Average Error: 48.1 → 16.7
Time: 32.1s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 6.05869271606677315 \cdot 10^{302}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k\right) \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \ell\right) \cdot \ell\\ \end{array}\]
\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 6.05869271606677315 \cdot 10^{302}:\\
\;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k\right) \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\

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

\end{array}
double f(double t, double l, double k) {
        double r106898 = 2.0;
        double r106899 = t;
        double r106900 = 3.0;
        double r106901 = pow(r106899, r106900);
        double r106902 = l;
        double r106903 = r106902 * r106902;
        double r106904 = r106901 / r106903;
        double r106905 = k;
        double r106906 = sin(r106905);
        double r106907 = r106904 * r106906;
        double r106908 = tan(r106905);
        double r106909 = r106907 * r106908;
        double r106910 = 1.0;
        double r106911 = r106905 / r106899;
        double r106912 = pow(r106911, r106898);
        double r106913 = r106910 + r106912;
        double r106914 = r106913 - r106910;
        double r106915 = r106909 * r106914;
        double r106916 = r106898 / r106915;
        return r106916;
}
↓
double f(double t, double l, double k) {
        double r106917 = l;
        double r106918 = r106917 * r106917;
        double r106919 = 6.058692716066773e+302;
        bool r106920 = r106918 <= r106919;
        double r106921 = 2.0;
        double r106922 = 1.0;
        double r106923 = sqrt(r106922);
        double r106924 = k;
        double r106925 = 2.0;
        double r106926 = r106921 / r106925;
        double r106927 = pow(r106924, r106926);
        double r106928 = r106923 / r106927;
        double r106929 = 1.0;
        double r106930 = pow(r106928, r106929);
        double r106931 = t;
        double r106932 = pow(r106931, r106929);
        double r106933 = r106927 * r106932;
        double r106934 = r106922 / r106933;
        double r106935 = pow(r106934, r106929);
        double r106936 = cos(r106924);
        double r106937 = r106935 * r106936;
        double r106938 = pow(r106917, r106925);
        double r106939 = sin(r106924);
        double r106940 = pow(r106939, r106925);
        double r106941 = r106938 / r106940;
        double r106942 = r106937 * r106941;
        double r106943 = r106930 * r106942;
        double r106944 = r106921 * r106943;
        double r106945 = 3.0;
        double r106946 = pow(r106931, r106945);
        double r106947 = r106921 / r106946;
        double r106948 = tan(r106924);
        double r106949 = r106939 * r106948;
        double r106950 = r106924 / r106931;
        double r106951 = pow(r106950, r106921);
        double r106952 = r106949 * r106951;
        double r106953 = r106947 / r106952;
        double r106954 = r106953 * r106917;
        double r106955 = r106954 * r106917;
        double r106956 = r106920 ? r106944 : r106955;
        return r106956;
}

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) < 6.058692716066773e+302

    1. Initial program 45.1

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

    2. Simplified36.5

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

    3. Taylor expanded around inf 14.2

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

    5. Applied sqr-pow14.2

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

    6. Applied associate-*l*11.8

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

    8. Applied add-sqr-sqrt11.8

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

    9. Applied times-frac11.6

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

    10. Applied unpow-prod-down11.6

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

    11. Applied associate-*l*9.8

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

    12. Simplified9.8

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

    14. Applied *-un-lft-identity9.8

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

    15. Applied unpow-prod-down9.8

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

    16. Applied times-frac9.8

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

    17. Applied associate-*r*9.8

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

    18. Simplified9.8

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

    if 6.058692716066773e+302 < (* l l)

    1. Initial program 63.7

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

    2. Simplified63.5

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

    4. Applied associate-*r*52.5

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

  4. Final simplification16.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 6.05869271606677315 \cdot 10^{302}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\sqrt{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \cos k\right) \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \ell\right) \cdot \ell\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))