Average Error: 48.7 → 9.2
Time: 1.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.677360797658749296700885414039032241489 \cdot 10^{73} \lor \neg \left(t \le -3851782140446467\right) \land \left(t \le 3.61662912678721611260546656818814338713 \cdot 10^{-28} \lor \neg \left(t \le 1.851317435285136661897085166390213799089 \cdot 10^{96}\right)\right):\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \left(2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\right) \cdot \left(\frac{1}{\tan k} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{2}{{t}^{3}}}{\frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right)\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.677360797658749296700885414039032241489 \cdot 10^{73} \lor \neg \left(t \le -3851782140446467\right) \land \left(t \le 3.61662912678721611260546656818814338713 \cdot 10^{-28} \lor \neg \left(t \le 1.851317435285136661897085166390213799089 \cdot 10^{96}\right)\right):\\
\;\;\;\;\left(\frac{\ell}{\sin k} \cdot \left(2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\right) \cdot \left(\frac{1}{\tan k} \cdot \ell\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{2}{{t}^{3}}}{\frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right)\\

\end{array}
double f(double t, double l, double k) {
        double r120465 = 2.0;
        double r120466 = t;
        double r120467 = 3.0;
        double r120468 = pow(r120466, r120467);
        double r120469 = l;
        double r120470 = r120469 * r120469;
        double r120471 = r120468 / r120470;
        double r120472 = k;
        double r120473 = sin(r120472);
        double r120474 = r120471 * r120473;
        double r120475 = tan(r120472);
        double r120476 = r120474 * r120475;
        double r120477 = 1.0;
        double r120478 = r120472 / r120466;
        double r120479 = pow(r120478, r120465);
        double r120480 = r120477 + r120479;
        double r120481 = r120480 - r120477;
        double r120482 = r120476 * r120481;
        double r120483 = r120465 / r120482;
        return r120483;
}
↓
double f(double t, double l, double k) {
        double r120484 = t;
        double r120485 = -1.6773607976587493e+73;
        bool r120486 = r120484 <= r120485;
        double r120487 = -3851782140446467.0;
        bool r120488 = r120484 <= r120487;
        double r120489 = !r120488;
        double r120490 = 3.616629126787216e-28;
        bool r120491 = r120484 <= r120490;
        double r120492 = 1.8513174352851367e+96;
        bool r120493 = r120484 <= r120492;
        double r120494 = !r120493;
        bool r120495 = r120491 || r120494;
        bool r120496 = r120489 && r120495;
        bool r120497 = r120486 || r120496;
        double r120498 = l;
        double r120499 = k;
        double r120500 = sin(r120499);
        double r120501 = r120498 / r120500;
        double r120502 = 2.0;
        double r120503 = 1.0;
        double r120504 = 2.0;
        double r120505 = r120502 / r120504;
        double r120506 = pow(r120499, r120505);
        double r120507 = 1.0;
        double r120508 = pow(r120484, r120507);
        double r120509 = r120506 * r120508;
        double r120510 = r120503 / r120509;
        double r120511 = r120510 / r120506;
        double r120512 = pow(r120511, r120507);
        double r120513 = r120502 * r120512;
        double r120514 = r120501 * r120513;
        double r120515 = tan(r120499);
        double r120516 = r120503 / r120515;
        double r120517 = r120516 * r120498;
        double r120518 = r120514 * r120517;
        double r120519 = r120499 / r120484;
        double r120520 = pow(r120519, r120505);
        double r120521 = r120503 / r120520;
        double r120522 = r120498 / r120515;
        double r120523 = 3.0;
        double r120524 = pow(r120484, r120523);
        double r120525 = r120502 / r120524;
        double r120526 = r120500 / r120498;
        double r120527 = r120525 / r120526;
        double r120528 = r120527 / r120520;
        double r120529 = r120522 * r120528;
        double r120530 = r120521 * r120529;
        double r120531 = r120497 ? r120518 : r120530;
        return r120531;
}

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. Split input into 2 regimes

  2. if t < -1.6773607976587493e+73 or -3851782140446467.0 < t < 3.616629126787216e-28 or 1.8513174352851367e+96 < t

    1. Initial program 51.7

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

    2. Simplified41.7

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

    3. Taylor expanded around inf 16.0

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

    4. Simplified15.9

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

    6. Applied sqr-pow15.9

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

    7. Applied associate-/r*10.3

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

    9. Applied div-inv10.3

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

    11. Applied *-un-lft-identity10.3

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

    12. Applied add-sqr-sqrt10.3

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

    13. Applied times-frac10.3

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

    14. Applied associate-/l*10.3

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

    15. Simplified10.2

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

    if -1.6773607976587493e+73 < t < -3851782140446467.0 or 3.616629126787216e-28 < t < 1.8513174352851367e+96

    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. Simplified12.3

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

    4. Applied sqr-pow12.3

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

    5. Applied *-un-lft-identity12.3

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

    6. Applied times-frac6.0

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

    7. Applied associate-*l*4.1

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

    8. Simplified4.0

      \[\leadsto \frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \color{blue}{\left(\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{2}{{t}^{3}}}{\frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.677360797658749296700885414039032241489 \cdot 10^{73} \lor \neg \left(t \le -3851782140446467\right) \land \left(t \le 3.61662912678721611260546656818814338713 \cdot 10^{-28} \lor \neg \left(t \le 1.851317435285136661897085166390213799089 \cdot 10^{96}\right)\right):\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \left(2 \cdot {\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)\right) \cdot \left(\frac{1}{\tan k} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{2}{{t}^{3}}}{\frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{\left(\frac{2}{2}\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))