Average Error: 47.3 → 1.2
Time: 1.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\frac{\sqrt{\sqrt{2}}}{\frac{\tan k}{\sqrt{\sqrt{2}}}} \cdot \ell\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\]
\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}
↓
\frac{\left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\frac{\sqrt{\sqrt{2}}}{\frac{\tan k}{\sqrt{\sqrt{2}}}} \cdot \ell\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}
double f(double t, double l, double k) {
        double r3751985 = 2.0;
        double r3751986 = t;
        double r3751987 = 3.0;
        double r3751988 = pow(r3751986, r3751987);
        double r3751989 = l;
        double r3751990 = r3751989 * r3751989;
        double r3751991 = r3751988 / r3751990;
        double r3751992 = k;
        double r3751993 = sin(r3751992);
        double r3751994 = r3751991 * r3751993;
        double r3751995 = tan(r3751992);
        double r3751996 = r3751994 * r3751995;
        double r3751997 = 1.0;
        double r3751998 = r3751992 / r3751986;
        double r3751999 = pow(r3751998, r3751985);
        double r3752000 = r3751997 + r3751999;
        double r3752001 = r3752000 - r3751997;
        double r3752002 = r3751996 * r3752001;
        double r3752003 = r3751985 / r3752002;
        return r3752003;
}
↓
double f(double t, double l, double k) {
        double r3752004 = 2.0;
        double r3752005 = sqrt(r3752004);
        double r3752006 = cbrt(r3752005);
        double r3752007 = k;
        double r3752008 = r3752006 / r3752007;
        double r3752009 = sqrt(r3752005);
        double r3752010 = tan(r3752007);
        double r3752011 = r3752010 / r3752009;
        double r3752012 = r3752009 / r3752011;
        double r3752013 = l;
        double r3752014 = r3752012 * r3752013;
        double r3752015 = r3752008 * r3752014;
        double r3752016 = r3752006 * r3752006;
        double r3752017 = r3752015 * r3752016;
        double r3752018 = t;
        double r3752019 = r3752013 / r3752007;
        double r3752020 = r3752018 / r3752019;
        double r3752021 = sin(r3752007);
        double r3752022 = r3752020 * r3752021;
        double r3752023 = r3752017 / r3752022;
        return r3752023;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.3

    \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]

  2. Simplified16.7

    \[\leadsto \color{blue}{\frac{\frac{2}{\frac{t}{\ell}}}{\frac{\left(\frac{k}{t} \cdot t\right) \cdot \left(\frac{k}{t} \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \tan k\right)}}\]
  3. Using strategy rm

  4. Applied div-inv16.7

    \[\leadsto \frac{\frac{2}{\color{blue}{t \cdot \frac{1}{\ell}}}}{\frac{\left(\frac{k}{t} \cdot t\right) \cdot \left(\frac{k}{t} \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \tan k\right)}\]

  5. Applied add-sqr-sqrt16.9

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{t \cdot \frac{1}{\ell}}}{\frac{\left(\frac{k}{t} \cdot t\right) \cdot \left(\frac{k}{t} \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \tan k\right)}\]

  6. Applied times-frac16.7

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{2}}{t} \cdot \frac{\sqrt{2}}{\frac{1}{\ell}}}}{\frac{\left(\frac{k}{t} \cdot t\right) \cdot \left(\frac{k}{t} \cdot t\right)}{\ell} \cdot \left(\sin k \cdot \tan k\right)}\]

  7. Applied times-frac15.3

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{2}}{t}}{\frac{\left(\frac{k}{t} \cdot t\right) \cdot \left(\frac{k}{t} \cdot t\right)}{\ell}} \cdot \frac{\frac{\sqrt{2}}{\frac{1}{\ell}}}{\sin k \cdot \tan k}}\]

  8. Simplified7.4

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{t \cdot \frac{k}{1}}{\frac{\ell}{\frac{k}{1}}}}} \cdot \frac{\frac{\sqrt{2}}{\frac{1}{\ell}}}{\sin k \cdot \tan k}\]

  9. Simplified5.3

    \[\leadsto \frac{\sqrt{2}}{\frac{t \cdot \frac{k}{1}}{\frac{\ell}{\frac{k}{1}}}} \cdot \color{blue}{\left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)}\]
  10. Using strategy rm

  11. Applied associate-/r/5.3

    \[\leadsto \frac{\sqrt{2}}{\frac{t \cdot \frac{k}{1}}{\color{blue}{\frac{\ell}{k} \cdot 1}}} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\]

  12. Applied times-frac4.5

    \[\leadsto \frac{\sqrt{2}}{\color{blue}{\frac{t}{\frac{\ell}{k}} \cdot \frac{\frac{k}{1}}{1}}} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\]

  13. Applied add-cube-cbrt4.5

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \sqrt[3]{\sqrt{2}}}}{\frac{t}{\frac{\ell}{k}} \cdot \frac{\frac{k}{1}}{1}} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\]

  14. Applied times-frac4.2

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \frac{\sqrt[3]{\sqrt{2}}}{\frac{\frac{k}{1}}{1}}\right)} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\]

  15. Applied associate-*l*3.1

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{\frac{\frac{k}{1}}{1}} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\right)}\]

  16. Simplified3.1

    \[\leadsto \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\frac{\sqrt{2}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\right)}\]
  17. Using strategy rm

  18. Applied add-sqr-sqrt3.0

    \[\leadsto \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\frac{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}{\tan k} \cdot \frac{\ell}{\sin k}\right)\right)\]

  19. Applied associate-/l*3.0

    \[\leadsto \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\color{blue}{\frac{\sqrt{\sqrt{2}}}{\frac{\tan k}{\sqrt{\sqrt{2}}}}} \cdot \frac{\ell}{\sin k}\right)\right)\]
  20. Using strategy rm

  21. Applied associate-*r/3.0

    \[\leadsto \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \color{blue}{\frac{\frac{\sqrt{\sqrt{2}}}{\frac{\tan k}{\sqrt{\sqrt{2}}}} \cdot \ell}{\sin k}}\right)\]

  22. Applied associate-*r/3.0

    \[\leadsto \frac{\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}}{\frac{t}{\frac{\ell}{k}}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\frac{\sqrt{\sqrt{2}}}{\frac{\tan k}{\sqrt{\sqrt{2}}}} \cdot \ell\right)}{\sin k}}\]

  23. Applied frac-times1.2

    \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right) \cdot \left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\frac{\sqrt{\sqrt{2}}}{\frac{\tan k}{\sqrt{\sqrt{2}}}} \cdot \ell\right)\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}}\]

  24. Final simplification1.2

    \[\leadsto \frac{\left(\frac{\sqrt[3]{\sqrt{2}}}{k} \cdot \left(\frac{\sqrt{\sqrt{2}}}{\frac{\tan k}{\sqrt{\sqrt{2}}}} \cdot \ell\right)\right) \cdot \left(\sqrt[3]{\sqrt{2}} \cdot \sqrt[3]{\sqrt{2}}\right)}{\frac{t}{\frac{\ell}{k}} \cdot \sin k}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))