Average Error: 32.4 → 12.6
Time: 1.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\left(\frac{\sqrt{\frac{\sqrt{2}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{t} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \frac{\sqrt{\frac{\sqrt{2}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{\frac{\tan k}{\frac{\ell}{t}}}\right) \cdot \frac{\ell}{t}\]
\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(\left(\frac{\sqrt{\frac{\sqrt{2}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{t} \cdot \frac{\sqrt{2}}{\sin k}\right) \cdot \frac{\sqrt{\frac{\sqrt{2}}{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{\frac{\tan k}{\frac{\ell}{t}}}\right) \cdot \frac{\ell}{t}
double f(double t, double l, double k) {
        double r3224313 = 2.0;
        double r3224314 = t;
        double r3224315 = 3.0;
        double r3224316 = pow(r3224314, r3224315);
        double r3224317 = l;
        double r3224318 = r3224317 * r3224317;
        double r3224319 = r3224316 / r3224318;
        double r3224320 = k;
        double r3224321 = sin(r3224320);
        double r3224322 = r3224319 * r3224321;
        double r3224323 = tan(r3224320);
        double r3224324 = r3224322 * r3224323;
        double r3224325 = 1.0;
        double r3224326 = r3224320 / r3224314;
        double r3224327 = pow(r3224326, r3224313);
        double r3224328 = r3224325 + r3224327;
        double r3224329 = r3224328 + r3224325;
        double r3224330 = r3224324 * r3224329;
        double r3224331 = r3224313 / r3224330;
        return r3224331;
}
↓
double f(double t, double l, double k) {
        double r3224332 = 2.0;
        double r3224333 = sqrt(r3224332);
        double r3224334 = k;
        double r3224335 = t;
        double r3224336 = r3224334 / r3224335;
        double r3224337 = r3224336 * r3224336;
        double r3224338 = r3224332 + r3224337;
        double r3224339 = r3224333 / r3224338;
        double r3224340 = sqrt(r3224339);
        double r3224341 = r3224340 / r3224335;
        double r3224342 = sin(r3224334);
        double r3224343 = r3224333 / r3224342;
        double r3224344 = r3224341 * r3224343;
        double r3224345 = tan(r3224334);
        double r3224346 = l;
        double r3224347 = r3224346 / r3224335;
        double r3224348 = r3224345 / r3224347;
        double r3224349 = r3224340 / r3224348;
        double r3224350 = r3224344 * r3224349;
        double r3224351 = r3224350 * r3224347;
        return r3224351;
}

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 32.4

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

  2. Simplified17.7

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

  4. Applied associate-*l/16.7

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

  5. Applied associate-*r/15.3

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

  6. Applied associate-/r/14.0

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

  8. Applied *-un-lft-identity14.0

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

  9. Applied add-sqr-sqrt14.1

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

  10. Applied times-frac14.1

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

  11. Applied times-frac13.8

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

  12. Simplified13.8

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

  14. Applied add-sqr-sqrt13.9

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

  15. Applied times-frac12.6

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

  16. Applied associate-*r*12.6

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

  17. Final simplification12.6

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

Reproduce

herbie shell --seed 2019139 
(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))))