Average Error: 48.0 → 16.1
Time: 1.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -7.34197474785895879644805628021741244835 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 4.162486401097287709225438879584039218636 \cdot 10^{148}:\\ \;\;\;\;\left(\left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}\right)}^{2}} \cdot \frac{\cos k}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\\ \end{array}\]
\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 \le -7.34197474785895879644805628021741244835 \cdot 10^{153}:\\
\;\;\;\;\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{elif}\;\ell \le 4.162486401097287709225438879584039218636 \cdot 10^{148}:\\
\;\;\;\;\left(\left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\

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

\end{array}
double f(double t, double l, double k) {
        double r101041 = 2.0;
        double r101042 = t;
        double r101043 = 3.0;
        double r101044 = pow(r101042, r101043);
        double r101045 = l;
        double r101046 = r101045 * r101045;
        double r101047 = r101044 / r101046;
        double r101048 = k;
        double r101049 = sin(r101048);
        double r101050 = r101047 * r101049;
        double r101051 = tan(r101048);
        double r101052 = r101050 * r101051;
        double r101053 = 1.0;
        double r101054 = r101048 / r101042;
        double r101055 = pow(r101054, r101041);
        double r101056 = r101053 + r101055;
        double r101057 = r101056 - r101053;
        double r101058 = r101052 * r101057;
        double r101059 = r101041 / r101058;
        return r101059;
}
↓
double f(double t, double l, double k) {
        double r101060 = l;
        double r101061 = -7.341974747858959e+153;
        bool r101062 = r101060 <= r101061;
        double r101063 = 2.0;
        double r101064 = t;
        double r101065 = cbrt(r101064);
        double r101066 = r101065 * r101065;
        double r101067 = 3.0;
        double r101068 = pow(r101066, r101067);
        double r101069 = r101068 / r101060;
        double r101070 = pow(r101065, r101067);
        double r101071 = r101070 / r101060;
        double r101072 = k;
        double r101073 = sin(r101072);
        double r101074 = r101071 * r101073;
        double r101075 = r101069 * r101074;
        double r101076 = tan(r101072);
        double r101077 = r101075 * r101076;
        double r101078 = r101063 / r101077;
        double r101079 = r101072 / r101064;
        double r101080 = pow(r101079, r101063);
        double r101081 = r101078 / r101080;
        double r101082 = 4.1624864010972877e+148;
        bool r101083 = r101060 <= r101082;
        double r101084 = 1.0;
        double r101085 = 1.0;
        double r101086 = pow(r101064, r101085);
        double r101087 = r101084 / r101086;
        double r101088 = 2.0;
        double r101089 = r101063 / r101088;
        double r101090 = pow(r101072, r101089);
        double r101091 = r101087 / r101090;
        double r101092 = pow(r101091, r101085);
        double r101093 = cos(r101072);
        double r101094 = pow(r101060, r101088);
        double r101095 = r101093 * r101094;
        double r101096 = pow(r101073, r101088);
        double r101097 = r101095 / r101096;
        double r101098 = r101092 * r101097;
        double r101099 = r101084 / r101090;
        double r101100 = pow(r101099, r101085);
        double r101101 = r101098 * r101100;
        double r101102 = r101101 * r101063;
        double r101103 = pow(r101064, r101067);
        double r101104 = r101103 / r101060;
        double r101105 = r101104 / r101060;
        double r101106 = r101105 * r101096;
        double r101107 = r101063 / r101106;
        double r101108 = cbrt(r101079);
        double r101109 = r101108 * r101108;
        double r101110 = pow(r101109, r101063);
        double r101111 = r101107 / r101110;
        double r101112 = pow(r101108, r101063);
        double r101113 = r101093 / r101112;
        double r101114 = r101111 * r101113;
        double r101115 = r101083 ? r101102 : r101114;
        double r101116 = r101062 ? r101081 : r101115;
        return r101116;
}

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 3 regimes

  2. if l < -7.341974747858959e+153

    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. Simplified64.0

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

    4. Applied add-cube-cbrt64.0

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

    5. Applied unpow-prod-down64.0

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

    6. Applied times-frac50.5

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

    7. Applied associate-*l*50.5

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

    if -7.341974747858959e+153 < l < 4.1624864010972877e+148

    1. Initial program 45.1

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

    2. Simplified36.2

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

    3. Taylor expanded around inf 14.0

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

    5. Applied sqr-pow14.0

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

    6. Applied associate-*l*11.6

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

    8. Applied add-sqr-sqrt11.6

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

    9. Applied times-frac11.5

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

    10. Applied unpow-prod-down11.5

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

    11. Applied associate-*l*9.7

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

    12. Simplified9.7

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

    14. Applied div-inv9.7

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

    15. Simplified9.7

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

    if 4.1624864010972877e+148 < l

    1. Initial program 63.6

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

    2. Simplified63.3

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

    4. Applied add-cube-cbrt63.3

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

    5. Applied unpow-prod-down63.3

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

    6. Applied tan-quot63.3

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

    7. Applied associate-*r/63.3

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

    8. Applied associate-/r/63.3

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

    9. Applied times-frac63.3

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

    10. Simplified51.1

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -7.34197474785895879644805628021741244835 \cdot 10^{153}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\ell \le 4.162486401097287709225438879584039218636 \cdot 10^{148}:\\ \;\;\;\;\left(\left({\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\cos k \cdot {\ell}^{2}}{{\left(\sin k\right)}^{2}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{{t}^{3}}{\ell}}{\ell} \cdot {\left(\sin k\right)}^{2}}}{{\left(\sqrt[3]{\frac{k}{t}} \cdot \sqrt[3]{\frac{k}{t}}\right)}^{2}} \cdot \frac{\cos k}{{\left(\sqrt[3]{\frac{k}{t}}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))