Average Error: 48.0 → 18.2
Time: 1.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -3.4500322461818804 \cdot 10^{114}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{{\ell}^{2}}{\sin k}\right)\right)\\ \end{array}\]
\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}\;\ell \le -3.4500322461818804 \cdot 10^{114}:\\
\;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r111067 = 2.0;
        double r111068 = t;
        double r111069 = 3.0;
        double r111070 = pow(r111068, r111069);
        double r111071 = l;
        double r111072 = r111071 * r111071;
        double r111073 = r111070 / r111072;
        double r111074 = k;
        double r111075 = sin(r111074);
        double r111076 = r111073 * r111075;
        double r111077 = tan(r111074);
        double r111078 = r111076 * r111077;
        double r111079 = 1.0;
        double r111080 = r111074 / r111068;
        double r111081 = pow(r111080, r111067);
        double r111082 = r111079 + r111081;
        double r111083 = r111082 - r111079;
        double r111084 = r111078 * r111083;
        double r111085 = r111067 / r111084;
        return r111085;
}
↓
double f(double t, double l, double k) {
        double r111086 = l;
        double r111087 = -3.4500322461818804e+114;
        bool r111088 = r111086 <= r111087;
        double r111089 = 2.0;
        double r111090 = t;
        double r111091 = cbrt(r111090);
        double r111092 = r111091 * r111091;
        double r111093 = 3.0;
        double r111094 = pow(r111092, r111093);
        double r111095 = r111094 / r111086;
        double r111096 = pow(r111091, r111093);
        double r111097 = r111096 / r111086;
        double r111098 = k;
        double r111099 = sin(r111098);
        double r111100 = r111097 * r111099;
        double r111101 = r111095 * r111100;
        double r111102 = tan(r111098);
        double r111103 = r111101 * r111102;
        double r111104 = r111089 / r111103;
        double r111105 = r111098 / r111090;
        double r111106 = pow(r111105, r111089);
        double r111107 = r111104 / r111106;
        double r111108 = 1.0;
        double r111109 = 2.0;
        double r111110 = r111089 / r111109;
        double r111111 = pow(r111098, r111110);
        double r111112 = r111108 / r111111;
        double r111113 = 1.0;
        double r111114 = pow(r111112, r111113);
        double r111115 = pow(r111090, r111113);
        double r111116 = r111111 * r111115;
        double r111117 = r111108 / r111116;
        double r111118 = pow(r111117, r111113);
        double r111119 = cos(r111098);
        double r111120 = r111119 / r111099;
        double r111121 = r111118 * r111120;
        double r111122 = pow(r111086, r111109);
        double r111123 = r111122 / r111099;
        double r111124 = r111121 * r111123;
        double r111125 = r111114 * r111124;
        double r111126 = r111089 * r111125;
        double r111127 = r111088 ? r111107 : r111126;
        return r111127;
}

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 < -3.4500322461818804e+114

    1. Initial program 60.7

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

    2. Simplified59.7

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

    4. Applied add-cube-cbrt59.8

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

    5. Applied unpow-prod-down59.8

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

    6. Applied times-frac48.5

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

    7. Applied associate-*l*48.5

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

    if -3.4500322461818804e+114 < l

    1. Initial program 46.7

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

    2. Simplified38.4

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

    3. Taylor expanded around inf 18.6

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

    5. Applied sqr-pow18.6

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

    6. Applied associate-*l*16.7

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

    8. Applied *-un-lft-identity16.7

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

    9. Applied times-frac16.5

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

    10. Applied unpow-prod-down16.5

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

    11. Applied associate-*l*15.1

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

    13. Applied unpow215.1

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

    14. Applied times-frac14.7

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

    15. Applied associate-*r*14.8

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

  4. Final simplification18.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -3.4500322461818804 \cdot 10^{114}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}}\right)}^{1} \cdot \frac{\cos k}{\sin k}\right) \cdot \frac{{\ell}^{2}}{\sin k}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(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))))