Average Error: 0.5 → 0.6
Time: 8.5s
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 r258764 = 1.0;
        double r258765 = 5.0;
        double r258766 = v;
        double r258767 = r258766 * r258766;
        double r258768 = r258765 * r258767;
        double r258769 = r258764 - r258768;
        double r258770 = atan2(1.0, 0.0);
        double r258771 = t;
        double r258772 = r258770 * r258771;
        double r258773 = 2.0;
        double r258774 = 3.0;
        double r258775 = r258774 * r258767;
        double r258776 = r258764 - r258775;
        double r258777 = r258773 * r258776;
        double r258778 = sqrt(r258777);
        double r258779 = r258772 * r258778;
        double r258780 = r258764 - r258767;
        double r258781 = r258779 * r258780;
        double r258782 = r258769 / r258781;
        return r258782;
}
↓
double f(double v, double t) {
        double r258783 = 1.5;
        double r258784 = v;
        double r258785 = 2.0;
        double r258786 = pow(r258784, r258785);
        double r258787 = t;
        double r258788 = 2.0;
        double r258789 = sqrt(r258788);
        double r258790 = 1.0;
        double r258791 = sqrt(r258790);
        double r258792 = atan2(1.0, 0.0);
        double r258793 = r258791 * r258792;
        double r258794 = r258789 * r258793;
        double r258795 = r258787 * r258794;
        double r258796 = r258786 / r258795;
        double r258797 = r258783 * r258796;
        double r258798 = r258789 * r258792;
        double r258799 = r258787 * r258798;
        double r258800 = r258791 / r258799;
        double r258801 = r258790 * r258800;
        double r258802 = 4.0;
        double r258803 = pow(r258784, r258802);
        double r258804 = r258803 / r258795;
        double r258805 = r258783 * r258804;
        double r258806 = 1.125;
        double r258807 = 3.0;
        double r258808 = pow(r258791, r258807);
        double r258809 = r258808 * r258792;
        double r258810 = r258789 * r258809;
        double r258811 = r258787 * r258810;
        double r258812 = r258803 / r258811;
        double r258813 = r258806 * r258812;
        double r258814 = r258805 + r258813;
        double r258815 = 4.0;
        double r258816 = r258786 * r258791;
        double r258817 = r258816 / r258799;
        double r258818 = r258803 * r258791;
        double r258819 = r258818 / r258799;
        double r258820 = r258817 + r258819;
        double r258821 = r258815 * r258820;
        double r258822 = r258814 + r258821;
        double r258823 = r258801 - r258822;
        double r258824 = r258797 + r258823;
        return r258824;
}

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 2019362 
(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)))))