Average Error: 48.1 → 3.9
Time: 1.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \frac{{\left(\frac{\sqrt[3]{\frac{1}{{t}^{1}}} \cdot \sqrt[3]{\frac{1}{{t}^{1}}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\ell \cdot \cos k\right) \cdot {\left(\frac{\sqrt[3]{\frac{1}{{t}^{1}}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)}{\sin k \cdot \frac{\sin k}{\ell}}\]
\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 \cdot \frac{{\left(\frac{\sqrt[3]{\frac{1}{{t}^{1}}} \cdot \sqrt[3]{\frac{1}{{t}^{1}}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left(\ell \cdot \cos k\right) \cdot {\left(\frac{\sqrt[3]{\frac{1}{{t}^{1}}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)}{\sin k \cdot \frac{\sin k}{\ell}}
double f(double t, double l, double k) {
        double r151131 = 2.0;
        double r151132 = t;
        double r151133 = 3.0;
        double r151134 = pow(r151132, r151133);
        double r151135 = l;
        double r151136 = r151135 * r151135;
        double r151137 = r151134 / r151136;
        double r151138 = k;
        double r151139 = sin(r151138);
        double r151140 = r151137 * r151139;
        double r151141 = tan(r151138);
        double r151142 = r151140 * r151141;
        double r151143 = 1.0;
        double r151144 = r151138 / r151132;
        double r151145 = pow(r151144, r151131);
        double r151146 = r151143 + r151145;
        double r151147 = r151146 - r151143;
        double r151148 = r151142 * r151147;
        double r151149 = r151131 / r151148;
        return r151149;
}
↓
double f(double t, double l, double k) {
        double r151150 = 2.0;
        double r151151 = 1.0;
        double r151152 = t;
        double r151153 = 1.0;
        double r151154 = pow(r151152, r151153);
        double r151155 = r151151 / r151154;
        double r151156 = cbrt(r151155);
        double r151157 = r151156 * r151156;
        double r151158 = k;
        double r151159 = 2.0;
        double r151160 = r151150 / r151159;
        double r151161 = pow(r151158, r151160);
        double r151162 = r151157 / r151161;
        double r151163 = pow(r151162, r151153);
        double r151164 = l;
        double r151165 = cos(r151158);
        double r151166 = r151164 * r151165;
        double r151167 = r151156 / r151161;
        double r151168 = pow(r151167, r151153);
        double r151169 = r151166 * r151168;
        double r151170 = r151163 * r151169;
        double r151171 = sin(r151158);
        double r151172 = r151171 / r151164;
        double r151173 = r151171 * r151172;
        double r151174 = r151170 / r151173;
        double r151175 = r151150 * r151174;
        return r151175;
}

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 48.1

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

  2. Simplified40.4

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

  3. Taylor expanded around inf 22.1

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

  5. Applied add-sqr-sqrt43.2

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

  6. Applied unpow-prod-down43.2

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

  7. Applied times-frac43.2

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

  8. Simplified43.2

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

  9. Simplified21.0

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

  11. Applied frac-times20.1

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

  12. Applied associate-*r/15.8

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

  13. Simplified15.5

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

  15. Applied sqr-pow15.5

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

  16. Applied add-cube-cbrt15.8

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

  17. Applied times-frac10.8

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

  18. Applied unpow-prod-down10.8

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

  19. Applied associate-*l*3.9

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

  20. Simplified3.9

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

  21. Final simplification3.9

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

Reproduce

herbie shell --seed 2019179 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))