Average Error: 0.4 → 0.4
Time: 5.7s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1 - \sqrt{5 \cdot \left(v \cdot v\right)} \cdot \sqrt{5 \cdot \left(v \cdot v\right)}}{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}} \cdot \left(1 - v \cdot v\right)}\]
\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{1 - \sqrt{5 \cdot \left(v \cdot v\right)} \cdot \sqrt{5 \cdot \left(v \cdot v\right)}}{\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 \left(1 - v \cdot v\right)}
double f(double v, double t) {
        double r223831 = 1.0;
        double r223832 = 5.0;
        double r223833 = v;
        double r223834 = r223833 * r223833;
        double r223835 = r223832 * r223834;
        double r223836 = r223831 - r223835;
        double r223837 = atan2(1.0, 0.0);
        double r223838 = t;
        double r223839 = r223837 * r223838;
        double r223840 = 2.0;
        double r223841 = 3.0;
        double r223842 = r223841 * r223834;
        double r223843 = r223831 - r223842;
        double r223844 = r223840 * r223843;
        double r223845 = sqrt(r223844);
        double r223846 = r223839 * r223845;
        double r223847 = r223831 - r223834;
        double r223848 = r223846 * r223847;
        double r223849 = r223836 / r223848;
        return r223849;
}
↓
double f(double v, double t) {
        double r223850 = 1.0;
        double r223851 = 5.0;
        double r223852 = v;
        double r223853 = r223852 * r223852;
        double r223854 = r223851 * r223853;
        double r223855 = sqrt(r223854);
        double r223856 = r223855 * r223855;
        double r223857 = r223850 - r223856;
        double r223858 = atan2(1.0, 0.0);
        double r223859 = t;
        double r223860 = r223858 * r223859;
        double r223861 = 2.0;
        double r223862 = r223850 * r223850;
        double r223863 = 3.0;
        double r223864 = r223863 * r223853;
        double r223865 = r223864 * r223864;
        double r223866 = r223862 - r223865;
        double r223867 = r223861 * r223866;
        double r223868 = sqrt(r223867);
        double r223869 = r223860 * r223868;
        double r223870 = r223850 + r223864;
        double r223871 = sqrt(r223870);
        double r223872 = r223869 / r223871;
        double r223873 = r223850 - r223853;
        double r223874 = r223872 * r223873;
        double r223875 = r223857 / r223874;
        return r223875;
}

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 add-sqr-sqrt0.4

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

  5. Applied flip--0.4

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

  6. Applied associate-*r/0.4

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

  7. Applied sqrt-div0.4

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

  8. Applied associate-*r/0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019356 +o rules:numerics
(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)))))