Average Error: 0.5 → 0.3
Time: 24.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}}}{1 \cdot 1 - {v}^{4}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]
\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)}
↓
\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}}}{1 \cdot 1 - {v}^{4}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)
double f(double v, double t) {
        double r175727 = 1.0;
        double r175728 = 5.0;
        double r175729 = v;
        double r175730 = r175729 * r175729;
        double r175731 = r175728 * r175730;
        double r175732 = r175727 - r175731;
        double r175733 = atan2(1.0, 0.0);
        double r175734 = t;
        double r175735 = r175733 * r175734;
        double r175736 = 2.0;
        double r175737 = 3.0;
        double r175738 = r175737 * r175730;
        double r175739 = r175727 - r175738;
        double r175740 = r175736 * r175739;
        double r175741 = sqrt(r175740);
        double r175742 = r175735 * r175741;
        double r175743 = r175727 - r175730;
        double r175744 = r175742 * r175743;
        double r175745 = r175732 / r175744;
        return r175745;
}
↓
double f(double v, double t) {
        double r175746 = 1.0;
        double r175747 = 5.0;
        double r175748 = v;
        double r175749 = r175748 * r175748;
        double r175750 = r175747 * r175749;
        double r175751 = r175746 - r175750;
        double r175752 = atan2(1.0, 0.0);
        double r175753 = r175751 / r175752;
        double r175754 = t;
        double r175755 = r175753 / r175754;
        double r175756 = 2.0;
        double r175757 = r175746 * r175746;
        double r175758 = 3.0;
        double r175759 = r175758 * r175758;
        double r175760 = 4.0;
        double r175761 = pow(r175748, r175760);
        double r175762 = r175759 * r175761;
        double r175763 = r175757 - r175762;
        double r175764 = r175756 * r175763;
        double r175765 = sqrt(r175764);
        double r175766 = r175755 / r175765;
        double r175767 = r175757 - r175761;
        double r175768 = r175766 / r175767;
        double r175769 = r175758 * r175749;
        double r175770 = r175746 + r175769;
        double r175771 = sqrt(r175770);
        double r175772 = r175746 + r175749;
        double r175773 = r175771 * r175772;
        double r175774 = r175768 * r175773;
        return r175774;
}

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. Using strategy rm

  3. Applied flip--0.5

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

  4. Applied flip--0.5

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

  5. Applied associate-*r/0.5

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

  6. Applied sqrt-div0.5

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

  7. Applied associate-*r/0.5

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

  8. Applied frac-times0.5

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

  9. Applied associate-/r/0.5

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

  10. Simplified0.5

    \[\leadsto \color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}\right)}}{1 \cdot 1 - {v}^{4}}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]
  11. Using strategy rm

  12. Applied associate-/r*0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}}}}{1 \cdot 1 - {v}^{4}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]
  13. Using strategy rm

  14. Applied associate-/r*0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}}}}{1 \cdot 1 - {v}^{4}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

  15. Final simplification0.3

    \[\leadsto \frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{t}}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}}}{1 \cdot 1 - {v}^{4}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left(1 + v \cdot v\right)\right)\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))