Average Error: 47.4 → 2.5
Time: 1.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\ell}{k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{\sin k}{\frac{\ell}{k}}}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}\]
\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)}
↓
\left(\frac{\ell}{k} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{\sin k}{\frac{\ell}{k}}}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}
double f(double t, double l, double k) {
        double r3022242 = 2.0;
        double r3022243 = t;
        double r3022244 = 3.0;
        double r3022245 = pow(r3022243, r3022244);
        double r3022246 = l;
        double r3022247 = r3022246 * r3022246;
        double r3022248 = r3022245 / r3022247;
        double r3022249 = k;
        double r3022250 = sin(r3022249);
        double r3022251 = r3022248 * r3022250;
        double r3022252 = tan(r3022249);
        double r3022253 = r3022251 * r3022252;
        double r3022254 = 1.0;
        double r3022255 = r3022249 / r3022243;
        double r3022256 = pow(r3022255, r3022242);
        double r3022257 = r3022254 + r3022256;
        double r3022258 = r3022257 - r3022254;
        double r3022259 = r3022253 * r3022258;
        double r3022260 = r3022242 / r3022259;
        return r3022260;
}
↓
double f(double t, double l, double k) {
        double r3022261 = l;
        double r3022262 = k;
        double r3022263 = r3022261 / r3022262;
        double r3022264 = 2.0;
        double r3022265 = t;
        double r3022266 = r3022264 / r3022265;
        double r3022267 = cbrt(r3022266);
        double r3022268 = tan(r3022262);
        double r3022269 = cbrt(r3022268);
        double r3022270 = r3022267 / r3022269;
        double r3022271 = r3022270 * r3022270;
        double r3022272 = sin(r3022262);
        double r3022273 = r3022272 / r3022263;
        double r3022274 = r3022271 / r3022273;
        double r3022275 = r3022263 * r3022274;
        double r3022276 = r3022275 * r3022270;
        return r3022276;
}

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 47.4

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

  2. Simplified22.1

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

  4. Applied associate-*r*18.2

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

  6. Applied *-un-lft-identity18.2

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

  7. Applied times-frac18.2

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

  8. Applied times-frac18.0

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

  9. Simplified14.4

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

  10. Taylor expanded around inf 23.0

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

  11. Simplified7.7

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

  13. Applied add-cube-cbrt8.0

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

  14. Applied add-cube-cbrt8.2

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

  15. Applied times-frac8.2

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

  16. Applied associate-*r*7.7

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

  17. Simplified2.5

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

  18. Final simplification2.5

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

Reproduce

herbie shell --seed 2019151 +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))))