Average Error: 0.5 → 0.3
Time: 23.0s
Precision: 64
\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
↓
\[\left(\left(\frac{1}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}} \cdot \frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{t}\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}\]
\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)}
↓
\left(\left(\frac{1}{\sqrt{2 \cdot \left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right)}} \cdot \frac{\frac{\sqrt{5 \cdot \left(v \cdot v\right)} + \sqrt{1}}{\pi}}{t}\right) \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \frac{\sqrt{1} - \sqrt{5 \cdot \left(v \cdot v\right)}}{1 - v \cdot v}
double f(double v, double t) {
        double r219968 = 1.0;
        double r219969 = 5.0;
        double r219970 = v;
        double r219971 = r219970 * r219970;
        double r219972 = r219969 * r219971;
        double r219973 = r219968 - r219972;
        double r219974 = atan2(1.0, 0.0);
        double r219975 = t;
        double r219976 = r219974 * r219975;
        double r219977 = 2.0;
        double r219978 = 3.0;
        double r219979 = r219978 * r219971;
        double r219980 = r219968 - r219979;
        double r219981 = r219977 * r219980;
        double r219982 = sqrt(r219981);
        double r219983 = r219976 * r219982;
        double r219984 = r219968 - r219971;
        double r219985 = r219983 * r219984;
        double r219986 = r219973 / r219985;
        return r219986;
}
↓
double f(double v, double t) {
        double r219987 = 1.0;
        double r219988 = 2.0;
        double r219989 = 1.0;
        double r219990 = r219989 * r219989;
        double r219991 = 3.0;
        double r219992 = r219991 * r219991;
        double r219993 = v;
        double r219994 = 4.0;
        double r219995 = pow(r219993, r219994);
        double r219996 = r219992 * r219995;
        double r219997 = r219990 - r219996;
        double r219998 = r219988 * r219997;
        double r219999 = sqrt(r219998);
        double r220000 = r219987 / r219999;
        double r220001 = 5.0;
        double r220002 = r219993 * r219993;
        double r220003 = r220001 * r220002;
        double r220004 = sqrt(r220003);
        double r220005 = sqrt(r219989);
        double r220006 = r220004 + r220005;
        double r220007 = atan2(1.0, 0.0);
        double r220008 = r220006 / r220007;
        double r220009 = t;
        double r220010 = r220008 / r220009;
        double r220011 = r220000 * r220010;
        double r220012 = r219991 * r220002;
        double r220013 = r219989 + r220012;
        double r220014 = sqrt(r220013);
        double r220015 = r220011 * r220014;
        double r220016 = r220005 - r220004;
        double r220017 = r219989 - r220002;
        double r220018 = r220016 / r220017;
        double r220019 = r220015 * r220018;
        return r220019;
}

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 add-log-exp0.5

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

  5. Applied add-sqr-sqrt0.5

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

  6. Applied add-sqr-sqrt0.5

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

  7. Applied difference-of-squares0.5

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

  8. Applied times-frac0.5

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

  9. Simplified1.1

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

  10. Simplified0.5

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

  12. Applied flip--0.5

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

  13. Applied associate-*r/0.5

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

  14. Applied sqrt-div0.5

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

  15. Applied associate-*r/0.5

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

  16. Applied associate-/r/0.5

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

  17. Simplified0.3

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

  19. Applied *-un-lft-identity0.3

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

  20. Applied *-un-lft-identity0.3

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

  21. Applied times-frac0.3

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

  22. Applied times-frac0.3

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

  23. Simplified0.3

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

  24. Final simplification0.3

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

Reproduce

herbie shell --seed 2019326 +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)))))