Average Error: 47.1 → 1.7
Time: 6.3m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -1.6308807875264197 \cdot 10^{-147}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{k} \cdot \left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\cos k}{t}\right)\right)\\ \mathbf{elif}\;\ell \le 3.733655982048962 \cdot 10^{-233}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{\left(\frac{\sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{\sin k}}} \cdot \frac{\sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{k}{\frac{\ell}{\sin k}}} \cdot \frac{\sqrt[3]{\frac{\ell}{\sin k}}}{\sqrt[3]{k}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{k} \cdot \left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\cos k}{t}\right)\right)\\ \end{array}\]
\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}\;\ell \le -1.6308807875264197 \cdot 10^{-147}:\\
\;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{k} \cdot \left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\cos k}{t}\right)\right)\\

\mathbf{elif}\;\ell \le 3.733655982048962 \cdot 10^{-233}:\\
\;\;\;\;\left(\frac{\frac{\cos k}{t}}{\left(\frac{\sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{\sin k}}} \cdot \frac{\sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{k}{\frac{\ell}{\sin k}}} \cdot \frac{\sqrt[3]{\frac{\ell}{\sin k}}}{\sqrt[3]{k}}\right) \cdot 2\\

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

\end{array}
double f(double t, double l, double k) {
        double r45565297 = 2.0;
        double r45565298 = t;
        double r45565299 = 3.0;
        double r45565300 = pow(r45565298, r45565299);
        double r45565301 = l;
        double r45565302 = r45565301 * r45565301;
        double r45565303 = r45565300 / r45565302;
        double r45565304 = k;
        double r45565305 = sin(r45565304);
        double r45565306 = r45565303 * r45565305;
        double r45565307 = tan(r45565304);
        double r45565308 = r45565306 * r45565307;
        double r45565309 = 1.0;
        double r45565310 = r45565304 / r45565298;
        double r45565311 = pow(r45565310, r45565297);
        double r45565312 = r45565309 + r45565311;
        double r45565313 = r45565312 - r45565309;
        double r45565314 = r45565308 * r45565313;
        double r45565315 = r45565297 / r45565314;
        return r45565315;
}
↓
double f(double t, double l, double k) {
        double r45565316 = l;
        double r45565317 = -1.6308807875264197e-147;
        bool r45565318 = r45565316 <= r45565317;
        double r45565319 = 2.0;
        double r45565320 = k;
        double r45565321 = sin(r45565320);
        double r45565322 = r45565316 / r45565321;
        double r45565323 = r45565322 / r45565320;
        double r45565324 = r45565320 * r45565321;
        double r45565325 = r45565316 / r45565324;
        double r45565326 = cos(r45565320);
        double r45565327 = t;
        double r45565328 = r45565326 / r45565327;
        double r45565329 = r45565325 * r45565328;
        double r45565330 = r45565323 * r45565329;
        double r45565331 = r45565319 * r45565330;
        double r45565332 = 3.733655982048962e-233;
        bool r45565333 = r45565316 <= r45565332;
        double r45565334 = cbrt(r45565320);
        double r45565335 = cbrt(r45565322);
        double r45565336 = r45565334 / r45565335;
        double r45565337 = r45565336 * r45565336;
        double r45565338 = r45565320 / r45565322;
        double r45565339 = r45565337 * r45565338;
        double r45565340 = r45565328 / r45565339;
        double r45565341 = r45565335 / r45565334;
        double r45565342 = r45565340 * r45565341;
        double r45565343 = r45565342 * r45565319;
        double r45565344 = r45565333 ? r45565343 : r45565331;
        double r45565345 = r45565318 ? r45565331 : r45565344;
        return r45565345;
}

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 < -1.6308807875264197e-147 or 3.733655982048962e-233 < l

    1. Initial program 47.6

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

    2. Simplified32.2

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

    3. Taylor expanded around -inf 25.0

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

    4. Simplified8.3

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

    6. Applied associate-*r*1.1

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

    8. Applied associate-/r*1.0

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

    if -1.6308807875264197e-147 < l < 3.733655982048962e-233

    1. Initial program 45.5

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

    2. Simplified24.7

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

    3. Taylor expanded around -inf 19.4

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

    4. Simplified8.5

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

    6. Applied associate-*r*5.7

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

    8. Applied associate-/r*5.7

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

    10. Applied add-cube-cbrt5.9

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

    11. Applied add-cube-cbrt5.9

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

    12. Applied times-frac5.9

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

    13. Applied associate-*r*5.9

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

    14. Simplified3.9

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

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -1.6308807875264197 \cdot 10^{-147}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{k} \cdot \left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\cos k}{t}\right)\right)\\ \mathbf{elif}\;\ell \le 3.733655982048962 \cdot 10^{-233}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t}}{\left(\frac{\sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{\sin k}}} \cdot \frac{\sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{\sin k}}}\right) \cdot \frac{k}{\frac{\ell}{\sin k}}} \cdot \frac{\sqrt[3]{\frac{\ell}{\sin k}}}{\sqrt[3]{k}}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\sin k}}{k} \cdot \left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\cos k}{t}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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))))