Average Error: 48.5 → 10.4
Time: 29.0s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -1.1514252442459826 \cdot 10^{-102} \lor \neg \left(k \le 7.02952442766517907 \cdot 10^{-84} \lor \neg \left(k \le 1.90701078281367166 \cdot 10^{146}\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]
\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}\;k \le -1.1514252442459826 \cdot 10^{-102} \lor \neg \left(k \le 7.02952442766517907 \cdot 10^{-84} \lor \neg \left(k \le 1.90701078281367166 \cdot 10^{146}\right)\right):\\
\;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r97234 = 2.0;
        double r97235 = t;
        double r97236 = 3.0;
        double r97237 = pow(r97235, r97236);
        double r97238 = l;
        double r97239 = r97238 * r97238;
        double r97240 = r97237 / r97239;
        double r97241 = k;
        double r97242 = sin(r97241);
        double r97243 = r97240 * r97242;
        double r97244 = tan(r97241);
        double r97245 = r97243 * r97244;
        double r97246 = 1.0;
        double r97247 = r97241 / r97235;
        double r97248 = pow(r97247, r97234);
        double r97249 = r97246 + r97248;
        double r97250 = r97249 - r97246;
        double r97251 = r97245 * r97250;
        double r97252 = r97234 / r97251;
        return r97252;
}
↓
double f(double t, double l, double k) {
        double r97253 = k;
        double r97254 = -1.1514252442459826e-102;
        bool r97255 = r97253 <= r97254;
        double r97256 = 7.029524427665179e-84;
        bool r97257 = r97253 <= r97256;
        double r97258 = 1.9070107828136717e+146;
        bool r97259 = r97253 <= r97258;
        double r97260 = !r97259;
        bool r97261 = r97257 || r97260;
        double r97262 = !r97261;
        bool r97263 = r97255 || r97262;
        double r97264 = 2.0;
        double r97265 = 1.0;
        double r97266 = cbrt(r97265);
        double r97267 = r97266 * r97266;
        double r97268 = pow(r97253, r97264);
        double r97269 = r97267 / r97268;
        double r97270 = 1.0;
        double r97271 = pow(r97269, r97270);
        double r97272 = t;
        double r97273 = pow(r97272, r97270);
        double r97274 = r97266 / r97273;
        double r97275 = pow(r97274, r97270);
        double r97276 = cos(r97253);
        double r97277 = l;
        double r97278 = r97276 * r97277;
        double r97279 = r97275 * r97278;
        double r97280 = r97271 * r97279;
        double r97281 = sin(r97253);
        double r97282 = fabs(r97281);
        double r97283 = r97282 / r97277;
        double r97284 = r97282 * r97283;
        double r97285 = r97280 / r97284;
        double r97286 = r97264 * r97285;
        double r97287 = 2.0;
        double r97288 = r97264 / r97287;
        double r97289 = pow(r97253, r97288);
        double r97290 = r97289 * r97273;
        double r97291 = r97289 * r97290;
        double r97292 = r97265 / r97291;
        double r97293 = pow(r97292, r97270);
        double r97294 = r97293 * r97278;
        double r97295 = r97294 / r97284;
        double r97296 = r97264 * r97295;
        double r97297 = r97263 ? r97286 : r97296;
        return r97297;
}

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 k < -1.1514252442459826e-102 or 7.029524427665179e-84 < k < 1.9070107828136717e+146

    1. Initial program 49.0

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

    2. Simplified39.7

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

    3. Taylor expanded around inf 18.8

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

    5. Applied add-sqr-sqrt18.8

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

    6. Applied times-frac18.8

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

    7. Simplified18.8

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

    8. Simplified18.0

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

    10. Applied frac-times17.6

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

    11. Applied associate-*r/11.7

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

    13. Applied add-cube-cbrt11.7

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

    14. Applied times-frac11.4

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

    15. Applied unpow-prod-down11.4

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

    16. Applied associate-*l*8.4

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

    if -1.1514252442459826e-102 < k < 7.029524427665179e-84 or 1.9070107828136717e+146 < k

    1. Initial program 47.5

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

    2. Simplified43.4

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

    3. Taylor expanded around inf 30.7

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

    5. Applied add-sqr-sqrt30.7

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

    6. Applied times-frac30.7

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

    7. Simplified30.7

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

    8. Simplified28.6

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

    10. Applied frac-times26.8

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

    11. Applied associate-*r/24.7

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

    13. Applied sqr-pow24.7

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

    14. Applied associate-*l*14.6

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -1.1514252442459826 \cdot 10^{-102} \lor \neg \left(k \le 7.02952442766517907 \cdot 10^{-84} \lor \neg \left(k \le 1.90701078281367166 \cdot 10^{146}\right)\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(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))))