Average Error: 46.8 → 1.2
Time: 4.6m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{2}{\frac{\frac{k}{\ell}}{\frac{1}{t}}}}{\sin k}}{\frac{k}{\ell} \cdot \tan k}\]
\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{\frac{\frac{2}{\frac{\frac{k}{\ell}}{\frac{1}{t}}}}{\sin k}}{\frac{k}{\ell} \cdot \tan k}
double f(double t, double l, double k) {
        double r11408233 = 2.0;
        double r11408234 = t;
        double r11408235 = 3.0;
        double r11408236 = pow(r11408234, r11408235);
        double r11408237 = l;
        double r11408238 = r11408237 * r11408237;
        double r11408239 = r11408236 / r11408238;
        double r11408240 = k;
        double r11408241 = sin(r11408240);
        double r11408242 = r11408239 * r11408241;
        double r11408243 = tan(r11408240);
        double r11408244 = r11408242 * r11408243;
        double r11408245 = 1.0;
        double r11408246 = r11408240 / r11408234;
        double r11408247 = pow(r11408246, r11408233);
        double r11408248 = r11408245 + r11408247;
        double r11408249 = r11408248 - r11408245;
        double r11408250 = r11408244 * r11408249;
        double r11408251 = r11408233 / r11408250;
        return r11408251;
}
↓
double f(double t, double l, double k) {
        double r11408252 = 2.0;
        double r11408253 = k;
        double r11408254 = l;
        double r11408255 = r11408253 / r11408254;
        double r11408256 = 1.0;
        double r11408257 = t;
        double r11408258 = r11408256 / r11408257;
        double r11408259 = r11408255 / r11408258;
        double r11408260 = r11408252 / r11408259;
        double r11408261 = sin(r11408253);
        double r11408262 = r11408260 / r11408261;
        double r11408263 = tan(r11408253);
        double r11408264 = r11408255 * r11408263;
        double r11408265 = r11408262 / r11408264;
        return r11408265;
}

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 46.8

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

  2. Simplified26.3

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

  3. Taylor expanded around 0 14.1

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

  4. Simplified14.1

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

  6. Applied *-un-lft-identity14.1

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

  7. Applied div-inv14.2

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

  8. Applied *-un-lft-identity14.2

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

  9. Applied times-frac10.1

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

  10. Applied times-frac2.8

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

  11. Applied *-un-lft-identity2.8

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

  12. Applied times-frac2.5

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

  13. Applied times-frac1.6

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

  14. Simplified1.6

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

  16. Applied associate-*l/1.6

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

  17. Applied associate-/l/1.2

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

  18. Final simplification1.2

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

Reproduce

herbie shell --seed 2019133 
(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))))