2 + -6 \cdot \left(v \cdot v\right)}\right) \cdot t} \cdot \frac{1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{5 \cdot \left(v \cdot v\right) + 1}\right)\]
\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(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1\right) \cdot \left(\frac{\frac{1}{1 - \left(\left(v \cdot v\right) \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot v\right)}}{\left(\pi \cdot \sqrt{2 + -6 \cdot \left(v \cdot v\right)}\right) \cdot t} \cdot \frac{1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{5 \cdot \left(v \cdot v\right) + 1}\right)
double f(double v, double t) {
        double r39388001 = 1.0;
        double r39388002 = 5.0;
        double r39388003 = v;
        double r39388004 = r39388003 * r39388003;
        double r39388005 = r39388002 * r39388004;
        double r39388006 = r39388001 - r39388005;
        double r39388007 = atan2(1.0, 0.0);
        double r39388008 = t;
        double r39388009 = r39388007 * r39388008;
        double r39388010 = 2.0;
        double r39388011 = 3.0;
        double r39388012 = r39388011 * r39388004;
        double r39388013 = r39388001 - r39388012;
        double r39388014 = r39388010 * r39388013;
        double r39388015 = sqrt(r39388014);
        double r39388016 = r39388009 * r39388015;
        double r39388017 = r39388001 - r39388004;
        double r39388018 = r39388016 * r39388017;
        double r39388019 = r39388006 / r39388018;
        return r39388019;
}
↓
double f(double v, double t) {
        double r39388020 = v;
        double r39388021 = r39388020 * r39388020;
        double r39388022 = r39388021 * r39388021;
        double r39388023 = r39388021 + r39388022;
        double r39388024 = 1.0;
        double r39388025 = r39388023 + r39388024;
        double r39388026 = r39388021 * r39388020;
        double r39388027 = r39388026 * r39388026;
        double r39388028 = r39388024 - r39388027;
        double r39388029 = r39388024 / r39388028;
        double r39388030 = atan2(1.0, 0.0);
        double r39388031 = 2.0;
        double r39388032 = -6.0;
        double r39388033 = r39388032 * r39388021;
        double r39388034 = r39388031 + r39388033;
        double r39388035 = sqrt(r39388034);
        double r39388036 = r39388030 * r39388035;
        double r39388037 = t;
        double r39388038 = r39388036 * r39388037;
        double r39388039 = r39388029 / r39388038;
        double r39388040 = 5.0;
        double r39388041 = r39388040 * r39388021;
        double r39388042 = r39388041 * r39388041;
        double r39388043 = r39388024 - r39388042;
        double r39388044 = r39388041 + r39388024;
        double r39388045 = r39388043 / r39388044;
        double r39388046 = r39388039 * r39388045;
        double r39388047 = r39388025 * r39388046;
        return r39388047;
}

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 flip3--0.4

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

  6. Applied associate-*r/0.4

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

  7. Applied associate-/r/0.4

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

  8. Simplified0.4

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

  10. Applied flip--0.4

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

  11. Applied associate-/l/0.4

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

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

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

  14. Applied times-frac0.4

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

  15. Simplified0.4

    \[\leadsto \left(\color{blue}{\frac{\frac{1}{1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)}}{t \cdot \left(\sqrt{\left(-v \cdot v\right) \cdot 6 + 2} \cdot \pi\right)}} \cdot \frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{1 + 5 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 + \left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)\]

  16. Final simplification0.4

    \[\leadsto \left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1\right) \cdot \left(\frac{\frac{1}{1 - \left(\left(v \cdot v\right) \cdot v\right) \cdot \left(\left(v \cdot v\right) \cdot v\right)}}{\left(\pi \cdot \sqrt{2 + -6 \cdot \left(v \cdot v\right)}\right) \cdot t} \cdot \frac{1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{5 \cdot \left(v \cdot v\right) + 1}\right)\]

Reproduce

herbie shell --seed 2019124 
(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)))))