Average Error: 32.5 → 17.9
Time: 1.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.361561607826163938590433986165123404617 \cdot 10^{74} \lor \neg \left(\ell \le -3.671469527357900756356766403755497030844 \cdot 10^{-125} \lor \neg \left(\ell \le 1.147150844170901392374916916178894499446 \cdot 10^{-119}\right) \land \ell \le 2.547304444790666703643914625432659827925 \cdot 10^{119}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{t}^{\left(\frac{1}{3} \cdot 3\right)}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array}\]
\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}\;\ell \le -1.361561607826163938590433986165123404617 \cdot 10^{74} \lor \neg \left(\ell \le -3.671469527357900756356766403755497030844 \cdot 10^{-125} \lor \neg \left(\ell \le 1.147150844170901392374916916178894499446 \cdot 10^{-119}\right) \land \ell \le 2.547304444790666703643914625432659827925 \cdot 10^{119}\right):\\
\;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{t}^{\left(\frac{1}{3} \cdot 3\right)}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\

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

\end{array}
double f(double t, double l, double k) {
        double r125344 = 2.0;
        double r125345 = t;
        double r125346 = 3.0;
        double r125347 = pow(r125345, r125346);
        double r125348 = l;
        double r125349 = r125348 * r125348;
        double r125350 = r125347 / r125349;
        double r125351 = k;
        double r125352 = sin(r125351);
        double r125353 = r125350 * r125352;
        double r125354 = tan(r125351);
        double r125355 = r125353 * r125354;
        double r125356 = 1.0;
        double r125357 = r125351 / r125345;
        double r125358 = pow(r125357, r125344);
        double r125359 = r125356 + r125358;
        double r125360 = r125359 + r125356;
        double r125361 = r125355 * r125360;
        double r125362 = r125344 / r125361;
        return r125362;
}
↓
double f(double t, double l, double k) {
        double r125363 = l;
        double r125364 = -1.361561607826164e+74;
        bool r125365 = r125363 <= r125364;
        double r125366 = -3.671469527357901e-125;
        bool r125367 = r125363 <= r125366;
        double r125368 = 1.1471508441709014e-119;
        bool r125369 = r125363 <= r125368;
        double r125370 = !r125369;
        double r125371 = 2.5473044447906667e+119;
        bool r125372 = r125363 <= r125371;
        bool r125373 = r125370 && r125372;
        bool r125374 = r125367 || r125373;
        double r125375 = !r125374;
        bool r125376 = r125365 || r125375;
        double r125377 = 2.0;
        double r125378 = t;
        double r125379 = cbrt(r125378);
        double r125380 = r125379 * r125379;
        double r125381 = 3.0;
        double r125382 = 2.0;
        double r125383 = r125381 / r125382;
        double r125384 = pow(r125380, r125383);
        double r125385 = cbrt(r125363);
        double r125386 = r125385 * r125385;
        double r125387 = r125384 / r125386;
        double r125388 = r125384 / r125385;
        double r125389 = r125387 * r125388;
        double r125390 = 0.3333333333333333;
        double r125391 = r125390 * r125381;
        double r125392 = pow(r125378, r125391);
        double r125393 = r125392 / r125363;
        double r125394 = r125389 * r125393;
        double r125395 = k;
        double r125396 = sin(r125395);
        double r125397 = r125394 * r125396;
        double r125398 = tan(r125395);
        double r125399 = 1.0;
        double r125400 = r125395 / r125378;
        double r125401 = pow(r125400, r125377);
        double r125402 = r125399 + r125401;
        double r125403 = r125402 + r125399;
        double r125404 = r125398 * r125403;
        double r125405 = r125397 * r125404;
        double r125406 = r125377 / r125405;
        double r125407 = 3.0;
        double r125408 = pow(r125378, r125407);
        double r125409 = pow(r125396, r125382);
        double r125410 = r125408 * r125409;
        double r125411 = cos(r125395);
        double r125412 = pow(r125363, r125382);
        double r125413 = r125411 * r125412;
        double r125414 = r125410 / r125413;
        double r125415 = r125377 * r125414;
        double r125416 = 1.0;
        double r125417 = -1.0;
        double r125418 = pow(r125417, r125381);
        double r125419 = r125416 / r125418;
        double r125420 = pow(r125419, r125399);
        double r125421 = pow(r125395, r125382);
        double r125422 = r125421 * r125409;
        double r125423 = r125378 * r125422;
        double r125424 = r125423 / r125413;
        double r125425 = r125420 * r125424;
        double r125426 = r125415 - r125425;
        double r125427 = r125377 / r125426;
        double r125428 = r125376 ? r125406 : r125427;
        return r125428;
}

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 l < -1.361561607826164e+74 or -3.671469527357901e-125 < l < 1.1471508441709014e-119 or 2.5473044447906667e+119 < l

    1. Initial program 36.8

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

    3. Applied add-cube-cbrt36.8

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

    4. Applied unpow-prod-down36.8

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

    5. Applied times-frac26.8

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt26.8

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

    8. Applied sqr-pow26.8

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

    9. Applied times-frac18.9

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
    10. Using strategy rm

    11. Applied associate-*l*18.6

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]
    12. Using strategy rm

    13. Applied pow1/341.3

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

    14. Applied pow-pow18.5

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

    if -1.361561607826164e+74 < l < -3.671469527357901e-125 or 1.1471508441709014e-119 < l < 2.5473044447906667e+119

    1. Initial program 25.5

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

    2. Taylor expanded around -inf 16.8

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.361561607826163938590433986165123404617 \cdot 10^{74} \lor \neg \left(\ell \le -3.671469527357900756356766403755497030844 \cdot 10^{-125} \lor \neg \left(\ell \le 1.147150844170901392374916916178894499446 \cdot 10^{-119}\right) \land \ell \le 2.547304444790666703643914625432659827925 \cdot 10^{119}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right) \cdot \frac{{t}^{\left(\frac{1}{3} \cdot 3\right)}}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}} - {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{t \cdot \left({k}^{2} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))