Average Error: 32.8 → 16.8
Time: 2.6m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le 1.1180053642670318 \cdot 10^{-203}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \cdot \sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right) \cdot \sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}}\\ \end{array}\]
\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 1.1180053642670318 \cdot 10^{-203}:\\
\;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\left(\sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \cdot \sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right) \cdot \sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}}\\

\end{array}
double f(double t, double l, double k) {
        double r275010 = 2.0;
        double r275011 = t;
        double r275012 = 3.0;
        double r275013 = pow(r275011, r275012);
        double r275014 = l;
        double r275015 = r275014 * r275014;
        double r275016 = r275013 / r275015;
        double r275017 = k;
        double r275018 = sin(r275017);
        double r275019 = r275016 * r275018;
        double r275020 = tan(r275017);
        double r275021 = r275019 * r275020;
        double r275022 = 1.0;
        double r275023 = r275017 / r275011;
        double r275024 = pow(r275023, r275010);
        double r275025 = r275022 + r275024;
        double r275026 = r275025 + r275022;
        double r275027 = r275021 * r275026;
        double r275028 = r275010 / r275027;
        return r275028;
}
↓
double f(double t, double l, double k) {
        double r275029 = t;
        double r275030 = 1.1180053642670318e-203;
        bool r275031 = r275029 <= r275030;
        double r275032 = 2.0;
        double r275033 = k;
        double r275034 = 2.0;
        double r275035 = pow(r275033, r275034);
        double r275036 = sin(r275033);
        double r275037 = pow(r275036, r275034);
        double r275038 = r275029 * r275037;
        double r275039 = r275035 * r275038;
        double r275040 = cos(r275033);
        double r275041 = l;
        double r275042 = pow(r275041, r275034);
        double r275043 = r275040 * r275042;
        double r275044 = r275039 / r275043;
        double r275045 = 3.0;
        double r275046 = pow(r275029, r275045);
        double r275047 = r275046 * r275037;
        double r275048 = r275047 / r275043;
        double r275049 = r275032 * r275048;
        double r275050 = r275044 + r275049;
        double r275051 = r275032 / r275050;
        double r275052 = 3.0;
        double r275053 = r275052 / r275034;
        double r275054 = pow(r275029, r275053);
        double r275055 = r275054 / r275041;
        double r275056 = r275055 * r275036;
        double r275057 = tan(r275033);
        double r275058 = 1.0;
        double r275059 = r275033 / r275029;
        double r275060 = pow(r275059, r275032);
        double r275061 = r275058 + r275060;
        double r275062 = r275061 + r275058;
        double r275063 = r275057 * r275062;
        double r275064 = r275056 * r275063;
        double r275065 = r275055 * r275064;
        double r275066 = cbrt(r275065);
        double r275067 = r275066 * r275066;
        double r275068 = r275067 * r275066;
        double r275069 = r275032 / r275068;
        double r275070 = r275031 ? r275051 : r275069;
        return r275070;
}

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 < 1.1180053642670318e-203

    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. Taylor expanded around inf 43.0

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

    if 1.1180053642670318e-203 < t

    1. Initial program 28.6

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

    3. Applied sqr-pow28.6

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

    4. Applied times-frac17.6

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

    5. Applied associate-*l*14.9

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

    7. Applied associate-*l*14.8

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

    9. Applied associate-*l*13.1

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

    11. Applied add-cube-cbrt13.3

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

  4. Final simplification16.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 1.1180053642670318 \cdot 10^{-203}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}} + 2 \cdot \frac{{t}^{3} \cdot {\left(\sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)} \cdot \sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}\right) \cdot \sqrt[3]{\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))