Average Error: 0.5 → 0.3
Time: 23.6s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\frac{1 - v \cdot \left(v \cdot 5\right)}{\pi}}{t \cdot \sqrt{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}}}{1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1\right)\]
\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{1 - v \cdot \left(v \cdot 5\right)}{\pi}}{t \cdot \sqrt{\left(1 - 3 \cdot \left(v \cdot v\right)\right) \cdot 2}}}{1 - \left(v \cdot \left(v \cdot v\right)\right) \cdot \left(v \cdot \left(v \cdot v\right)\right)} \cdot \left(\left(v \cdot v + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) + 1\right)
double f(double v, double t) {
        double r3335709 = 1.0;
        double r3335710 = 5.0;
        double r3335711 = v;
        double r3335712 = r3335711 * r3335711;
        double r3335713 = r3335710 * r3335712;
        double r3335714 = r3335709 - r3335713;
        double r3335715 = atan2(1.0, 0.0);
        double r3335716 = t;
        double r3335717 = r3335715 * r3335716;
        double r3335718 = 2.0;
        double r3335719 = 3.0;
        double r3335720 = r3335719 * r3335712;
        double r3335721 = r3335709 - r3335720;
        double r3335722 = r3335718 * r3335721;
        double r3335723 = sqrt(r3335722);
        double r3335724 = r3335717 * r3335723;
        double r3335725 = r3335709 - r3335712;
        double r3335726 = r3335724 * r3335725;
        double r3335727 = r3335714 / r3335726;
        return r3335727;
}
↓
double f(double v, double t) {
        double r3335728 = 1.0;
        double r3335729 = v;
        double r3335730 = 5.0;
        double r3335731 = r3335729 * r3335730;
        double r3335732 = r3335729 * r3335731;
        double r3335733 = r3335728 - r3335732;
        double r3335734 = atan2(1.0, 0.0);
        double r3335735 = r3335733 / r3335734;
        double r3335736 = t;
        double r3335737 = 3.0;
        double r3335738 = r3335729 * r3335729;
        double r3335739 = r3335737 * r3335738;
        double r3335740 = r3335728 - r3335739;
        double r3335741 = 2.0;
        double r3335742 = r3335740 * r3335741;
        double r3335743 = sqrt(r3335742);
        double r3335744 = r3335736 * r3335743;
        double r3335745 = r3335735 / r3335744;
        double r3335746 = r3335729 * r3335738;
        double r3335747 = r3335746 * r3335746;
        double r3335748 = r3335728 - r3335747;
        double r3335749 = r3335745 / r3335748;
        double r3335750 = r3335738 * r3335738;
        double r3335751 = r3335738 + r3335750;
        double r3335752 = r3335751 + r3335728;
        double r3335753 = r3335749 * r3335752;
        return r3335753;
}

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 associate-*l*0.5

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

  5. Applied flip3--0.5

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

  6. Applied associate-*r/0.5

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

  7. Applied associate-/r/0.5

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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)))))