Average Error: 46.7 → 7.0
Time: 2.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{1}{k} \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{1}{t}}\right) \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\cos k}{\sin k \cdot k} \cdot 2\right)\right)\]
\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)}
↓
\left(\frac{1}{k} \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{1}{t}}\right) \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\cos k}{\sin k \cdot k} \cdot 2\right)\right)
double f(double t, double l, double k) {
        double r2468165 = 2.0;
        double r2468166 = t;
        double r2468167 = 3.0;
        double r2468168 = pow(r2468166, r2468167);
        double r2468169 = l;
        double r2468170 = r2468169 * r2468169;
        double r2468171 = r2468168 / r2468170;
        double r2468172 = k;
        double r2468173 = sin(r2468172);
        double r2468174 = r2468171 * r2468173;
        double r2468175 = tan(r2468172);
        double r2468176 = r2468174 * r2468175;
        double r2468177 = 1.0;
        double r2468178 = r2468172 / r2468166;
        double r2468179 = pow(r2468178, r2468165);
        double r2468180 = r2468177 + r2468179;
        double r2468181 = r2468180 - r2468177;
        double r2468182 = r2468176 * r2468181;
        double r2468183 = r2468165 / r2468182;
        return r2468183;
}
↓
double f(double t, double l, double k) {
        double r2468184 = 1.0;
        double r2468185 = k;
        double r2468186 = r2468184 / r2468185;
        double r2468187 = l;
        double r2468188 = t;
        double r2468189 = r2468187 / r2468188;
        double r2468190 = sin(r2468185);
        double r2468191 = r2468189 / r2468190;
        double r2468192 = r2468184 / r2468188;
        double r2468193 = r2468191 / r2468192;
        double r2468194 = r2468186 * r2468193;
        double r2468195 = cos(r2468185);
        double r2468196 = r2468190 * r2468185;
        double r2468197 = r2468195 / r2468196;
        double r2468198 = 2.0;
        double r2468199 = r2468197 * r2468198;
        double r2468200 = r2468189 * r2468199;
        double r2468201 = r2468194 * r2468200;
        return r2468201;
}

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 46.7

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

  2. Simplified30.4

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

  4. Applied times-frac20.2

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

  6. Applied *-un-lft-identity20.2

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

  7. Applied *-un-lft-identity20.2

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

  8. Applied times-frac19.3

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

  9. Applied times-frac13.1

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

  10. Applied associate-*r*11.7

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

  11. Taylor expanded around inf 11.0

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

  13. Applied div-inv11.0

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

  14. Applied *-un-lft-identity11.0

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

  15. Applied *-un-lft-identity11.0

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

  16. Applied *-un-lft-identity11.0

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

  17. Applied times-frac11.0

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

  18. Applied times-frac11.0

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

  19. Applied times-frac7.0

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

  20. Simplified7.0

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

  21. Final simplification7.0

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

Reproduce

herbie shell --seed 2019152 +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))))