Average Error: 0.4 → 0.5
Time: 9.0s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{1}{\frac{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}} \cdot \left(1 - v \cdot v\right)} - \frac{5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
\frac{1 - 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}{\frac{\pi \cdot \left(t \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}} \cdot \left(1 - v \cdot v\right)} - \frac{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)}
double f(double v, double t) {
        double r225708 = 1.0;
        double r225709 = 5.0;
        double r225710 = v;
        double r225711 = r225710 * r225710;
        double r225712 = r225709 * r225711;
        double r225713 = r225708 - r225712;
        double r225714 = atan2(1.0, 0.0);
        double r225715 = t;
        double r225716 = r225714 * r225715;
        double r225717 = 2.0;
        double r225718 = 3.0;
        double r225719 = r225718 * r225711;
        double r225720 = r225708 - r225719;
        double r225721 = r225717 * r225720;
        double r225722 = sqrt(r225721);
        double r225723 = r225716 * r225722;
        double r225724 = r225708 - r225711;
        double r225725 = r225723 * r225724;
        double r225726 = r225713 / r225725;
        return r225726;
}
↓
double f(double v, double t) {
        double r225727 = 1.0;
        double r225728 = atan2(1.0, 0.0);
        double r225729 = t;
        double r225730 = 2.0;
        double r225731 = 3.0;
        double r225732 = pow(r225727, r225731);
        double r225733 = 3.0;
        double r225734 = v;
        double r225735 = r225734 * r225734;
        double r225736 = r225733 * r225735;
        double r225737 = pow(r225736, r225731);
        double r225738 = r225732 - r225737;
        double r225739 = r225730 * r225738;
        double r225740 = sqrt(r225739);
        double r225741 = r225729 * r225740;
        double r225742 = r225728 * r225741;
        double r225743 = r225727 * r225727;
        double r225744 = r225736 * r225736;
        double r225745 = r225727 * r225736;
        double r225746 = r225744 + r225745;
        double r225747 = r225743 + r225746;
        double r225748 = sqrt(r225747);
        double r225749 = r225742 / r225748;
        double r225750 = r225727 - r225735;
        double r225751 = r225749 * r225750;
        double r225752 = r225727 / r225751;
        double r225753 = 5.0;
        double r225754 = r225753 * r225735;
        double r225755 = r225728 * r225729;
        double r225756 = r225727 - r225736;
        double r225757 = r225730 * r225756;
        double r225758 = sqrt(r225757);
        double r225759 = r225755 * r225758;
        double r225760 = r225759 * r225750;
        double r225761 = r225754 / r225760;
        double r225762 = r225752 - r225761;
        return r225762;
}

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 div-sub0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied flip3--0.4

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

  8. Applied associate-*r/0.4

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

  9. Applied sqrt-div0.5

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

  10. Applied associate-*r/0.5

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

  11. Applied associate-*r/0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020060 
(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)))))