Average Error: 47.9 → 18.4
Time: 31.6s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.1809042033014301 \cdot 10^{287}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{\sin k} \cdot \frac{1}{\sin k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \ell\right) \cdot \ell\\ \end{array}\]
\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 4.1809042033014301 \cdot 10^{287}:\\
\;\;\;\;2 \cdot \left({\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 \cdot {\ell}^{2}}{\sin k} \cdot \frac{1}{\sin k}\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 r114901 = 2.0;
        double r114902 = t;
        double r114903 = 3.0;
        double r114904 = pow(r114902, r114903);
        double r114905 = l;
        double r114906 = r114905 * r114905;
        double r114907 = r114904 / r114906;
        double r114908 = k;
        double r114909 = sin(r114908);
        double r114910 = r114907 * r114909;
        double r114911 = tan(r114908);
        double r114912 = r114910 * r114911;
        double r114913 = 1.0;
        double r114914 = r114908 / r114902;
        double r114915 = pow(r114914, r114901);
        double r114916 = r114913 + r114915;
        double r114917 = r114916 - r114913;
        double r114918 = r114912 * r114917;
        double r114919 = r114901 / r114918;
        return r114919;
}
↓
double f(double t, double l, double k) {
        double r114920 = l;
        double r114921 = r114920 * r114920;
        double r114922 = 4.18090420330143e+287;
        bool r114923 = r114921 <= r114922;
        double r114924 = 2.0;
        double r114925 = 1.0;
        double r114926 = k;
        double r114927 = 2.0;
        double r114928 = r114924 / r114927;
        double r114929 = pow(r114926, r114928);
        double r114930 = t;
        double r114931 = 1.0;
        double r114932 = pow(r114930, r114931);
        double r114933 = r114929 * r114932;
        double r114934 = r114929 * r114933;
        double r114935 = r114925 / r114934;
        double r114936 = pow(r114935, r114931);
        double r114937 = cos(r114926);
        double r114938 = pow(r114920, r114927);
        double r114939 = r114937 * r114938;
        double r114940 = sin(r114926);
        double r114941 = r114939 / r114940;
        double r114942 = r114925 / r114940;
        double r114943 = r114941 * r114942;
        double r114944 = r114936 * r114943;
        double r114945 = r114924 * r114944;
        double r114946 = 3.0;
        double r114947 = pow(r114930, r114946);
        double r114948 = r114924 / r114947;
        double r114949 = tan(r114926);
        double r114950 = r114940 * r114949;
        double r114951 = r114926 / r114930;
        double r114952 = pow(r114951, r114924);
        double r114953 = r114950 * r114952;
        double r114954 = r114948 / r114953;
        double r114955 = r114954 * r114920;
        double r114956 = r114955 * r114920;
        double r114957 = r114923 ? r114945 : r114956;
        return r114957;
}

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) < 4.18090420330143e+287

    1. Initial program 45.2

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

    2. Simplified36.7

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

    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*12.0

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

    8. Applied unpow212.0

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

    9. Applied associate-/r*11.6

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

    11. Applied div-inv11.6

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

    if 4.18090420330143e+287 < (* l l)

    1. Initial program 62.3

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

    2. Simplified62.2

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

    4. Applied associate-*r*54.3

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

  4. Final simplification18.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.1809042033014301 \cdot 10^{287}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{\sin k} \cdot \frac{1}{\sin k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{{t}^{3}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \cdot \ell\right) \cdot \ell\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))