Average Error: 0.5 → 0.6
Time: 10.9s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}\right) + 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)\]
\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)}
↓
1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}\right) + 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)
double f(double v, double t) {
        double r238856 = 1.0;
        double r238857 = 5.0;
        double r238858 = v;
        double r238859 = r238858 * r238858;
        double r238860 = r238857 * r238859;
        double r238861 = r238856 - r238860;
        double r238862 = atan2(1.0, 0.0);
        double r238863 = t;
        double r238864 = r238862 * r238863;
        double r238865 = 2.0;
        double r238866 = 3.0;
        double r238867 = r238866 * r238859;
        double r238868 = r238856 - r238867;
        double r238869 = r238865 * r238868;
        double r238870 = sqrt(r238869);
        double r238871 = r238864 * r238870;
        double r238872 = r238856 - r238859;
        double r238873 = r238871 * r238872;
        double r238874 = r238861 / r238873;
        return r238874;
}
↓
double f(double v, double t) {
        double r238875 = 1.5;
        double r238876 = v;
        double r238877 = 2.0;
        double r238878 = pow(r238876, r238877);
        double r238879 = t;
        double r238880 = 2.0;
        double r238881 = sqrt(r238880);
        double r238882 = 1.0;
        double r238883 = sqrt(r238882);
        double r238884 = atan2(1.0, 0.0);
        double r238885 = r238883 * r238884;
        double r238886 = r238881 * r238885;
        double r238887 = r238879 * r238886;
        double r238888 = r238878 / r238887;
        double r238889 = r238875 * r238888;
        double r238890 = r238881 * r238884;
        double r238891 = r238879 * r238890;
        double r238892 = r238883 / r238891;
        double r238893 = r238882 * r238892;
        double r238894 = 4.0;
        double r238895 = pow(r238876, r238894);
        double r238896 = r238895 / r238887;
        double r238897 = r238875 * r238896;
        double r238898 = 1.125;
        double r238899 = 3.0;
        double r238900 = pow(r238883, r238899);
        double r238901 = r238900 * r238884;
        double r238902 = r238881 * r238901;
        double r238903 = r238879 * r238902;
        double r238904 = r238895 / r238903;
        double r238905 = r238898 * r238904;
        double r238906 = r238897 + r238905;
        double r238907 = 4.0;
        double r238908 = r238878 * r238883;
        double r238909 = r238908 / r238891;
        double r238910 = r238895 * r238883;
        double r238911 = r238910 / r238891;
        double r238912 = r238909 + r238911;
        double r238913 = r238907 * r238912;
        double r238914 = r238906 + r238913;
        double r238915 = r238893 - r238914;
        double r238916 = r238889 + r238915;
        return r238916;
}

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. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right) - \left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)} + \left(4 \cdot \frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + 4 \cdot \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}\right) + 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)}\]

  4. Final simplification0.6

    \[\leadsto 1.5 \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + \left(1 \cdot \frac{\sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\left(1.5 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left(\sqrt{1} \cdot \pi\right)\right)} + 1.125 \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \left({\left(\sqrt{1}\right)}^{3} \cdot \pi\right)\right)}\right) + 4 \cdot \left(\frac{{v}^{2} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{{v}^{4} \cdot \sqrt{1}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020035 
(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)))))