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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.3440785796627555 \cdot 10^{+40}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

double f(double t, double l, double Om, double Omc) {
        double r3619701 = 1.0;
        double r3619702 = Om;
        double r3619703 = Omc;
        double r3619704 = r3619702 / r3619703;
        double r3619705 = 2.0;
        double r3619706 = pow(r3619704, r3619705);
        double r3619707 = r3619701 - r3619706;
        double r3619708 = t;
        double r3619709 = l;
        double r3619710 = r3619708 / r3619709;
        double r3619711 = pow(r3619710, r3619705);
        double r3619712 = r3619705 * r3619711;
        double r3619713 = r3619701 + r3619712;
        double r3619714 = r3619707 / r3619713;
        double r3619715 = sqrt(r3619714);
        double r3619716 = asin(r3619715);
        return r3619716;
}

↓

double f(double t, double l, double Om, double Omc) {
        double r3619717 = t;
        double r3619718 = l;
        double r3619719 = r3619717 / r3619718;
        double r3619720 = 2.3440785796627555e+40;
        bool r3619721 = r3619719 <= r3619720;
        double r3619722 = 1.0;
        double r3619723 = Om;
        double r3619724 = Omc;
        double r3619725 = r3619723 / r3619724;
        double r3619726 = r3619725 * r3619725;
        double r3619727 = r3619722 - r3619726;
        double r3619728 = r3619719 * r3619719;
        double r3619729 = 2.0;
        double r3619730 = fma(r3619728, r3619729, r3619722);
        double r3619731 = r3619727 / r3619730;
        double r3619732 = sqrt(r3619731);
        double r3619733 = asin(r3619732);
        double r3619734 = log1p(r3619733);
        double r3619735 = expm1(r3619734);
        double r3619736 = sqrt(r3619727);
        double r3619737 = sqrt(r3619729);
        double r3619738 = r3619717 * r3619737;
        double r3619739 = r3619738 / r3619718;
        double r3619740 = r3619736 / r3619739;
        double r3619741 = asin(r3619740);
        double r3619742 = r3619721 ? r3619735 : r3619741;
        return r3619742;
}

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

↓

\begin{array}{l}
\mathbf{if}\;\frac{t}{\ell} \le 2.3440785796627555 \cdot 10^{+40}:\\
\;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right))} - 1)^*\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\

\end{array}

Derivation

Split input into 2 regimes
if (/ t l) < 2.3440785796627555e+40
1. Initial program 6.7
  \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
2. Simplified6.7
  \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)}\]
3. Using strategy rm
4. Applied expm1-log1p-u6.7
  \[\leadsto \color{blue}{(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right))} - 1)^*}\]
if 2.3440785796627555e+40 < (/ t l)
1. Initial program 23.2
  \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
2. Simplified23.2
  \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)}\]
3. Using strategy rm
4. Applied sqrt-div23.2
  \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)}\]
5. Taylor expanded around -inf 1.0
  \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}}\right)\]
Recombined 2 regimes into one program.
Final simplification5.4
\[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.3440785796627555 \cdot 10^{+40}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Error

Derivation

`if (/ t l) < 2.3440785796627555e+40`

`if 2.3440785796627555e+40 < (/ t l)`

Reproduce