Average Error: 48.1 → 11.1
Time: 37.9s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.5430129409159728 \cdot 10^{185}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]
\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}\;\ell \cdot \ell \le 4.5430129409159728 \cdot 10^{185}:\\
\;\;\;\;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(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r172 = 2.0;
        double r173 = t;
        double r174 = 3.0;
        double r175 = pow(r173, r174);
        double r176 = l;
        double r177 = r176 * r176;
        double r178 = r175 / r177;
        double r179 = k;
        double r180 = sin(r179);
        double r181 = r178 * r180;
        double r182 = tan(r179);
        double r183 = r181 * r182;
        double r184 = 1.0;
        double r185 = r179 / r173;
        double r186 = pow(r185, r172);
        double r187 = r184 + r186;
        double r188 = r187 - r184;
        double r189 = r183 * r188;
        double r190 = r172 / r189;
        return r190;
}
↓
double f(double t, double l, double k) {
        double r191 = l;
        double r192 = r191 * r191;
        double r193 = 4.543012940915973e+185;
        bool r194 = r192 <= r193;
        double r195 = 2.0;
        double r196 = 1.0;
        double r197 = k;
        double r198 = 2.0;
        double r199 = r195 / r198;
        double r200 = pow(r197, r199);
        double r201 = t;
        double r202 = 1.0;
        double r203 = pow(r201, r202);
        double r204 = r200 * r203;
        double r205 = r200 * r204;
        double r206 = r196 / r205;
        double r207 = pow(r206, r202);
        double r208 = cos(r197);
        double r209 = sin(r197);
        double r210 = r208 / r209;
        double r211 = r210 * r191;
        double r212 = r207 * r211;
        double r213 = r209 / r191;
        double r214 = r212 / r213;
        double r215 = r195 * r214;
        double r216 = pow(r197, r195);
        double r217 = r196 / r216;
        double r218 = pow(r217, r202);
        double r219 = r196 / r203;
        double r220 = pow(r219, r202);
        double r221 = r220 * r211;
        double r222 = r218 * r221;
        double r223 = r222 / r213;
        double r224 = r195 * r223;
        double r225 = r194 ? r215 : r224;
        return r225;
}

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 l) < 4.543012940915973e+185

    1. Initial program 44.7

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

    2. Simplified35.3

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

    3. Taylor expanded around inf 12.6

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

    5. Applied add-sqr-sqrt39.0

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

    6. Applied unpow-prod-down39.0

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

    7. Applied times-frac39.0

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

    8. Simplified39.0

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

    9. Simplified11.3

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

    11. Applied associate-*r/10.4

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

    12. Applied associate-*r/8.5

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

    14. Applied sqr-pow8.5

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

    15. Applied associate-*l*4.3

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

    if 4.543012940915973e+185 < (* l l)

    1. Initial program 59.0

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

    2. Simplified56.7

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

    3. Taylor expanded around inf 51.6

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

    5. Applied add-sqr-sqrt57.9

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

    6. Applied unpow-prod-down57.9

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

    7. Applied times-frac57.9

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

    8. Simplified57.9

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

    9. Simplified51.6

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

    11. Applied associate-*r/51.6

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

    12. Applied associate-*r/39.0

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

    14. Applied *-un-lft-identity39.0

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

    15. Applied times-frac38.5

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

    16. Applied unpow-prod-down38.5

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

    17. Applied associate-*l*33.0

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

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.5430129409159728 \cdot 10^{185}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)}{\frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\sin k} \cdot \ell\right)\right)}{\frac{\sin k}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))))