Average Error: 46.8 → 4.2
Time: 5.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\left(\frac{1}{k} \cdot \frac{t}{t \cdot \frac{\sin k}{\ell}}\right) \cdot \frac{2 \cdot \frac{\ell}{k}}{t}}{\tan k}\]
\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{\left(\frac{1}{k} \cdot \frac{t}{t \cdot \frac{\sin k}{\ell}}\right) \cdot \frac{2 \cdot \frac{\ell}{k}}{t}}{\tan k}
double f(double t, double l, double k) {
        double r12019414 = 2.0;
        double r12019415 = t;
        double r12019416 = 3.0;
        double r12019417 = pow(r12019415, r12019416);
        double r12019418 = l;
        double r12019419 = r12019418 * r12019418;
        double r12019420 = r12019417 / r12019419;
        double r12019421 = k;
        double r12019422 = sin(r12019421);
        double r12019423 = r12019420 * r12019422;
        double r12019424 = tan(r12019421);
        double r12019425 = r12019423 * r12019424;
        double r12019426 = 1.0;
        double r12019427 = r12019421 / r12019415;
        double r12019428 = pow(r12019427, r12019414);
        double r12019429 = r12019426 + r12019428;
        double r12019430 = r12019429 - r12019426;
        double r12019431 = r12019425 * r12019430;
        double r12019432 = r12019414 / r12019431;
        return r12019432;
}
↓
double f(double t, double l, double k) {
        double r12019433 = 1.0;
        double r12019434 = k;
        double r12019435 = r12019433 / r12019434;
        double r12019436 = t;
        double r12019437 = sin(r12019434);
        double r12019438 = l;
        double r12019439 = r12019437 / r12019438;
        double r12019440 = r12019436 * r12019439;
        double r12019441 = r12019436 / r12019440;
        double r12019442 = r12019435 * r12019441;
        double r12019443 = 2.0;
        double r12019444 = r12019438 / r12019434;
        double r12019445 = r12019443 * r12019444;
        double r12019446 = r12019445 / r12019436;
        double r12019447 = r12019442 * r12019446;
        double r12019448 = tan(r12019434);
        double r12019449 = r12019447 / r12019448;
        return r12019449;
}

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. Simplified29.4

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

  4. Applied *-un-lft-identity29.4

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

  5. Applied times-frac29.3

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

  6. Applied times-frac17.7

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

  8. Applied associate-/r/17.7

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

  9. Applied associate-/r*10.9

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

  11. Applied *-un-lft-identity10.9

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

  12. Applied *-un-lft-identity10.9

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

  13. Applied times-frac10.9

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

  14. Applied *-un-lft-identity10.9

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

  15. Applied times-frac10.9

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

  16. Applied div-inv10.9

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

  17. Applied add-sqr-sqrt10.9

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

  18. Applied times-frac10.9

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

  19. Applied times-frac10.9

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

  20. Simplified10.9

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

  21. Simplified10.8

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

  22. Taylor expanded around inf 4.2

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

  23. Final simplification4.2

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

Reproduce

herbie shell --seed 2019135 
(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))))