Average Error: 48.0 → 19.2
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.6783233808706762 \cdot 10^{-155} \lor \neg \left(t \le 2.86903913255846773 \cdot 10^{-58}\right):\\ \;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{2}}{\sin k}\right) \cdot \frac{\cos k}{\sin k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]
\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}\;t \le 1.6783233808706762 \cdot 10^{-155} \lor \neg \left(t \le 2.86903913255846773 \cdot 10^{-58}\right):\\
\;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{2}}{\sin k}\right) \cdot \frac{\cos k}{\sin k}\right)\right) \cdot 2\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\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 r107044 = 2.0;
        double r107045 = t;
        double r107046 = 3.0;
        double r107047 = pow(r107045, r107046);
        double r107048 = l;
        double r107049 = r107048 * r107048;
        double r107050 = r107047 / r107049;
        double r107051 = k;
        double r107052 = sin(r107051);
        double r107053 = r107050 * r107052;
        double r107054 = tan(r107051);
        double r107055 = r107053 * r107054;
        double r107056 = 1.0;
        double r107057 = r107051 / r107045;
        double r107058 = pow(r107057, r107044);
        double r107059 = r107056 + r107058;
        double r107060 = r107059 - r107056;
        double r107061 = r107055 * r107060;
        double r107062 = r107044 / r107061;
        return r107062;
}
↓
double f(double t, double l, double k) {
        double r107063 = t;
        double r107064 = 1.6783233808706762e-155;
        bool r107065 = r107063 <= r107064;
        double r107066 = 2.8690391325584677e-58;
        bool r107067 = r107063 <= r107066;
        double r107068 = !r107067;
        bool r107069 = r107065 || r107068;
        double r107070 = 1.0;
        double r107071 = k;
        double r107072 = 2.0;
        double r107073 = 2.0;
        double r107074 = r107072 / r107073;
        double r107075 = pow(r107071, r107074);
        double r107076 = r107070 / r107075;
        double r107077 = 1.0;
        double r107078 = pow(r107076, r107077);
        double r107079 = pow(r107063, r107077);
        double r107080 = r107076 / r107079;
        double r107081 = pow(r107080, r107077);
        double r107082 = l;
        double r107083 = pow(r107082, r107073);
        double r107084 = sin(r107071);
        double r107085 = r107083 / r107084;
        double r107086 = r107081 * r107085;
        double r107087 = cos(r107071);
        double r107088 = r107087 / r107084;
        double r107089 = r107086 * r107088;
        double r107090 = r107078 * r107089;
        double r107091 = r107090 * r107072;
        double r107092 = 3.0;
        double r107093 = r107092 / r107073;
        double r107094 = pow(r107063, r107093);
        double r107095 = r107094 / r107082;
        double r107096 = r107095 * r107095;
        double r107097 = r107096 * r107084;
        double r107098 = tan(r107071);
        double r107099 = r107097 * r107098;
        double r107100 = r107072 / r107099;
        double r107101 = r107071 / r107063;
        double r107102 = pow(r107101, r107072);
        double r107103 = r107100 / r107102;
        double r107104 = r107069 ? r107091 : r107103;
        return r107104;
}

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 t < 1.6783233808706762e-155 or 2.8690391325584677e-58 < t

    1. Initial program 48.3

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

    2. Simplified40.1

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

    3. Taylor expanded around inf 22.2

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

    5. Applied sqr-pow22.2

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

    6. Applied associate-*l*20.4

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

    8. Applied add-cube-cbrt20.4

      \[\leadsto 2 \cdot \left({\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

    9. Applied times-frac20.2

      \[\leadsto 2 \cdot \left({\color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}} \cdot \frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

    10. Applied unpow-prod-down20.2

      \[\leadsto 2 \cdot \left(\color{blue}{\left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot {\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\]

    11. Applied associate-*l*18.9

      \[\leadsto 2 \cdot \color{blue}{\left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left({\left(\frac{\sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right)\right)}\]

    12. Simplified18.9

      \[\leadsto 2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left(\frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)}\right)\]
    13. Using strategy rm

    14. Applied sqr-pow18.9

      \[\leadsto 2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\cos k \cdot {\ell}^{2}}{\color{blue}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\sin k\right)}^{\left(\frac{2}{2}\right)}}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right)\]

    15. Applied times-frac18.5

      \[\leadsto 2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\color{blue}{\left(\frac{\cos k}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}} \cdot \frac{{\ell}^{2}}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}}\right)} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right)\]

    16. Applied associate-*l*18.6

      \[\leadsto 2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \color{blue}{\left(\frac{\cos k}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{{\ell}^{2}}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}} \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1}\right)\right)}\right)\]

    17. Simplified18.5

      \[\leadsto 2 \cdot \left({\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\cos k}{{\left(\sin k\right)}^{\left(\frac{2}{2}\right)}} \cdot \color{blue}{\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{2}}{\sin k}\right)}\right)\right)\]

    if 1.6783233808706762e-155 < t < 2.8690391325584677e-58

    1. Initial program 45.3

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

    2. Simplified44.8

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

    4. Applied sqr-pow44.8

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

    5. Applied times-frac28.3

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

  4. Final simplification19.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.6783233808706762 \cdot 10^{-155} \lor \neg \left(t \le 2.86903913255846773 \cdot 10^{-58}\right):\\ \;\;\;\;\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \frac{{\ell}^{2}}{\sin k}\right) \cdot \frac{\cos k}{\sin k}\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \frac{{t}^{\left(\frac{3}{2}\right)}}{\ell}\right) \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))