Toniolo and Linder, Equation (10+) Percentage Accurate: 55.0% → 82.6%
Time: 21.4s
Alternatives: 20
Speedup: TODO×
26.6% of points produce a very large (infinite) output. You may want to add a precondition. (more) Specification ? \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Enter valid numbers for all inputs
Local Percentage Accuracy vs ?
The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples. Accuracy vs Speed? The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs. Alternative 1? \[\begin{array}{l}
t_1 := {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 2 \cdot 10^{+260}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + t_1\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2.00000000000000013e260 Initial program 79.3%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/79.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/81.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/80.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/80.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative80.6%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified80.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Step-by-step derivation add-cube-cbrt80.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}}
\]
pow380.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}^{3}}}
\]
Applied egg-rr 87.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}}
\]
if 2.00000000000000013e260 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) Initial program 20.6%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/20.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/20.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/20.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/21.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative21.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified21.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 52.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative52.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*52.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative52.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow252.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*57.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified57.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/57.0%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 57.0%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u47.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef41.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*41.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative41.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 41.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def47.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p57.0%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
*-commutative57.0%
\[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right) \cdot \tan k}}
\]
associate-/r*57.0%
\[\leadsto \color{blue}{\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}}
\]
associate-*r*57.0%
\[\leadsto \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot 2}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
unpow257.0%
\[\leadsto \frac{\frac{\color{blue}{{\ell}^{2}} \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
*-commutative57.0%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(k \cdot \left(t \cdot k\right)\right) \cdot \sin k}}}{\tan k}
\]
*-commutative57.0%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(k \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \sin k}}{\tan k}
\]
associate-*r*52.5%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \sin k}}{\tan k}
\]
unpow252.5%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(\color{blue}{{k}^{2}} \cdot t\right) \cdot \sin k}}{\tan k}
\]
times-frac52.5%
\[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{2}{\sin k}}}{\tan k}
\]
associate-/r*50.6%
\[\leadsto \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow250.6%
\[\leadsto \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow250.6%
\[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
times-frac76.4%
\[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow276.4%
\[\leadsto \frac{\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
Simplified76.4%
\[\leadsto \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}}
\]
Recombined 2 regimes into one program. Final simplification82.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{+260}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\end{array}
\]
Alternative 2? \[\begin{array}{l}
t_1 := {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 2 \cdot 10^{+260}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2.00000000000000013e260 Initial program 79.3%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/79.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/81.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/80.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/80.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative80.6%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified80.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Step-by-step derivation add-cube-cbrt80.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{{t}^{3} \cdot \sin k} \cdot \sqrt[3]{{t}^{3} \cdot \sin k}\right) \cdot \sqrt[3]{{t}^{3} \cdot \sin k}\right)}\right)}
\]
pow380.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(\sqrt[3]{{t}^{3} \cdot \sin k}\right)}^{3}}\right)}
\]
cbrt-prod80.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\sin k}\right)}}^{3}\right)}
\]
unpow380.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\sqrt[3]{\color{blue}{\left(t \cdot t\right) \cdot t}} \cdot \sqrt[3]{\sin k}\right)}^{3}\right)}
\]
add-cbrt-cube84.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\color{blue}{t} \cdot \sqrt[3]{\sin k}\right)}^{3}\right)}
\]
Applied egg-rr 84.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}\right)}
\]
if 2.00000000000000013e260 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) Initial program 20.6%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/20.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/20.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/20.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/21.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative21.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified21.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 52.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative52.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*52.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative52.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow252.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*57.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified57.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/57.0%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 57.0%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u47.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef41.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*41.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative41.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 41.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def47.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p57.0%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
*-commutative57.0%
\[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right) \cdot \tan k}}
\]
associate-/r*57.0%
\[\leadsto \color{blue}{\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}}
\]
associate-*r*57.0%
\[\leadsto \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot 2}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
unpow257.0%
\[\leadsto \frac{\frac{\color{blue}{{\ell}^{2}} \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
*-commutative57.0%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(k \cdot \left(t \cdot k\right)\right) \cdot \sin k}}}{\tan k}
\]
*-commutative57.0%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(k \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \sin k}}{\tan k}
\]
associate-*r*52.5%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \sin k}}{\tan k}
\]
unpow252.5%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(\color{blue}{{k}^{2}} \cdot t\right) \cdot \sin k}}{\tan k}
\]
times-frac52.5%
\[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{2}{\sin k}}}{\tan k}
\]
associate-/r*50.6%
\[\leadsto \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow250.6%
\[\leadsto \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow250.6%
\[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
times-frac76.4%
\[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow276.4%
\[\leadsto \frac{\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
Simplified76.4%
\[\leadsto \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}}
\]
Recombined 2 regimes into one program. Final simplification80.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{+260}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\end{array}
\]
Alternative 3? \[\begin{array}{l}
t_1 := {\left(\frac{k}{t}\right)}^{2}\\
\mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 2 \cdot 10^{+260}:\\
\;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\end{array}
\]
Derivation Split input into 2 regimes if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 2.00000000000000013e260 Initial program 79.3%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/79.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/81.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/80.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/80.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative80.6%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative80.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified80.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u54.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)}
\]
expm1-udef52.1%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1}
\]
Applied egg-rr 52.1%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def54.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)}
\]
expm1-log1p80.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
associate-*l*83.2%
\[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}
\]
*-commutative83.2%
\[\leadsto \ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot {t}^{3}\right)}\right)}\right)
\]
Simplified83.2%
\[\leadsto \color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)\right)}\right)}
\]
if 2.00000000000000013e260 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) Initial program 20.6%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/20.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/20.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/20.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/21.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative21.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative21.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified21.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 52.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative52.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*52.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative52.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow252.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*57.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified57.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/57.0%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 57.0%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u47.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef41.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*41.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative41.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 41.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def47.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p57.0%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
*-commutative57.0%
\[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right) \cdot \tan k}}
\]
associate-/r*57.0%
\[\leadsto \color{blue}{\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}}
\]
associate-*r*57.0%
\[\leadsto \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot 2}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
unpow257.0%
\[\leadsto \frac{\frac{\color{blue}{{\ell}^{2}} \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
*-commutative57.0%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(k \cdot \left(t \cdot k\right)\right) \cdot \sin k}}}{\tan k}
\]
*-commutative57.0%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(k \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \sin k}}{\tan k}
\]
associate-*r*52.5%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \sin k}}{\tan k}
\]
unpow252.5%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(\color{blue}{{k}^{2}} \cdot t\right) \cdot \sin k}}{\tan k}
\]
times-frac52.5%
\[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{2}{\sin k}}}{\tan k}
\]
associate-/r*50.6%
\[\leadsto \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow250.6%
\[\leadsto \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow250.6%
\[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
times-frac76.4%
\[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow276.4%
\[\leadsto \frac{\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
Simplified76.4%
\[\leadsto \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}}
\]
Recombined 2 regimes into one program. Final simplification80.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 2 \cdot 10^{+260}:\\
\;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\end{array}
\]
Alternative 4? \[\begin{array}{l}
t_1 := \frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\
\mathbf{if}\;t \leq -2.5 \cdot 10^{-7}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq -3.7 \cdot 10^{-225}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\mathbf{elif}\;t \leq 2.8 \cdot 10^{-71}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\mathbf{else}:\\
\;\;\;\;t_1\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -2.49999999999999989e-7 or 2.8e-71 < t Initial program 61.9%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-*l*61.9%
\[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
+-commutative61.9%
\[\leadsto \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}
\]
Simplified61.9%
\[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}}
\]
Taylor expanded in k around 0 62.8%
\[\leadsto \frac{2}{\color{blue}{\frac{k \cdot {t}^{3}}{{\ell}^{2}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}
\]
Step-by-step derivation *-commutative62.8%
\[\leadsto \frac{2}{\frac{\color{blue}{{t}^{3} \cdot k}}{{\ell}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}
\]
unpow262.8%
\[\leadsto \frac{2}{\frac{{t}^{3} \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}
\]
times-frac70.4%
\[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}
\]
Simplified70.4%
\[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}
\]
if -2.49999999999999989e-7 < t < -3.69999999999999988e-225 Initial program 58.3%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/58.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/58.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/58.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/58.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative58.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/58.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*58.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative58.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*58.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative58.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified58.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative73.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*73.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative73.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow273.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*75.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified75.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/75.8%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 75.8%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u32.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef27.4%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*27.4%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative27.4%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 27.4%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def32.2%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p75.8%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
*-commutative75.8%
\[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right) \cdot \tan k}}
\]
associate-/r*75.7%
\[\leadsto \color{blue}{\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}}
\]
associate-*r*75.7%
\[\leadsto \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot 2}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
unpow275.7%
\[\leadsto \frac{\frac{\color{blue}{{\ell}^{2}} \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
*-commutative75.7%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(k \cdot \left(t \cdot k\right)\right) \cdot \sin k}}}{\tan k}
\]
*-commutative75.7%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(k \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \sin k}}{\tan k}
\]
associate-*r*73.6%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \sin k}}{\tan k}
\]
unpow273.6%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(\color{blue}{{k}^{2}} \cdot t\right) \cdot \sin k}}{\tan k}
\]
times-frac75.5%
\[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{2}{\sin k}}}{\tan k}
\]
associate-/r*75.7%
\[\leadsto \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow275.7%
\[\leadsto \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow275.7%
\[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
times-frac95.2%
\[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow295.2%
\[\leadsto \frac{\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
Simplified95.2%
\[\leadsto \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}}
\]
if -3.69999999999999988e-225 < t < 2.8e-71 Initial program 35.6%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/35.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/35.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/35.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/37.0%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative37.0%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/37.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*37.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative37.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*37.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative37.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified37.0%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 74.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative74.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*74.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative74.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow274.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*79.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified79.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/79.8%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 79.8%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u57.8%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef47.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*47.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative47.2%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 47.2%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def57.9%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p79.9%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
times-frac93.4%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\ell \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}
\]
*-commutative93.4%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{2 \cdot \ell}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}
\]
associate-*r*93.4%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(\sin k \cdot k\right) \cdot \left(t \cdot k\right)}}
\]
*-commutative93.4%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}}
\]
Simplified93.4%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \left(k \cdot t\right)}}
\]
Recombined 3 regimes into one program. Final simplification81.1%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -2.5 \cdot 10^{-7}:\\
\;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\
\mathbf{elif}\;t \leq -3.7 \cdot 10^{-225}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\mathbf{elif}\;t \leq 2.8 \cdot 10^{-71}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\
\end{array}
\]
Alternative 5? \[\begin{array}{l}
t_1 := \frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\mathbf{if}\;t \leq -5.1 \cdot 10^{-225}:\\
\;\;\;\;t_1\\
\mathbf{elif}\;t \leq 10^{-80}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\mathbf{elif}\;t \leq 5 \cdot 10^{+25}:\\
\;\;\;\;t_1\\
\mathbf{else}:\\
\;\;\;\;\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \sin k}\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -5.0999999999999999e-225 or 9.99999999999999961e-81 < t < 5.00000000000000024e25 Initial program 57.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/57.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/58.8%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/58.3%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/59.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative59.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/59.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*59.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative59.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*59.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative59.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified59.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 57.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative57.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*57.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative57.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow257.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*57.9%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified57.9%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/57.9%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 57.9%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u35.6%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef34.5%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*34.5%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative34.5%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 34.5%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def35.6%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p57.9%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
*-commutative57.9%
\[\leadsto \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right) \cdot \tan k}}
\]
associate-/r*59.4%
\[\leadsto \color{blue}{\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}}
\]
associate-*r*59.4%
\[\leadsto \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot 2}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
unpow259.4%
\[\leadsto \frac{\frac{\color{blue}{{\ell}^{2}} \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}
\]
*-commutative59.4%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(k \cdot \left(t \cdot k\right)\right) \cdot \sin k}}}{\tan k}
\]
*-commutative59.4%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(k \cdot \color{blue}{\left(k \cdot t\right)}\right) \cdot \sin k}}{\tan k}
\]
associate-*r*58.6%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot t\right)} \cdot \sin k}}{\tan k}
\]
unpow258.6%
\[\leadsto \frac{\frac{{\ell}^{2} \cdot 2}{\left(\color{blue}{{k}^{2}} \cdot t\right) \cdot \sin k}}{\tan k}
\]
times-frac59.4%
\[\leadsto \frac{\color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot t} \cdot \frac{2}{\sin k}}}{\tan k}
\]
associate-/r*59.5%
\[\leadsto \frac{\color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow259.5%
\[\leadsto \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow259.5%
\[\leadsto \frac{\frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
times-frac73.4%
\[\leadsto \frac{\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
unpow273.4%
\[\leadsto \frac{\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{t} \cdot \frac{2}{\sin k}}{\tan k}
\]
Simplified73.4%
\[\leadsto \color{blue}{\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}}
\]
if -5.0999999999999999e-225 < t < 9.99999999999999961e-81 Initial program 35.0%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/35.0%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/35.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/35.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/35.0%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative35.0%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/35.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*35.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative35.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*35.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative35.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified35.0%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative73.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*73.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative73.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow273.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*79.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified79.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/79.2%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 79.2%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u56.6%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef45.7%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*45.7%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative45.7%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 45.7%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def56.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p79.3%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
times-frac93.2%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\ell \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}
\]
*-commutative93.2%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{2 \cdot \ell}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}
\]
associate-*r*93.3%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(\sin k \cdot k\right) \cdot \left(t \cdot k\right)}}
\]
*-commutative93.3%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}}
\]
Simplified93.3%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \left(k \cdot t\right)}}
\]
if 5.00000000000000024e25 < t Initial program 69.1%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/69.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/70.8%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/69.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/69.2%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative69.2%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/69.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*69.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative69.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*69.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative69.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified69.2%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 68.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)}
\]
Taylor expanded in t around inf 68.2%
\[\leadsto \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{k \cdot \left(\sin k \cdot {t}^{3}\right)}}
\]
Step-by-step derivation times-frac69.4%
\[\leadsto \color{blue}{\frac{\cos k}{k} \cdot \frac{{\ell}^{2}}{\sin k \cdot {t}^{3}}}
\]
unpow269.4%
\[\leadsto \frac{\cos k}{k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot {t}^{3}}
\]
Simplified69.4%
\[\leadsto \color{blue}{\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot {t}^{3}}}
\]
Recombined 3 regimes into one program. Final simplification77.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -5.1 \cdot 10^{-225}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\mathbf{elif}\;t \leq 10^{-80}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\mathbf{elif}\;t \leq 5 \cdot 10^{+25}:\\
\;\;\;\;\frac{\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{2}{\sin k}}{\tan k}\\
\mathbf{else}:\\
\;\;\;\;\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \sin k}\\
\end{array}
\]
Alternative 6? \[\begin{array}{l}
\mathbf{if}\;t \leq -4.7 \cdot 10^{-7} \lor \neg \left(t \leq 1.35 \cdot 10^{+23}\right):\\
\;\;\;\;\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \sin k}\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\end{array}
\]
Derivation Split input into 2 regimes if t < -4.7e-7 or 1.3499999999999999e23 < t Initial program 61.7%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/61.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/64.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/62.8%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/62.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative62.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/62.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*62.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative62.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*62.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative62.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified62.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 62.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}\right)}
\]
Taylor expanded in t around inf 62.5%
\[\leadsto \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{k \cdot \left(\sin k \cdot {t}^{3}\right)}}
\]
Step-by-step derivation times-frac63.5%
\[\leadsto \color{blue}{\frac{\cos k}{k} \cdot \frac{{\ell}^{2}}{\sin k \cdot {t}^{3}}}
\]
unpow263.5%
\[\leadsto \frac{\cos k}{k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot {t}^{3}}
\]
Simplified63.5%
\[\leadsto \color{blue}{\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{\sin k \cdot {t}^{3}}}
\]
if -4.7e-7 < t < 1.3499999999999999e23 Initial program 48.2%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/48.2%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/48.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/48.1%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/49.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative49.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/49.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*49.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative49.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*49.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative49.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified49.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 72.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative72.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*72.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative72.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow272.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*75.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified75.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/75.7%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 75.7%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u47.7%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef40.9%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*40.9%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative40.9%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 40.9%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def47.8%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p75.7%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
times-frac86.0%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\ell \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}
\]
*-commutative86.0%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{2 \cdot \ell}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}
\]
associate-*r*86.0%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(\sin k \cdot k\right) \cdot \left(t \cdot k\right)}}
\]
*-commutative86.0%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}}
\]
Simplified86.0%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \left(k \cdot t\right)}}
\]
Recombined 2 regimes into one program. Final simplification75.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -4.7 \cdot 10^{-7} \lor \neg \left(t \leq 1.35 \cdot 10^{+23}\right):\\
\;\;\;\;\frac{\cos k}{k} \cdot \frac{\ell \cdot \ell}{{t}^{3} \cdot \sin k}\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\end{array}
\]
Alternative 7? \[\begin{array}{l}
\mathbf{if}\;\ell \cdot \ell \leq 10^{-307}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)\right)\right)\\
\mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{-239}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\
\mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+287}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}\right)\right)\\
\end{array}
\]
Derivation Split input into 4 regimes if (*.f64 l l) < 9.99999999999999909e-308 Initial program 52.9%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/52.9%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/52.8%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/51.9%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/51.2%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative51.2%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/51.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*51.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative51.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*51.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative51.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified51.2%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 52.2%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac56.8%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow256.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow256.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative56.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac75.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified75.9%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} + 0.3333333333333333 \cdot \ell\right)}\right)\right)
\]
Step-by-step derivation +-commutative73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(0.3333333333333333 \cdot \ell + \frac{\ell}{{k}^{2}}\right)}\right)\right)
\]
*-commutative73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\color{blue}{\ell \cdot 0.3333333333333333} + \frac{\ell}{{k}^{2}}\right)\right)\right)
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)\right)
\]
Simplified73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)}\right)\right)
\]
if 9.99999999999999909e-308 < (*.f64 l l) < 2.0000000000000002e-239 Initial program 81.3%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/81.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/93.3%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/93.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/93.8%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative93.8%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/93.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*93.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative93.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*93.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative93.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified93.8%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 88.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(2 \cdot \left(k \cdot {t}^{3}\right)\right)}}
\]
if 2.0000000000000002e-239 < (*.f64 l l) < 2.0000000000000002e287 Initial program 63.1%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/63.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/64.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/63.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/65.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative65.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/65.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*65.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative65.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*65.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative65.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified65.0%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 66.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative66.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*66.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative66.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow266.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*71.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified71.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
if 2.0000000000000002e287 < (*.f64 l l) Initial program 39.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/39.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/39.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/39.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/39.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative39.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/39.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*39.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative39.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*39.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative39.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified39.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 52.9%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac53.0%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow253.0%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow253.0%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative53.0%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac57.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified57.6%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 57.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} + 0.3333333333333333 \cdot \ell\right)}\right)\right)
\]
Step-by-step derivation +-commutative57.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(0.3333333333333333 \cdot \ell + \frac{\ell}{{k}^{2}}\right)}\right)\right)
\]
*-commutative57.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\color{blue}{\ell \cdot 0.3333333333333333} + \frac{\ell}{{k}^{2}}\right)\right)\right)
\]
unpow257.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)\right)
\]
Simplified57.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)}\right)\right)
\]
Taylor expanded in k around inf 52.9%
\[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
Step-by-step derivation *-commutative52.9%
\[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot t}\right)
\]
times-frac53.0%
\[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)}\right)
\]
unpow253.0%
\[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t}\right)\right)
\]
unpow253.0%
\[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t}\right)\right)
\]
times-frac60.4%
\[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t}\right)\right)
\]
unpow260.4%
\[\leadsto 2 \cdot \left(0.3333333333333333 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\cos k}{t}\right)\right)
\]
Simplified60.4%
\[\leadsto 2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}\right)\right)}
\]
Recombined 4 regimes into one program. Final simplification69.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;\ell \cdot \ell \leq 10^{-307}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)\right)\right)\\
\mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{-239}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\
\mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+287}:\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t}\right)\right)\\
\end{array}
\]
Alternative 8? \[\begin{array}{l}
\mathbf{if}\;t \leq -12000000000000 \lor \neg \left(t \leq 3.5 \cdot 10^{+17}\right):\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\end{array}
\]
Derivation Split input into 2 regimes if t < -1.2e13 or 3.5e17 < t Initial program 60.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/60.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/63.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/62.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/62.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative62.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/62.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*62.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative62.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*62.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative62.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified62.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 61.8%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(2 \cdot \left(k \cdot {t}^{3}\right)\right)}}
\]
if -1.2e13 < t < 3.5e17 Initial program 49.7%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/49.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/49.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/49.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/50.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative50.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/50.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*50.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative50.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*50.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative50.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified50.2%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 71.9%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)}}
\]
Step-by-step derivation *-commutative71.9%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot {k}^{2}\right)}}
\]
associate-*l*71.9%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(t \cdot {k}^{2}\right)\right)}}
\]
*-commutative71.9%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)}
\]
unpow271.9%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)}
\]
associate-*l*75.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)}
\]
Simplified75.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation associate-*r/75.3%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Applied egg-rr 75.3%
\[\leadsto \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}}
\]
Step-by-step derivation expm1-log1p-u47.1%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\right)}
\]
expm1-udef40.6%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1}
\]
associate-*l*40.6%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)} - 1
\]
*-commutative40.6%
\[\leadsto e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \color{blue}{\left(t \cdot k\right)}\right)\right)}\right)} - 1
\]
Applied egg-rr 40.6%
\[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)} - 1}
\]
Step-by-step derivation expm1-def47.1%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\right)\right)}
\]
expm1-log1p75.4%
\[\leadsto \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}}
\]
times-frac85.2%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{\ell \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}
\]
*-commutative85.2%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{\color{blue}{2 \cdot \ell}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}
\]
associate-*r*85.2%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\color{blue}{\left(\sin k \cdot k\right) \cdot \left(t \cdot k\right)}}
\]
*-commutative85.2%
\[\leadsto \frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \color{blue}{\left(k \cdot t\right)}}
\]
Simplified85.2%
\[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(\sin k \cdot k\right) \cdot \left(k \cdot t\right)}}
\]
Recombined 2 regimes into one program. Final simplification74.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -12000000000000 \lor \neg \left(t \leq 3.5 \cdot 10^{+17}\right):\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\left(k \cdot \sin k\right) \cdot \left(t \cdot k\right)}\\
\end{array}
\]
Alternative 9? \[\begin{array}{l}
\mathbf{if}\;t \leq -180000 \lor \neg \left(t \leq 8.5 \cdot 10^{-32}\right):\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)\right)\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if t < -1.8e5 or 8.5000000000000003e-32 < t Initial program 61.0%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/61.0%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/63.3%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/62.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/62.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative62.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/62.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*62.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative62.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*62.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative62.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified62.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 61.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\left(2 \cdot \left(k \cdot {t}^{3}\right)\right)}}
\]
if -1.8e5 < t < 8.5000000000000003e-32 Initial program 48.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/48.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/48.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/48.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/49.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative49.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/49.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*49.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative49.0%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*49.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative49.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified49.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 72.9%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac72.9%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow272.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow272.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative72.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac82.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified82.9%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 74.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} + 0.3333333333333333 \cdot \ell\right)}\right)\right)
\]
Step-by-step derivation +-commutative74.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(0.3333333333333333 \cdot \ell + \frac{\ell}{{k}^{2}}\right)}\right)\right)
\]
*-commutative74.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\color{blue}{\ell \cdot 0.3333333333333333} + \frac{\ell}{{k}^{2}}\right)\right)\right)
\]
unpow274.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)\right)
\]
Simplified74.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)}\right)\right)
\]
Recombined 2 regimes into one program. Final simplification68.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -180000 \lor \neg \left(t \leq 8.5 \cdot 10^{-32}\right):\\
\;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot k\right)\right)}\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)\right)\right)\\
\end{array}
\]
Alternative 10? \[\begin{array}{l}
t_1 := \frac{\ell}{k \cdot k}\\
\mathbf{if}\;t \leq -6.6 \cdot 10^{-90}:\\
\;\;\;\;\frac{\ell \cdot t_1}{{t}^{3}}\\
\mathbf{elif}\;t \leq 3.5 \cdot 10^{-71}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + t_1\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -6.6e-90 Initial program 60.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/60.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/62.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/61.9%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/61.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative61.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/61.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*61.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative61.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*61.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative61.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified61.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 51.6%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow251.6%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative51.6%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac56.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow256.5%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified56.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation associate-*l/57.5%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
Applied egg-rr 57.5%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
if -6.6e-90 < t < 3.4999999999999999e-71 Initial program 38.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/38.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/38.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/38.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/39.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative39.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/39.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*39.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative39.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*39.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative39.5%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified39.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.6%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac73.6%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow273.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow273.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative73.6%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac87.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified87.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 78.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} + 0.3333333333333333 \cdot \ell\right)}\right)\right)
\]
Step-by-step derivation +-commutative78.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(0.3333333333333333 \cdot \ell + \frac{\ell}{{k}^{2}}\right)}\right)\right)
\]
*-commutative78.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\color{blue}{\ell \cdot 0.3333333333333333} + \frac{\ell}{{k}^{2}}\right)\right)\right)
\]
unpow278.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)\right)
\]
Simplified78.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)}\right)\right)
\]
if 3.4999999999999999e-71 < t Initial program 67.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/67.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/68.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/67.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/68.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 57.5%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow257.5%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative57.5%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow264.0%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation associate-*r/65.4%
\[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}}
\]
Applied egg-rr 65.4%
\[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}}
\]
Recombined 3 regimes into one program. Final simplification67.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -6.6 \cdot 10^{-90}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}\\
\mathbf{elif}\;t \leq 3.5 \cdot 10^{-71}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \left(\ell \cdot 0.3333333333333333 + \frac{\ell}{k \cdot k}\right)\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\
\end{array}
\]
Alternative 11? \[\begin{array}{l}
t_1 := \frac{\ell}{k \cdot k}\\
\mathbf{if}\;t \leq -1 \cdot 10^{-62}:\\
\;\;\;\;\frac{\ell \cdot t_1}{{t}^{3}}\\
\mathbf{elif}\;t \leq 4.5 \cdot 10^{-72}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot t_1\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -1e-62 Initial program 58.8%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/58.8%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/61.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/60.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/60.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 49.1%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow249.1%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative49.1%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow254.5%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation associate-*l/55.6%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
Applied egg-rr 55.6%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
if -1e-62 < t < 4.5e-72 Initial program 41.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/41.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/42.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.8%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac73.7%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac86.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified86.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right)\right)
\]
Step-by-step derivation unpow277.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)
\]
Simplified77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right)\right)
\]
if 4.5e-72 < t Initial program 67.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/67.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/68.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/67.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/68.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 57.5%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow257.5%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative57.5%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow264.0%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation associate-*r/65.4%
\[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}}
\]
Applied egg-rr 65.4%
\[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}}
\]
Recombined 3 regimes into one program. Final simplification67.4%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1 \cdot 10^{-62}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}\\
\mathbf{elif}\;t \leq 4.5 \cdot 10^{-72}:\\
\;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\
\end{array}
\]
Alternative 12? \[\begin{array}{l}
t_1 := \frac{\ell}{k \cdot k}\\
\mathbf{if}\;t \leq -3.2 \cdot 10^{-65} \lor \neg \left(t \leq 2.75 \cdot 10^{-71}\right):\\
\;\;\;\;t_1 \cdot \left(\ell \cdot {t}^{-3}\right)\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot t_1\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if t < -3.1999999999999999e-65 or 2.7499999999999999e-71 < t Initial program 63.2%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/63.2%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/65.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/64.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/64.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative64.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/64.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*64.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative64.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*64.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative64.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified64.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 53.4%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow253.4%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative53.4%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac59.3%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow259.3%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified59.3%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation expm1-log1p-u51.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k}
\]
expm1-udef46.5%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k}
\]
div-inv46.5%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
pow-flip46.5%
\[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
metadata-eval46.5%
\[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
Applied egg-rr 46.5%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k}
\]
Step-by-step derivation expm1-def51.3%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k}
\]
expm1-log1p59.3%
\[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k}
\]
Simplified59.3%
\[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k}
\]
if -3.1999999999999999e-65 < t < 2.7499999999999999e-71 Initial program 41.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/41.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/42.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.8%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac73.7%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac86.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified86.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right)\right)
\]
Step-by-step derivation unpow277.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)
\]
Simplified77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right)\right)
\]
Taylor expanded in k around 0 75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} - 0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Step-by-step derivation sub-neg75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.5\right)\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
unpow275.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.5\right)\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
metadata-eval75.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \color{blue}{-0.5}\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Simplified75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{k \cdot k} + -0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Recombined 2 regimes into one program. Final simplification65.5%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -3.2 \cdot 10^{-65} \lor \neg \left(t \leq 2.75 \cdot 10^{-71}\right):\\
\;\;\;\;\frac{\ell}{k \cdot k} \cdot \left(\ell \cdot {t}^{-3}\right)\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\end{array}
\]
Alternative 13? \[\begin{array}{l}
t_1 := \frac{\ell}{k \cdot k}\\
\mathbf{if}\;t \leq -1.05 \cdot 10^{-63}:\\
\;\;\;\;t_1 \cdot \frac{\ell}{{t}^{3}}\\
\mathbf{elif}\;t \leq 1.8 \cdot 10^{-72}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot t_1\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;t_1 \cdot \left(\ell \cdot {t}^{-3}\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -1.05e-63 Initial program 58.8%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/58.8%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/61.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/60.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/60.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 49.1%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow249.1%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative49.1%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow254.5%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
if -1.05e-63 < t < 1.8e-72 Initial program 41.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/41.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/42.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.8%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac73.7%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac86.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified86.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right)\right)
\]
Step-by-step derivation unpow277.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)
\]
Simplified77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right)\right)
\]
Taylor expanded in k around 0 75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} - 0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Step-by-step derivation sub-neg75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.5\right)\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
unpow275.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.5\right)\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
metadata-eval75.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \color{blue}{-0.5}\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Simplified75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{k \cdot k} + -0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
if 1.8e-72 < t Initial program 67.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/67.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/68.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/67.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/68.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 57.5%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow257.5%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative57.5%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow264.0%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation expm1-log1p-u50.9%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k}
\]
expm1-udef45.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k}
\]
div-inv45.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
pow-flip45.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
metadata-eval45.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
Applied egg-rr 45.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k}
\]
Step-by-step derivation expm1-def51.0%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k}
\]
expm1-log1p64.0%
\[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k}
\]
Simplified64.0%
\[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k}
\]
Recombined 3 regimes into one program. Final simplification65.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -1.05 \cdot 10^{-63}:\\
\;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\
\mathbf{elif}\;t \leq 1.8 \cdot 10^{-72}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell}{k \cdot k} \cdot \left(\ell \cdot {t}^{-3}\right)\\
\end{array}
\]
Alternative 14? \[\begin{array}{l}
t_1 := \frac{\ell}{k \cdot k}\\
\mathbf{if}\;t \leq -2.6 \cdot 10^{-61}:\\
\;\;\;\;\frac{\ell \cdot t_1}{{t}^{3}}\\
\mathbf{elif}\;t \leq 1.12 \cdot 10^{-72}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot t_1\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;t_1 \cdot \left(\ell \cdot {t}^{-3}\right)\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -2.6000000000000001e-61 Initial program 58.8%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/58.8%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/61.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/60.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/60.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 49.1%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow249.1%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative49.1%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow254.5%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation associate-*l/55.6%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
Applied egg-rr 55.6%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
if -2.6000000000000001e-61 < t < 1.12000000000000005e-72 Initial program 41.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/41.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/42.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.8%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac73.7%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac86.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified86.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right)\right)
\]
Step-by-step derivation unpow277.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)
\]
Simplified77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right)\right)
\]
Taylor expanded in k around 0 75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} - 0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Step-by-step derivation sub-neg75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.5\right)\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
unpow275.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.5\right)\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
metadata-eval75.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \color{blue}{-0.5}\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Simplified75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{k \cdot k} + -0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
if 1.12000000000000005e-72 < t Initial program 67.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/67.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/68.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/67.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/68.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 57.5%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow257.5%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative57.5%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow264.0%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation expm1-log1p-u50.9%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)\right)} \cdot \frac{\ell}{k \cdot k}
\]
expm1-udef45.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\ell}{{t}^{3}}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k}
\]
div-inv45.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \frac{1}{{t}^{3}}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
pow-flip45.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot \color{blue}{{t}^{\left(-3\right)}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
metadata-eval45.3%
\[\leadsto \left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{\color{blue}{-3}}\right)} - 1\right) \cdot \frac{\ell}{k \cdot k}
\]
Applied egg-rr 45.3%
\[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)} - 1\right)} \cdot \frac{\ell}{k \cdot k}
\]
Step-by-step derivation expm1-def51.0%
\[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot {t}^{-3}\right)\right)} \cdot \frac{\ell}{k \cdot k}
\]
expm1-log1p64.0%
\[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k}
\]
Simplified64.0%
\[\leadsto \color{blue}{\left(\ell \cdot {t}^{-3}\right)} \cdot \frac{\ell}{k \cdot k}
\]
Recombined 3 regimes into one program. Final simplification65.9%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -2.6 \cdot 10^{-61}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}\\
\mathbf{elif}\;t \leq 1.12 \cdot 10^{-72}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell}{k \cdot k} \cdot \left(\ell \cdot {t}^{-3}\right)\\
\end{array}
\]
Alternative 15? \[\begin{array}{l}
t_1 := \frac{\ell}{k \cdot k}\\
\mathbf{if}\;t \leq -2.7 \cdot 10^{-63}:\\
\;\;\;\;\frac{\ell \cdot t_1}{{t}^{3}}\\
\mathbf{elif}\;t \leq 4.4 \cdot 10^{-71}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot t_1\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\
\end{array}
\]
Derivation Split input into 3 regimes if t < -2.7000000000000002e-63 Initial program 58.8%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/58.8%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/61.2%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/60.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/60.1%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative60.1%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified60.1%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 49.1%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow249.1%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative49.1%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow254.5%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified54.5%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation associate-*l/55.6%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
Applied egg-rr 55.6%
\[\leadsto \color{blue}{\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}}
\]
if -2.7000000000000002e-63 < t < 4.39999999999999995e-71 Initial program 41.4%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/41.4%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/41.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/42.3%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative42.2%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative42.3%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified42.3%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 73.8%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac73.7%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow273.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative73.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac86.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified86.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right)\right)
\]
Step-by-step derivation unpow277.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)
\]
Simplified77.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right)\right)
\]
Taylor expanded in k around 0 75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} - 0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Step-by-step derivation sub-neg75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.5\right)\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
unpow275.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.5\right)\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
metadata-eval75.0%
\[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \color{blue}{-0.5}\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Simplified75.0%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{k \cdot k} + -0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
if 4.39999999999999995e-71 < t Initial program 67.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/67.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/68.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/67.5%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/68.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative68.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified68.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around 0 57.5%
\[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}}
\]
Step-by-step derivation unpow257.5%
\[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}}
\]
*-commutative57.5%
\[\leadsto \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}}
\]
times-frac64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}}
\]
unpow264.0%
\[\leadsto \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}}
\]
Simplified64.0%
\[\leadsto \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}}
\]
Step-by-step derivation associate-*r/65.4%
\[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}}
\]
Applied egg-rr 65.4%
\[\leadsto \color{blue}{\frac{\frac{\ell}{{t}^{3}} \cdot \ell}{k \cdot k}}
\]
Recombined 3 regimes into one program. Final simplification66.3%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq -2.7 \cdot 10^{-63}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot k}}{{t}^{3}}\\
\mathbf{elif}\;t \leq 4.4 \cdot 10^{-71}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;\frac{\ell \cdot \frac{\ell}{{t}^{3}}}{k \cdot k}\\
\end{array}
\]
Alternative 16? \[\begin{array}{l}
\mathbf{if}\;t \leq 7.1 \cdot 10^{+105}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if t < 7.1000000000000003e105 Initial program 53.1%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/53.2%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/54.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/53.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/54.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative54.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified54.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 62.9%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac63.8%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow263.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow263.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative63.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac70.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified70.7%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 64.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right)\right)
\]
Step-by-step derivation unpow264.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)
\]
Simplified64.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right)\right)
\]
Taylor expanded in k around 0 63.6%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} - 0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Step-by-step derivation sub-neg63.6%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{{k}^{2}} + \left(-0.5\right)\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
unpow263.6%
\[\leadsto 2 \cdot \left(\left(\frac{1}{\color{blue}{k \cdot k}} + \left(-0.5\right)\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
metadata-eval63.6%
\[\leadsto 2 \cdot \left(\left(\frac{1}{k \cdot k} + \color{blue}{-0.5}\right) \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Simplified63.6%
\[\leadsto 2 \cdot \left(\color{blue}{\left(\frac{1}{k \cdot k} + -0.5\right)} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
if 7.1000000000000003e105 < t Initial program 61.7%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/61.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/61.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/61.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/61.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative61.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified61.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 33.1%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac33.3%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow233.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow233.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative33.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac38.1%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified38.1%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 29.6%
\[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
Step-by-step derivation associate--l+29.6%
\[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)}
\]
fma-def29.6%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
associate-/r*27.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
times-frac27.3%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
Simplified30.7%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}, \frac{{\left(\frac{\ell}{k \cdot k}\right)}^{2}}{t} - \frac{-0.3333333333333333}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)}
\]
Taylor expanded in k around inf 37.2%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}}
\]
Step-by-step derivation distribute-rgt-out--37.3%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}}
\]
metadata-eval37.3%
\[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}}
\]
*-commutative37.3%
\[\leadsto 2 \cdot \frac{\color{blue}{-0.16666666666666666 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}}
\]
unpow237.3%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}
\]
associate-*r/42.2%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}}{{k}^{2}}
\]
unpow242.2%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{\color{blue}{k \cdot k}}
\]
Simplified42.2%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k}}
\]
Step-by-step derivation times-frac54.6%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Applied egg-rr 54.6%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Recombined 2 regimes into one program. Final simplification62.2%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq 7.1 \cdot 10^{+105}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\right)\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)\\
\end{array}
\]
Alternative 17? \[\begin{array}{l}
\mathbf{if}\;t \leq 3 \cdot 10^{+104}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{1}{k \cdot k}\right)\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if t < 2.99999999999999969e104 Initial program 53.1%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/53.2%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/54.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/53.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/54.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative54.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified54.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 62.9%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac63.8%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow263.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow263.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative63.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac70.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified70.7%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 64.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{{k}^{2}}}\right)\right)
\]
Step-by-step derivation unpow264.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right)\right)
\]
Simplified64.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \color{blue}{\frac{\ell}{k \cdot k}}\right)\right)
\]
Taylor expanded in k around 0 63.0%
\[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{{k}^{2}}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Step-by-step derivation unpow263.0%
\[\leadsto 2 \cdot \left(\frac{1}{\color{blue}{k \cdot k}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
Simplified63.0%
\[\leadsto 2 \cdot \left(\color{blue}{\frac{1}{k \cdot k}} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)\right)
\]
if 2.99999999999999969e104 < t Initial program 61.7%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/61.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/61.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/61.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/61.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative61.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified61.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 33.1%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac33.3%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow233.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow233.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative33.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac38.1%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified38.1%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 29.6%
\[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
Step-by-step derivation associate--l+29.6%
\[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)}
\]
fma-def29.6%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
associate-/r*27.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
times-frac27.3%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
Simplified30.7%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}, \frac{{\left(\frac{\ell}{k \cdot k}\right)}^{2}}{t} - \frac{-0.3333333333333333}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)}
\]
Taylor expanded in k around inf 37.2%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}}
\]
Step-by-step derivation distribute-rgt-out--37.3%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}}
\]
metadata-eval37.3%
\[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}}
\]
*-commutative37.3%
\[\leadsto 2 \cdot \frac{\color{blue}{-0.16666666666666666 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}}
\]
unpow237.3%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}
\]
associate-*r/42.2%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}}{{k}^{2}}
\]
unpow242.2%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{\color{blue}{k \cdot k}}
\]
Simplified42.2%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k}}
\]
Step-by-step derivation times-frac54.6%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Applied egg-rr 54.6%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Recombined 2 regimes into one program. Final simplification61.6%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq 3 \cdot 10^{+104}:\\
\;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{1}{k \cdot k}\right)\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)\\
\end{array}
\]
Alternative 18? \[\begin{array}{l}
t_1 := \frac{\ell}{k \cdot k}\\
\mathbf{if}\;t \leq 8.2 \cdot 10^{+95}:\\
\;\;\;\;2 \cdot \frac{t_1 \cdot t_1}{t}\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)\\
\end{array}
\]
Derivation Split input into 2 regimes if t < 8.19999999999999972e95 Initial program 53.1%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/53.2%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/54.4%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/53.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/54.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative54.5%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative54.4%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified54.4%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 62.9%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac63.8%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow263.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow263.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative63.8%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac70.7%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified70.7%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 54.2%
\[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}}
\]
Step-by-step derivation associate-/r*54.6%
\[\leadsto 2 \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{4}}}{t}}
\]
unpow254.6%
\[\leadsto 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}}}{t}
\]
metadata-eval54.6%
\[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{{k}^{\color{blue}{\left(2 \cdot 2\right)}}}}{t}
\]
pow-sqr54.6%
\[\leadsto 2 \cdot \frac{\frac{\ell \cdot \ell}{\color{blue}{{k}^{2} \cdot {k}^{2}}}}{t}
\]
times-frac60.8%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{k}^{2}}}}{t}
\]
unpow260.8%
\[\leadsto 2 \cdot \frac{\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{k}^{2}}}{t}
\]
unpow160.8%
\[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k \cdot k}\right)}^{1}} \cdot \frac{\ell}{{k}^{2}}}{t}
\]
unpow260.8%
\[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k \cdot k}\right)}^{1} \cdot \frac{\ell}{\color{blue}{k \cdot k}}}{t}
\]
pow-plus60.8%
\[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{\ell}{k \cdot k}\right)}^{\left(1 + 1\right)}}}{t}
\]
metadata-eval60.8%
\[\leadsto 2 \cdot \frac{{\left(\frac{\ell}{k \cdot k}\right)}^{\color{blue}{2}}}{t}
\]
Simplified60.8%
\[\leadsto 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k \cdot k}\right)}^{2}}{t}}
\]
Step-by-step derivation unpow260.8%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}}}{t}
\]
Applied egg-rr 60.8%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}}}{t}
\]
if 8.19999999999999972e95 < t Initial program 61.7%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/61.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/61.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/61.7%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/61.7%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative61.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative61.7%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified61.7%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 33.1%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac33.3%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow233.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow233.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative33.3%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac38.1%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified38.1%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 29.6%
\[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
Step-by-step derivation associate--l+29.6%
\[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)}
\]
fma-def29.6%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
associate-/r*27.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.2%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
times-frac27.3%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
Simplified30.7%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}, \frac{{\left(\frac{\ell}{k \cdot k}\right)}^{2}}{t} - \frac{-0.3333333333333333}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)}
\]
Taylor expanded in k around inf 37.2%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}}
\]
Step-by-step derivation distribute-rgt-out--37.3%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}}
\]
metadata-eval37.3%
\[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}}
\]
*-commutative37.3%
\[\leadsto 2 \cdot \frac{\color{blue}{-0.16666666666666666 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}}
\]
unpow237.3%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}
\]
associate-*r/42.2%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}}{{k}^{2}}
\]
unpow242.2%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{\color{blue}{k \cdot k}}
\]
Simplified42.2%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k}}
\]
Step-by-step derivation times-frac54.6%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Applied egg-rr 54.6%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Recombined 2 regimes into one program. Final simplification59.8%
\[\leadsto \begin{array}{l}
\mathbf{if}\;t \leq 8.2 \cdot 10^{+95}:\\
\;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{k \cdot k}}{t}\\
\mathbf{else}:\\
\;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)\\
\end{array}
\]
Alternative 19? \[2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)
\]
Derivation Initial program 54.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/54.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/55.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/55.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/55.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative55.6%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified55.6%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 58.1%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac58.9%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow258.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow258.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative58.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac65.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified65.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 26.6%
\[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
Step-by-step derivation associate--l+26.6%
\[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)}
\]
fma-def26.6%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
associate-/r*27.9%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.9%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.9%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
times-frac28.0%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
Simplified33.0%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}, \frac{{\left(\frac{\ell}{k \cdot k}\right)}^{2}}{t} - \frac{-0.3333333333333333}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)}
\]
Taylor expanded in k around inf 28.1%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}}
\]
Step-by-step derivation distribute-rgt-out--29.7%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}}
\]
metadata-eval29.7%
\[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}}
\]
*-commutative29.7%
\[\leadsto 2 \cdot \frac{\color{blue}{-0.16666666666666666 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}}
\]
unpow229.7%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}
\]
associate-*r/31.1%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}}{{k}^{2}}
\]
unpow231.1%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{\color{blue}{k \cdot k}}
\]
Simplified31.1%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k}}
\]
Step-by-step derivation times-frac33.9%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Applied egg-rr 33.9%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)}
\]
Final simplification33.9%
\[\leadsto 2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k}\right)
\]
Alternative 20? \[-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}
\]
Derivation Initial program 54.5%
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
\]
Step-by-step derivation associate-/l/54.5%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}
\]
associate-*l/55.6%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k}
\]
associate-*l/55.0%
\[\leadsto \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}}
\]
associate-/r/55.6%
\[\leadsto \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)}
\]
*-commutative55.6%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}}
\]
associate-/l/55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}
\]
associate-*r*55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
*-commutative55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}}
\]
associate-*r*55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}}
\]
*-commutative55.6%
\[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}
\]
Simplified55.6%
\[\leadsto \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}
\]
Taylor expanded in k around inf 58.1%
\[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}}
\]
Step-by-step derivation times-frac58.9%
\[\leadsto 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)}
\]
unpow258.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)
\]
unpow258.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right)
\]
*-commutative58.9%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right)
\]
times-frac65.5%
\[\leadsto 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right)
\]
Simplified65.5%
\[\leadsto \color{blue}{2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\right)}
\]
Taylor expanded in k around 0 26.6%
\[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
Step-by-step derivation associate--l+26.6%
\[\leadsto 2 \cdot \color{blue}{\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \left(\frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)\right)}
\]
fma-def26.6%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{{\ell}^{2}}{{k}^{2} \cdot t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)}
\]
associate-/r*27.9%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \color{blue}{\frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.9%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
unpow227.9%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
times-frac28.0%
\[\leadsto 2 \cdot \mathsf{fma}\left(-0.5, \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t}, \frac{{\ell}^{2}}{{k}^{4} \cdot t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)
\]
Simplified33.0%
\[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(-0.5, \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}, \frac{{\left(\frac{\ell}{k \cdot k}\right)}^{2}}{t} - \frac{-0.3333333333333333}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)}
\]
Taylor expanded in k around inf 28.1%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.5 \cdot \frac{{\ell}^{2}}{t} - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}{{k}^{2}}}
\]
Step-by-step derivation distribute-rgt-out--29.7%
\[\leadsto 2 \cdot \frac{\color{blue}{\frac{{\ell}^{2}}{t} \cdot \left(-0.5 - -0.3333333333333333\right)}}{{k}^{2}}
\]
metadata-eval29.7%
\[\leadsto 2 \cdot \frac{\frac{{\ell}^{2}}{t} \cdot \color{blue}{-0.16666666666666666}}{{k}^{2}}
\]
*-commutative29.7%
\[\leadsto 2 \cdot \frac{\color{blue}{-0.16666666666666666 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}}
\]
unpow229.7%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}
\]
associate-*r/31.1%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)}}{{k}^{2}}
\]
unpow231.1%
\[\leadsto 2 \cdot \frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{\color{blue}{k \cdot k}}
\]
Simplified31.1%
\[\leadsto 2 \cdot \color{blue}{\frac{-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{k \cdot k}}
\]
Taylor expanded in l around 0 29.5%
\[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}}
\]
Step-by-step derivation associate-*r/29.5%
\[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{{k}^{2} \cdot t}}
\]
associate-/l/29.7%
\[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333 \cdot {\ell}^{2}}{t}}{{k}^{2}}}
\]
associate-*r/29.7%
\[\leadsto \frac{\color{blue}{-0.3333333333333333 \cdot \frac{{\ell}^{2}}{t}}}{{k}^{2}}
\]
associate-*r/29.7%
\[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{2}}}
\]
unpow229.7%
\[\leadsto -0.3333333333333333 \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{k}^{2}}
\]
associate-*r/31.1%
\[\leadsto -0.3333333333333333 \cdot \frac{\color{blue}{\ell \cdot \frac{\ell}{t}}}{{k}^{2}}
\]
unpow231.1%
\[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{t}}{\color{blue}{k \cdot k}}
\]
Simplified31.1%
\[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}}
\]
Final simplification31.1%
\[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}
\]
Reproduce ? herbie shell --seed 2023166
(FPCore (t l k)
:name "Toniolo and Linder, Equation (10+)"
:precision binary64
(/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))
