Average Error: 48.0 → 14.2
Time: 33.5s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 5.830478013546999449775592818541939019798 \cdot 10^{295}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right) \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\left|\sin k\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]
\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}\;\ell \cdot \ell \le 5.830478013546999449775592818541939019798 \cdot 10^{295}:\\
\;\;\;\;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(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right) \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\left|\sin k\right|}\right)\right)\\

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

\end{array}
double f(double t, double l, double k) {
        double r85052 = 2.0;
        double r85053 = t;
        double r85054 = 3.0;
        double r85055 = pow(r85053, r85054);
        double r85056 = l;
        double r85057 = r85056 * r85056;
        double r85058 = r85055 / r85057;
        double r85059 = k;
        double r85060 = sin(r85059);
        double r85061 = r85058 * r85060;
        double r85062 = tan(r85059);
        double r85063 = r85061 * r85062;
        double r85064 = 1.0;
        double r85065 = r85059 / r85053;
        double r85066 = pow(r85065, r85052);
        double r85067 = r85064 + r85066;
        double r85068 = r85067 - r85064;
        double r85069 = r85063 * r85068;
        double r85070 = r85052 / r85069;
        return r85070;
}
↓
double f(double t, double l, double k) {
        double r85071 = l;
        double r85072 = r85071 * r85071;
        double r85073 = 5.830478013546999e+295;
        bool r85074 = r85072 <= r85073;
        double r85075 = 2.0;
        double r85076 = 1.0;
        double r85077 = k;
        double r85078 = 2.0;
        double r85079 = r85075 / r85078;
        double r85080 = pow(r85077, r85079);
        double r85081 = t;
        double r85082 = 1.0;
        double r85083 = pow(r85081, r85082);
        double r85084 = r85080 * r85083;
        double r85085 = r85080 * r85084;
        double r85086 = r85076 / r85085;
        double r85087 = pow(r85086, r85082);
        double r85088 = cos(r85077);
        double r85089 = sin(r85077);
        double r85090 = fabs(r85089);
        double r85091 = r85088 / r85090;
        double r85092 = r85091 * r85071;
        double r85093 = r85078 / r85078;
        double r85094 = pow(r85071, r85093);
        double r85095 = r85094 / r85090;
        double r85096 = r85092 * r85095;
        double r85097 = r85087 * r85096;
        double r85098 = r85075 * r85097;
        double r85099 = cbrt(r85081);
        double r85100 = r85099 * r85099;
        double r85101 = 3.0;
        double r85102 = pow(r85100, r85101);
        double r85103 = r85102 / r85071;
        double r85104 = pow(r85099, r85101);
        double r85105 = r85104 / r85071;
        double r85106 = r85103 * r85105;
        double r85107 = r85106 * r85089;
        double r85108 = tan(r85077);
        double r85109 = r85107 * r85108;
        double r85110 = r85075 / r85109;
        double r85111 = r85077 / r85081;
        double r85112 = pow(r85111, r85075);
        double r85113 = r85110 / r85112;
        double r85114 = r85074 ? r85098 : r85113;
        return r85114;
}

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) < 5.830478013546999e+295

    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. Simplified36.0

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

    3. Taylor expanded around inf 14.1

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

    5. Applied sqr-pow14.1

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

    6. Applied associate-*l*11.7

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

    8. Applied add-sqr-sqrt11.7

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

    9. Applied times-frac11.7

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

    10. Simplified11.7

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

    11. Simplified11.4

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

    13. Applied *-un-lft-identity11.4

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

    14. Applied sqr-pow11.4

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

    15. Applied times-frac9.4

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

    16. Applied associate-*r*7.6

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

    17. Simplified7.6

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

    if 5.830478013546999e+295 < (* l l)

    1. Initial program 63.4

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

    2. Simplified63.2

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

    4. Applied add-cube-cbrt63.2

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

    5. Applied unpow-prod-down63.2

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

    6. Applied times-frac49.9

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

  4. Final simplification14.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 5.830478013546999449775592818541939019798 \cdot 10^{295}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right) \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\left|\sin k\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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))))