Average Error: 0.4 → 0.3
Time: 10.1s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\frac{\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{\pi}}{t \cdot \sqrt{2 \cdot \left(1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r328726 = 1.0;
        double r328727 = 5.0;
        double r328728 = v;
        double r328729 = r328728 * r328728;
        double r328730 = r328727 * r328729;
        double r328731 = r328726 - r328730;
        double r328732 = atan2(1.0, 0.0);
        double r328733 = t;
        double r328734 = r328732 * r328733;
        double r328735 = 2.0;
        double r328736 = 3.0;
        double r328737 = r328736 * r328729;
        double r328738 = r328726 - r328737;
        double r328739 = r328735 * r328738;
        double r328740 = sqrt(r328739);
        double r328741 = r328734 * r328740;
        double r328742 = r328726 - r328729;
        double r328743 = r328741 * r328742;
        double r328744 = r328731 / r328743;
        return r328744;
}
↓
double f(double v, double t) {
        double r328745 = 1.0;
        double r328746 = 5.0;
        double r328747 = v;
        double r328748 = r328747 * r328747;
        double r328749 = r328746 * r328748;
        double r328750 = r328745 - r328749;
        double r328751 = sqrt(r328750);
        double r328752 = atan2(1.0, 0.0);
        double r328753 = r328751 / r328752;
        double r328754 = t;
        double r328755 = 2.0;
        double r328756 = 3.0;
        double r328757 = r328756 * r328748;
        double r328758 = exp(r328757);
        double r328759 = log(r328758);
        double r328760 = r328745 - r328759;
        double r328761 = r328755 * r328760;
        double r328762 = sqrt(r328761);
        double r328763 = r328754 * r328762;
        double r328764 = r328753 / r328763;
        double r328765 = r328745 - r328748;
        double r328766 = r328751 / r328765;
        double r328767 = r328764 * r328766;
        return r328767;
}

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-log-exp0.4

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

  5. Applied associate-*l*0.4

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

  7. Applied add-sqr-sqrt0.8

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

  8. Applied associate-*l*0.7

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

  10. Applied add-sqr-sqrt0.7

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

  11. Applied times-frac0.7

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

  12. Simplified0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019318 
(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)))))