Average Error: 47.2 → 1.4
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{-2}{-\sin k} \cdot \frac{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{k}}}}{\frac{\tan k}{\frac{1}{\frac{\sqrt[3]{t}}{\frac{\ell}{k}}}}}\]
\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{-2}{-\sin k} \cdot \frac{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{k}}}}{\frac{\tan k}{\frac{1}{\frac{\sqrt[3]{t}}{\frac{\ell}{k}}}}}
double f(double t, double l, double k) {
        double r2661163 = 2.0;
        double r2661164 = t;
        double r2661165 = 3.0;
        double r2661166 = pow(r2661164, r2661165);
        double r2661167 = l;
        double r2661168 = r2661167 * r2661167;
        double r2661169 = r2661166 / r2661168;
        double r2661170 = k;
        double r2661171 = sin(r2661170);
        double r2661172 = r2661169 * r2661171;
        double r2661173 = tan(r2661170);
        double r2661174 = r2661172 * r2661173;
        double r2661175 = 1.0;
        double r2661176 = r2661170 / r2661164;
        double r2661177 = pow(r2661176, r2661163);
        double r2661178 = r2661175 + r2661177;
        double r2661179 = r2661178 - r2661175;
        double r2661180 = r2661174 * r2661179;
        double r2661181 = r2661163 / r2661180;
        return r2661181;
}
↓
double f(double t, double l, double k) {
        double r2661182 = -2.0;
        double r2661183 = k;
        double r2661184 = sin(r2661183);
        double r2661185 = -r2661184;
        double r2661186 = r2661182 / r2661185;
        double r2661187 = 1.0;
        double r2661188 = t;
        double r2661189 = cbrt(r2661188);
        double r2661190 = r2661189 * r2661189;
        double r2661191 = l;
        double r2661192 = r2661191 / r2661183;
        double r2661193 = r2661190 / r2661192;
        double r2661194 = r2661187 / r2661193;
        double r2661195 = tan(r2661183);
        double r2661196 = r2661189 / r2661192;
        double r2661197 = r2661187 / r2661196;
        double r2661198 = r2661195 / r2661197;
        double r2661199 = r2661194 / r2661198;
        double r2661200 = r2661186 * r2661199;
        return r2661200;
}

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 47.2

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

  2. Simplified21.1

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

  4. Applied frac-2neg21.1

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

  5. Simplified16.7

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

  7. Applied distribute-lft-neg-in16.7

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

  8. Applied div-inv16.7

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

  9. Applied times-frac16.5

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

  10. Simplified14.8

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

  11. Taylor expanded around inf 21.6

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

  12. Simplified8.3

    \[\leadsto \frac{-2}{-\sin k} \cdot \frac{\frac{1}{\color{blue}{\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}}}{\tan k}\]
  13. Using strategy rm

  14. Applied add-cube-cbrt8.6

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

  15. Applied times-frac3.2

    \[\leadsto \frac{-2}{-\sin k} \cdot \frac{\frac{1}{\color{blue}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{k}} \cdot \frac{\sqrt[3]{t}}{\frac{\ell}{k}}}}}{\tan k}\]

  16. Applied *-un-lft-identity3.2

    \[\leadsto \frac{-2}{-\sin k} \cdot \frac{\frac{\color{blue}{1 \cdot 1}}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{k}} \cdot \frac{\sqrt[3]{t}}{\frac{\ell}{k}}}}{\tan k}\]

  17. Applied times-frac2.8

    \[\leadsto \frac{-2}{-\sin k} \cdot \frac{\color{blue}{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{k}}} \cdot \frac{1}{\frac{\sqrt[3]{t}}{\frac{\ell}{k}}}}}{\tan k}\]

  18. Applied associate-/l*1.4

    \[\leadsto \frac{-2}{-\sin k} \cdot \color{blue}{\frac{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{k}}}}{\frac{\tan k}{\frac{1}{\frac{\sqrt[3]{t}}{\frac{\ell}{k}}}}}}\]

  19. Final simplification1.4

    \[\leadsto \frac{-2}{-\sin k} \cdot \frac{\frac{1}{\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\frac{\ell}{k}}}}{\frac{\tan k}{\frac{1}{\frac{\sqrt[3]{t}}{\frac{\ell}{k}}}}}\]

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(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))))