Average Error: 48.1 → 2.7
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -4.921872527115797496752548577916202694591 \cdot 10^{-130} \lor \neg \left(k \le 2.033695468655643953129386056528103986017 \cdot 10^{-153}\right):\\ \;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\ell}{\left|k\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\cos k}{\sin k} \cdot \frac{{\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1}}{\sin k}\right) \cdot \left(\frac{\ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right)\right) \cdot 2\\ \end{array}\]
\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}\;k \le -4.921872527115797496752548577916202694591 \cdot 10^{-130} \lor \neg \left(k \le 2.033695468655643953129386056528103986017 \cdot 10^{-153}\right):\\
\;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\ell}{\left|k\right|}\right)\\

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

\end{array}
double f(double t, double l, double k) {
        double r116145 = 2.0;
        double r116146 = t;
        double r116147 = 3.0;
        double r116148 = pow(r116146, r116147);
        double r116149 = l;
        double r116150 = r116149 * r116149;
        double r116151 = r116148 / r116150;
        double r116152 = k;
        double r116153 = sin(r116152);
        double r116154 = r116151 * r116153;
        double r116155 = tan(r116152);
        double r116156 = r116154 * r116155;
        double r116157 = 1.0;
        double r116158 = r116152 / r116146;
        double r116159 = pow(r116158, r116145);
        double r116160 = r116157 + r116159;
        double r116161 = r116160 - r116157;
        double r116162 = r116156 * r116161;
        double r116163 = r116145 / r116162;
        return r116163;
}
↓
double f(double t, double l, double k) {
        double r116164 = k;
        double r116165 = -4.9218725271157975e-130;
        bool r116166 = r116164 <= r116165;
        double r116167 = 2.033695468655644e-153;
        bool r116168 = r116164 <= r116167;
        double r116169 = !r116168;
        bool r116170 = r116166 || r116169;
        double r116171 = 2.0;
        double r116172 = 1.0;
        double r116173 = -1.0;
        double r116174 = pow(r116173, r116171);
        double r116175 = t;
        double r116176 = 1.0;
        double r116177 = pow(r116175, r116176);
        double r116178 = r116174 * r116177;
        double r116179 = r116172 / r116178;
        double r116180 = pow(r116179, r116176);
        double r116181 = cos(r116164);
        double r116182 = sin(r116164);
        double r116183 = 2.0;
        double r116184 = pow(r116182, r116183);
        double r116185 = r116181 / r116184;
        double r116186 = r116180 * r116185;
        double r116187 = l;
        double r116188 = fabs(r116164);
        double r116189 = r116187 / r116188;
        double r116190 = r116186 * r116189;
        double r116191 = r116190 * r116189;
        double r116192 = r116171 * r116191;
        double r116193 = r116181 / r116182;
        double r116194 = r116180 / r116182;
        double r116195 = r116193 * r116194;
        double r116196 = r116189 * r116189;
        double r116197 = r116195 * r116196;
        double r116198 = r116197 * r116171;
        double r116199 = r116170 ? r116192 : r116198;
        return r116199;
}

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 < -4.9218725271157975e-130 or 2.033695468655644e-153 < k

    1. Initial program 47.3

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

    2. Simplified36.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \frac{{t}^{3}}{\ell}}}{\frac{\sin k}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} - 0\right)}}\]

    3. Taylor expanded around -inf 21.9

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

    5. Applied times-frac20.5

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

    6. Applied associate-*r*20.0

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

    8. Applied add-sqr-sqrt20.0

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

    9. Applied sqr-pow20.0

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

    10. Applied times-frac15.3

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

    11. Simplified15.3

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

    12. Simplified7.6

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

    14. Applied associate-*r*2.0

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

    15. Simplified2.0

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

    if -4.9218725271157975e-130 < k < 2.033695468655644e-153

    1. Initial program 64.0

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

    2. Simplified58.3

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \frac{{t}^{3}}{\ell}}}{\frac{\sin k}{\ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} - 0\right)}}\]

    3. Taylor expanded around -inf 64.0

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

    5. Applied times-frac63.0

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

    6. Applied associate-*r*60.4

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

    8. Applied add-sqr-sqrt60.4

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

    9. Applied sqr-pow60.4

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

    10. Applied times-frac55.4

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

    11. Simplified55.4

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

    12. Simplified55.3

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

    14. Applied sqr-pow55.3

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

    15. Applied *-un-lft-identity55.3

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

    16. Applied times-frac55.3

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

    17. Applied associate-*r*16.9

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

    18. Simplified16.8

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

  4. Final simplification2.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -4.921872527115797496752548577916202694591 \cdot 10^{-130} \lor \neg \left(k \le 2.033695468655643953129386056528103986017 \cdot 10^{-153}\right):\\ \;\;\;\;2 \cdot \left(\left(\left({\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{{\left(\sin k\right)}^{2}}\right) \cdot \frac{\ell}{\left|k\right|}\right) \cdot \frac{\ell}{\left|k\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{\cos k}{\sin k} \cdot \frac{{\left(\frac{1}{{-1}^{2} \cdot {t}^{1}}\right)}^{1}}{\sin k}\right) \cdot \left(\frac{\ell}{\left|k\right|} \cdot \frac{\ell}{\left|k\right|}\right)\right) \cdot 2\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))