Toniolo and Linder, Equation (2)

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999985e135

   1. Initial program 55.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around -inf 89.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg89.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

      2. *-commutative89.5%

         \[\leadsto \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]

      3. distribute-rgt-neg-in89.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right)} \]

      4. unpow289.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      5. unpow289.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      6. times-frac99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      7. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      8. associate-/l*97.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)\right) \]

      9. associate-/r/99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{t} \cdot \ell}\right)\right) \]

   4. Simplified99.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\right)\right)} \]

      2. expm1-udef43.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\right)} - 1} \]

   6. Applied egg-rr43.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def97.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p97.4%

         \[\leadsto \color{blue}{-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)} \]

      3. associate-/r/99.5%

         \[\leadsto -\sin^{-1} \color{blue}{\left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{t} \cdot \ell\right)} \]

      4. unpow1/299.5%

         \[\leadsto -\sin^{-1} \left(\frac{\color{blue}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}{t} \cdot \ell\right) \]

      5. *-commutative99.5%

         \[\leadsto -\sin^{-1} \left(\frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{t} \cdot \ell\right) \]

   8. Simplified99.5%

      \[\leadsto \color{blue}{-\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t} \cdot \ell\right)} \]

   if -3.99999999999999985e135 < (/.f64 t l) < 2.00000000000000012e63

   1. Initial program 99.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right)}}\right) \]

      3. frac-times99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{1 \cdot t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]

      4. *-un-lft-identity99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\color{blue}{t}}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   3. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]
   4. Step-by-step derivation

      1. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      3. un-div-inv99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   5. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   if 2.00000000000000012e63 < (/.f64 t l)

   1. Initial program 64.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around inf 91.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative91.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

      2. unpow291.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      3. unpow291.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      4. times-frac99.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      5. unpow299.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      6. associate-/l*98.7%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Simplified98.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
   5. Step-by-step derivation

      1. div-inv98.8%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{1}{\frac{t}{\ell}}\right)}\right) \]

      2. clear-num99.6%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\sqrt{0.5} \cdot \color{blue}{\frac{\ell}{t}}\right)\right) \]

   6. Applied egg-rr99.6%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+135}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+63}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\right)\\ \end{array} \]

1. Initial program 87.1%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. Step-by-step derivation

   1. sqrt-div87.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

   2. add-sqr-sqrt87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

   3. hypot-1-def87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

   4. *-commutative87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

   5. sqrt-prod87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

   6. unpow287.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

   7. sqrt-prod53.4%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

   8. add-sqr-sqrt99.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

3. Applied egg-rr99.0%

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

4. Final simplification99.0%

   \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999985e135

   1. Initial program 55.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div55.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow255.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt97.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr97.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 43.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative43.8%

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

      2. *-commutative43.8%

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

      3. unpow243.8%

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

      4. rem-square-sqrt43.8%

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

      5. associate-*l/43.8%

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

      6. *-commutative43.8%

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

      7. fma-def43.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow243.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow243.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac55.7%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified55.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   if -3.99999999999999985e135 < (/.f64 t l) < 1e21

   1. Initial program 99.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow299.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right)}}\right) \]

      3. frac-times99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{1 \cdot t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]

      4. *-un-lft-identity99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\color{blue}{t}}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   3. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]
   4. Step-by-step derivation

      1. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      3. un-div-inv99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   5. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   if 1e21 < (/.f64 t l)

   1. Initial program 70.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around inf 93.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative93.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

      2. unpow293.2%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      3. unpow293.2%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      4. times-frac99.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      5. unpow299.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      6. associate-/l*98.8%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Simplified98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-udef34.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]

      3. associate-*r/34.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)}\right)} - 1 \]

      4. pow1/234.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      5. pow1/234.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5} \cdot \color{blue}{{0.5}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      6. pow-prod-down34.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

   6. Applied egg-rr34.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p98.8%

         \[\leadsto \color{blue}{\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)} \]

      3. associate-/r/99.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{t} \cdot \ell\right)} \]

      4. unpow1/299.4%

         \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}{t} \cdot \ell\right) \]

      5. *-commutative99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{t} \cdot \ell\right) \]

   8. Simplified99.4%

      \[\leadsto \color{blue}{\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t} \cdot \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+21}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999985e135

   1. Initial program 55.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around -inf 89.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg89.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

      2. *-commutative89.5%

         \[\leadsto \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]

      3. distribute-rgt-neg-in89.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right)} \]

      4. unpow289.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      5. unpow289.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      6. times-frac99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      7. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]

      8. associate-/l*97.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)\right) \]

      9. associate-/r/99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{t} \cdot \ell}\right)\right) \]

   4. Simplified99.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\right)\right)} \]

      2. expm1-udef43.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\right)} - 1} \]

   6. Applied egg-rr43.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def97.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p97.4%

         \[\leadsto \color{blue}{-\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)} \]

      3. associate-/r/99.5%

         \[\leadsto -\sin^{-1} \color{blue}{\left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{t} \cdot \ell\right)} \]

      4. unpow1/299.5%

         \[\leadsto -\sin^{-1} \left(\frac{\color{blue}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}{t} \cdot \ell\right) \]

      5. *-commutative99.5%

         \[\leadsto -\sin^{-1} \left(\frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{t} \cdot \ell\right) \]

   8. Simplified99.5%

      \[\leadsto \color{blue}{-\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t} \cdot \ell\right)} \]

   if -3.99999999999999985e135 < (/.f64 t l) < 1e21

   1. Initial program 99.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow299.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right)}}\right) \]

      3. frac-times99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{1 \cdot t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]

      4. *-un-lft-identity99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\color{blue}{t}}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   3. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]
   4. Step-by-step derivation

      1. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      3. un-div-inv99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   5. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   if 1e21 < (/.f64 t l)

   1. Initial program 70.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around inf 93.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. *-commutative93.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

      2. unpow293.2%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      3. unpow293.2%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      4. times-frac99.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      5. unpow299.4%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]

      6. associate-/l*98.8%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Simplified98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-udef34.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]

      3. associate-*r/34.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)}\right)} - 1 \]

      4. pow1/234.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      5. pow1/234.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5} \cdot \color{blue}{{0.5}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      6. pow-prod-down34.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

   6. Applied egg-rr34.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def98.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p98.8%

         \[\leadsto \color{blue}{\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)} \]

      3. associate-/r/99.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{t} \cdot \ell\right)} \]

      4. unpow1/299.4%

         \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}{t} \cdot \ell\right) \]

      5. *-commutative99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{t} \cdot \ell\right) \]

   8. Simplified99.4%

      \[\leadsto \color{blue}{\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t} \cdot \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+135}:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+21}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.9999999999999999e155

   1. Initial program 49.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div49.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt49.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def49.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative49.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod49.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow249.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt97.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr97.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 49.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative49.4%

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

      2. *-commutative49.4%

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

      3. unpow249.4%

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

      4. rem-square-sqrt49.4%

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

      5. associate-*l/49.4%

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

      6. *-commutative49.4%

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

      7. fma-def49.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow249.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow249.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac49.4%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified49.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   if -4.9999999999999999e155 < (/.f64 t l) < 1.99999999999999989e72

   1. Initial program 99.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right)}}\right) \]

      3. frac-times96.9%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{1 \cdot t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]

      4. *-un-lft-identity96.9%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\color{blue}{t}}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   3. Applied egg-rr96.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]
   4. Step-by-step derivation

      1. unpow296.9%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      2. clear-num96.9%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      3. un-div-inv96.9%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   5. Applied egg-rr96.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   6. Taylor expanded in t around 0 79.4%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \color{blue}{\frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   7. Step-by-step derivation

      1. unpow279.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}}}\right) \]

      2. unpow279.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}}}\right) \]

      3. times-frac99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]

   8. Simplified99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]

   if 1.99999999999999989e72 < (/.f64 t l)

   1. Initial program 62.2%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div62.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt62.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def62.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative62.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod62.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow262.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 44.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative44.6%

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

      2. *-commutative44.6%

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

      3. unpow244.6%

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

      4. rem-square-sqrt44.6%

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

      5. associate-*l/44.6%

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

      6. *-commutative44.6%

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

      7. fma-def44.6%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow244.6%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow244.6%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac62.2%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified62.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999985e135

   1. Initial program 55.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div55.5%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod55.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow255.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt97.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr97.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 43.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative43.8%

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

      2. *-commutative43.8%

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

      3. unpow243.8%

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

      4. rem-square-sqrt43.8%

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

      5. associate-*l/43.8%

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

      6. *-commutative43.8%

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

      7. fma-def43.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow243.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow243.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac55.7%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified55.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   if -3.99999999999999985e135 < (/.f64 t l) < 2.00000000000000008e36

   1. Initial program 99.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right)}}\right) \]

      3. frac-times99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{1 \cdot t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]

      4. *-un-lft-identity99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\color{blue}{t}}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   3. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{t}{\frac{\ell}{t} \cdot \ell}}}}\right) \]
   4. Step-by-step derivation

      1. unpow299.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      2. clear-num99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      3. un-div-inv99.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   5. Applied egg-rr99.5%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   if 2.00000000000000008e36 < (/.f64 t l)

   1. Initial program 68.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div68.0%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt68.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def68.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative68.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod67.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow267.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 46.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative46.4%

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

      2. *-commutative46.4%

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

      3. unpow246.4%

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

      4. rem-square-sqrt46.5%

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

      5. associate-*l/46.5%

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

      6. *-commutative46.5%

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

      7. fma-def46.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow246.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow246.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac68.1%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified68.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e9

   1. Initial program 73.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div72.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod72.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow272.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 45.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative45.0%

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

      2. *-commutative45.0%

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

      3. unpow245.0%

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

      4. rem-square-sqrt45.1%

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

      5. associate-*l/45.1%

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

      6. *-commutative45.1%

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

      7. fma-def45.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.0%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   if -2e9 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 99.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around 0 85.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. unpow285.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]

      2. unpow285.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]

      3. times-frac98.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]

      4. unpow298.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

   4. Simplified98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow298.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]

      2. clear-num98.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{1}{\frac{Omc}{Om}}} \cdot \frac{Om}{Omc}}\right) \]

      3. clear-num98.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{1}{\frac{Omc}{Om}} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}}\right) \]

      4. frac-times98.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{1 \cdot 1}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}}}}\right) \]

      5. metadata-eval98.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{1}}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}}}\right) \]

   6. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{1}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}}}}\right) \]

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 73.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div73.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod73.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow273.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 48.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative48.6%

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

      2. *-commutative48.6%

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

      3. unpow248.6%

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

      4. rem-square-sqrt48.8%

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

      5. associate-*l/48.8%

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

      6. *-commutative48.8%

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

      7. fma-def48.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.3%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 + \frac{-1}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e9

   1. Initial program 73.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div72.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod72.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow272.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 45.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative45.0%

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

      2. *-commutative45.0%

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

      3. unpow245.0%

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

      4. rem-square-sqrt45.1%

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

      5. associate-*l/45.1%

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

      6. *-commutative45.1%

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

      7. fma-def45.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.0%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   if -2e9 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 99.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]

   2. Taylor expanded in t around 0 85.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. unpow285.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]

      2. unpow285.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]

      3. times-frac98.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]

      4. unpow298.9%

         \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

   4. Simplified98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]
   5. Step-by-step derivation

      1. unpow299.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      2. clear-num99.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

      3. un-div-inv99.4%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}{1 + 2 \cdot \frac{t}{\frac{\ell}{t} \cdot \ell}}}\right) \]

   6. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}\right) \]

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 73.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div73.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod73.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow273.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 48.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative48.6%

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

      2. *-commutative48.6%

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

      3. unpow248.6%

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

      4. rem-square-sqrt48.8%

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

      5. associate-*l/48.8%

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

      6. *-commutative48.8%

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

      7. fma-def48.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.3%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e9

   1. Initial program 73.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div72.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod72.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow272.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 45.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative45.0%

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

      2. *-commutative45.0%

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

      3. unpow245.0%

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

      4. rem-square-sqrt45.1%

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

      5. associate-*l/45.1%

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

      6. *-commutative45.1%

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

      7. fma-def45.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.0%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg99.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

      2. associate-/l*98.2%

         \[\leadsto \sin^{-1} \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   9. Simplified98.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]

   if -2e9 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 99.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div99.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow299.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod61.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 90.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative90.1%

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

      2. *-commutative90.1%

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

      3. unpow290.1%

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

      4. rem-square-sqrt90.1%

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

      5. associate-*l/90.1%

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

      6. *-commutative90.1%

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

      7. fma-def90.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow290.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow290.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac98.5%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified98.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 89.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(1 + -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg89.7%

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

      2. unsub-neg89.7%

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

      3. unpow289.7%

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

      4. unpow289.7%

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

      5. times-frac98.1%

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

      6. unpow298.1%

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

   9. Simplified98.1%

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

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 73.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div73.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod73.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow273.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 48.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative48.6%

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

      2. *-commutative48.6%

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

      3. unpow248.6%

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

      4. rem-square-sqrt48.8%

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

      5. associate-*l/48.8%

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

      6. *-commutative48.8%

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

      7. fma-def48.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.3%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e9

   1. Initial program 73.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div72.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod72.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow272.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 45.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative45.0%

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

      2. *-commutative45.0%

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

      3. unpow245.0%

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

      4. rem-square-sqrt45.1%

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

      5. associate-*l/45.1%

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

      6. *-commutative45.1%

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

      7. fma-def45.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.0%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   if -2e9 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 99.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div99.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow299.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod61.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 90.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative90.1%

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

      2. *-commutative90.1%

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

      3. unpow290.1%

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

      4. rem-square-sqrt90.1%

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

      5. associate-*l/90.1%

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

      6. *-commutative90.1%

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

      7. fma-def90.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow290.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow290.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac98.5%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified98.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 89.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(1 + -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg89.7%

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

      2. unsub-neg89.7%

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

      3. unpow289.7%

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

      4. unpow289.7%

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

      5. times-frac98.1%

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

      6. unpow298.1%

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

   9. Simplified98.1%

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

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 73.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div73.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod73.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow273.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 48.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative48.6%

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

      2. *-commutative48.6%

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

      3. unpow248.6%

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

      4. rem-square-sqrt48.8%

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

      5. associate-*l/48.8%

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

      6. *-commutative48.8%

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

      7. fma-def48.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.3%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Split input into 2 regimes

2. if (/.f64 t l) < -1.19999999999999993e205 or 0.71999999999999997 < (/.f64 t l)

   1. Initial program 71.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div71.3%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt71.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def71.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative71.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod71.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow271.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod67.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 54.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative54.5%

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

      2. *-commutative54.5%

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

      3. unpow254.5%

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

      4. rem-square-sqrt54.6%

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

      5. associate-*l/54.6%

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

      6. *-commutative54.6%

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

      7. fma-def54.6%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow254.6%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow254.6%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac71.3%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified71.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 88.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
   8. Step-by-step derivation

      1. associate-/l*88.3%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]

   9. Simplified88.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]

   if -1.19999999999999993e205 < (/.f64 t l) < 0.71999999999999997

   1. Initial program 94.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div94.1%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt94.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def94.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative94.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod94.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow294.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod47.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 76.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative76.5%

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

      2. *-commutative76.5%

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

      3. unpow276.5%

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

      4. rem-square-sqrt76.5%

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

      5. associate-*l/76.5%

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

      6. *-commutative76.5%

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

      7. fma-def76.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow276.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow276.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac93.4%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified93.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 76.7%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.2 \cdot 10^{+205} \lor \neg \left(\frac{t}{\ell} \leq 0.72\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.9999999999999999e214

   1. Initial program 67.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div67.1%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt67.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def67.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative67.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod67.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow267.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt96.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr96.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 67.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative67.1%

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

      2. *-commutative67.1%

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

      3. unpow267.1%

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

      4. rem-square-sqrt67.1%

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

      5. associate-*l/67.1%

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

      6. *-commutative67.1%

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

      7. fma-def67.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow267.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow267.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac67.1%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified67.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 66.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
   8. Step-by-step derivation

      1. associate-/l*66.6%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]

   9. Simplified66.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]

   if -1.9999999999999999e214 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 94.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div94.1%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt94.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def94.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative94.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod94.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow294.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod47.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 76.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative76.5%

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

      2. *-commutative76.5%

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

      3. unpow276.5%

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

      4. rem-square-sqrt76.5%

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

      5. associate-*l/76.5%

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

      6. *-commutative76.5%

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

      7. fma-def76.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow276.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow276.5%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac93.4%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified93.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 76.7%

      \[\leadsto \sin^{-1} \color{blue}{1} \]

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 73.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div73.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod73.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow273.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 48.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative48.6%

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

      2. *-commutative48.6%

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

      3. unpow248.6%

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

      4. rem-square-sqrt48.8%

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

      5. associate-*l/48.8%

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

      6. *-commutative48.8%

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

      7. fma-def48.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.3%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+214}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e9

   1. Initial program 73.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div72.9%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative72.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod72.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow272.8%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod0.0%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.3%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 45.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative45.0%

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

      2. *-commutative45.0%

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

      3. unpow245.0%

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

      4. rem-square-sqrt45.1%

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

      5. associate-*l/45.1%

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

      6. *-commutative45.1%

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

      7. fma-def45.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow245.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.0%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around -inf 99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg99.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

      2. associate-/l*98.2%

         \[\leadsto \sin^{-1} \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   9. Simplified98.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]

   if -2e9 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 99.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div99.4%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow299.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod61.5%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt99.4%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr99.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 90.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative90.1%

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

      2. *-commutative90.1%

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

      3. unpow290.1%

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

      4. rem-square-sqrt90.1%

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

      5. associate-*l/90.1%

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

      6. *-commutative90.1%

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

      7. fma-def90.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow290.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow290.1%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac98.5%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified98.5%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around 0 98.0%

      \[\leadsto \sin^{-1} \color{blue}{1} \]

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 73.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. sqrt-div73.2%

         \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

      2. add-sqr-sqrt73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

      3. hypot-1-def73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

      4. *-commutative73.2%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

      5. sqrt-prod73.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

      6. unpow273.1%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

      7. sqrt-prod98.6%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

      8. add-sqr-sqrt98.9%

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Applied egg-rr98.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   4. Taylor expanded in Om around 0 48.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
   5. Step-by-step derivation

      1. +-commutative48.6%

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

      2. *-commutative48.6%

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

      3. unpow248.6%

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

      4. rem-square-sqrt48.8%

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

      5. associate-*l/48.8%

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

      6. *-commutative48.8%

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

      7. fma-def48.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

      8. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

      9. unpow248.8%

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

      10. times-frac73.3%

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

   6. Simplified73.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

   7. Taylor expanded in t around inf 98.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

1. Initial program 87.1%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. Step-by-step derivation

   1. sqrt-div87.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]

   2. add-sqr-sqrt87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]

   3. hypot-1-def87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]

   4. *-commutative87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]

   5. sqrt-prod87.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]

   6. unpow287.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]

   7. sqrt-prod53.4%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]

   8. add-sqr-sqrt99.0%

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

3. Applied egg-rr99.0%

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

4. Taylor expanded in Om around 0 69.7%

   \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right)} \]
5. Step-by-step derivation

   1. +-commutative69.7%

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

   2. *-commutative69.7%

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

   3. unpow269.7%

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

   4. rem-square-sqrt69.8%

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

   5. associate-*l/69.8%

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

   6. *-commutative69.8%

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

   7. fma-def69.8%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{{t}^{2}}{{\ell}^{2}}, 1\right)}}}\right) \]

   8. unpow269.8%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}, 1\right)}}\right) \]

   9. unpow269.8%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, 1\right)}}\right) \]

   10. times-frac86.6%

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, 1\right)}}\right) \]

6. Simplified86.6%

   \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)} \]

7. Taylor expanded in t around 0 54.4%

   \[\leadsto \sin^{-1} \color{blue}{1} \]

8. Final simplification54.4%

   \[\leadsto \sin^{-1} 1 \]

herbie shell --seed 2023166 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))