Average Error: 0.4 → 0.5
Time: 23.3s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\frac{1}{\pi \cdot \sqrt{2}} \cdot \frac{1}{t} - \frac{5}{2} \cdot \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right)\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}\]
\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(\frac{1}{\pi \cdot \sqrt{2}} \cdot \frac{1}{t} - \frac{5}{2} \cdot \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right)\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}
double f(double v, double t) {
        double r5565617 = 1.0;
        double r5565618 = 5.0;
        double r5565619 = v;
        double r5565620 = r5565619 * r5565619;
        double r5565621 = r5565618 * r5565620;
        double r5565622 = r5565617 - r5565621;
        double r5565623 = atan2(1.0, 0.0);
        double r5565624 = t;
        double r5565625 = r5565623 * r5565624;
        double r5565626 = 2.0;
        double r5565627 = 3.0;
        double r5565628 = r5565627 * r5565620;
        double r5565629 = r5565617 - r5565628;
        double r5565630 = r5565626 * r5565629;
        double r5565631 = sqrt(r5565630);
        double r5565632 = r5565625 * r5565631;
        double r5565633 = r5565617 - r5565620;
        double r5565634 = r5565632 * r5565633;
        double r5565635 = r5565622 / r5565634;
        return r5565635;
}
↓
double f(double v, double t) {
        double r5565636 = 1.0;
        double r5565637 = atan2(1.0, 0.0);
        double r5565638 = 2.0;
        double r5565639 = sqrt(r5565638);
        double r5565640 = r5565637 * r5565639;
        double r5565641 = r5565636 / r5565640;
        double r5565642 = t;
        double r5565643 = r5565636 / r5565642;
        double r5565644 = r5565641 * r5565643;
        double r5565645 = 2.5;
        double r5565646 = v;
        double r5565647 = r5565639 * r5565642;
        double r5565648 = r5565646 / r5565647;
        double r5565649 = r5565646 / r5565637;
        double r5565650 = r5565648 * r5565649;
        double r5565651 = r5565645 * r5565650;
        double r5565652 = r5565644 - r5565651;
        double r5565653 = 4.0;
        double r5565654 = pow(r5565646, r5565653);
        double r5565655 = 6.625;
        double r5565656 = r5565654 * r5565655;
        double r5565657 = r5565656 / r5565640;
        double r5565658 = r5565657 / r5565642;
        double r5565659 = r5565652 - r5565658;
        return r5565659;
}

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. Taylor expanded around 0 0.5

    \[\leadsto \color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)} - \left(\frac{53}{8} \cdot \frac{{v}^{4}}{t \cdot \left(\sqrt{2} \cdot \pi\right)} + \frac{5}{2} \cdot \frac{{v}^{2}}{t \cdot \left(\sqrt{2} \cdot \pi\right)}\right)}\]

  3. Simplified0.6

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{\sqrt{2} \cdot t}}{\pi} - \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right) \cdot \frac{5}{2}\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}}\]

  4. Taylor expanded around 0 0.5

    \[\leadsto \left(\color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)}} - \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right) \cdot \frac{5}{2}\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}\]

  5. Taylor expanded around -inf 0.5

    \[\leadsto \left(\color{blue}{\frac{1}{t \cdot \left(\sqrt{2} \cdot \pi\right)}} - \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right) \cdot \frac{5}{2}\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}\]

  6. Simplified0.5

    \[\leadsto \left(\color{blue}{\frac{\frac{1}{t}}{\pi \cdot \sqrt{2}}} - \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right) \cdot \frac{5}{2}\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}\]
  7. Using strategy rm

  8. Applied div-inv0.5

    \[\leadsto \left(\color{blue}{\frac{1}{t} \cdot \frac{1}{\pi \cdot \sqrt{2}}} - \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right) \cdot \frac{5}{2}\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}\]

  9. Final simplification0.5

    \[\leadsto \left(\frac{1}{\pi \cdot \sqrt{2}} \cdot \frac{1}{t} - \frac{5}{2} \cdot \left(\frac{v}{\sqrt{2} \cdot t} \cdot \frac{v}{\pi}\right)\right) - \frac{\frac{{v}^{4} \cdot \frac{53}{8}}{\pi \cdot \sqrt{2}}}{t}\]

Reproduce

herbie shell --seed 2019134 
(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)))))