Average Error: 46.8 → 2.6
Time: 1.8m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\frac{\sqrt{\sqrt[3]{\sqrt{2}}}}{\tan k}}{\frac{\sqrt[3]{\sqrt[3]{t}}}{\sqrt[3]{\ell}}} \cdot \frac{\frac{\left|\sqrt[3]{\sqrt{2}}\right|}{\sin k}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\left(\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{\frac{\sqrt{2}}{k}}{\sqrt[3]{t}}\right)\right) \cdot \frac{\frac{\sqrt{\sqrt{2}}}{k}}{\frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}\right)\]
\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{\frac{\sqrt{\sqrt[3]{\sqrt{2}}}}{\tan k}}{\frac{\sqrt[3]{\sqrt[3]{t}}}{\sqrt[3]{\ell}}} \cdot \frac{\frac{\left|\sqrt[3]{\sqrt{2}}\right|}{\sin k}}{\sqrt[3]{\sqrt[3]{t} \cdot \sqrt[3]{t}}}\right) \cdot \left(\left(\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{\frac{\sqrt{2}}{k}}{\sqrt[3]{t}}\right)\right) \cdot \frac{\frac{\sqrt{\sqrt{2}}}{k}}{\frac{\sqrt[3]{t}}{\sqrt[3]{\ell}}}\right)
double f(double t, double l, double k) {
        double r3300559 = 2.0;
        double r3300560 = t;
        double r3300561 = 3.0;
        double r3300562 = pow(r3300560, r3300561);
        double r3300563 = l;
        double r3300564 = r3300563 * r3300563;
        double r3300565 = r3300562 / r3300564;
        double r3300566 = k;
        double r3300567 = sin(r3300566);
        double r3300568 = r3300565 * r3300567;
        double r3300569 = tan(r3300566);
        double r3300570 = r3300568 * r3300569;
        double r3300571 = 1.0;
        double r3300572 = r3300566 / r3300560;
        double r3300573 = pow(r3300572, r3300559);
        double r3300574 = r3300571 + r3300573;
        double r3300575 = r3300574 - r3300571;
        double r3300576 = r3300570 * r3300575;
        double r3300577 = r3300559 / r3300576;
        return r3300577;
}
↓
double f(double t, double l, double k) {
        double r3300578 = 2.0;
        double r3300579 = sqrt(r3300578);
        double r3300580 = cbrt(r3300579);
        double r3300581 = sqrt(r3300580);
        double r3300582 = k;
        double r3300583 = tan(r3300582);
        double r3300584 = r3300581 / r3300583;
        double r3300585 = t;
        double r3300586 = cbrt(r3300585);
        double r3300587 = cbrt(r3300586);
        double r3300588 = l;
        double r3300589 = cbrt(r3300588);
        double r3300590 = r3300587 / r3300589;
        double r3300591 = r3300584 / r3300590;
        double r3300592 = fabs(r3300580);
        double r3300593 = sin(r3300582);
        double r3300594 = r3300592 / r3300593;
        double r3300595 = r3300586 * r3300586;
        double r3300596 = cbrt(r3300595);
        double r3300597 = r3300594 / r3300596;
        double r3300598 = r3300591 * r3300597;
        double r3300599 = r3300579 / r3300582;
        double r3300600 = r3300599 / r3300586;
        double r3300601 = r3300589 * r3300600;
        double r3300602 = r3300588 * r3300601;
        double r3300603 = sqrt(r3300579);
        double r3300604 = r3300603 / r3300582;
        double r3300605 = r3300586 / r3300589;
        double r3300606 = r3300604 / r3300605;
        double r3300607 = r3300602 * r3300606;
        double r3300608 = r3300598 * r3300607;
        return r3300608;
}

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. Simplified29.6

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

  4. Applied associate-*r/30.1

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

  5. Applied frac-times40.8

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

  6. Applied associate-/r/40.8

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

  7. Applied times-frac36.9

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

  8. Simplified17.7

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

  10. Applied add-cube-cbrt17.9

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

  11. Applied add-cube-cbrt17.9

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

  12. Applied times-frac18.0

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

  13. Applied add-sqr-sqrt18.0

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

  14. Applied times-frac17.7

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

  15. Applied times-frac9.6

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

  16. Applied associate-*l*6.3

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

  17. Simplified6.8

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

  19. Applied times-frac6.8

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

  20. Applied add-sqr-sqrt6.8

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

  21. Applied sqrt-prod6.8

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

  22. Applied times-frac6.8

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

  23. Applied times-frac4.1

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

  24. Applied associate-*l*4.4

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

  26. Applied *-un-lft-identity4.4

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

  27. Applied add-cube-cbrt4.4

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

  28. Applied cbrt-prod4.4

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

  29. Applied times-frac4.4

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

  30. Applied add-cube-cbrt4.4

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

  31. Applied sqrt-prod4.4

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

  32. Applied times-frac4.4

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

  33. Applied times-frac2.6

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

  34. Simplified2.6

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

  35. Final simplification2.6

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

Reproduce

herbie shell --seed 2019141 +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))))