Average Error: 10.0 → 5.6
Time: 25.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.092158378202942 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{t}{\ell} \cdot \left(2 \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 5.092158378202942 \cdot 10^{+135}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{t}{\ell} \cdot \left(2 \cdot \frac{t}{\ell}\right) + 1}}\right)\\

\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 r1662732 = 1.0;
        double r1662733 = Om;
        double r1662734 = Omc;
        double r1662735 = r1662733 / r1662734;
        double r1662736 = 2.0;
        double r1662737 = pow(r1662735, r1662736);
        double r1662738 = r1662732 - r1662737;
        double r1662739 = t;
        double r1662740 = l;
        double r1662741 = r1662739 / r1662740;
        double r1662742 = pow(r1662741, r1662736);
        double r1662743 = r1662736 * r1662742;
        double r1662744 = r1662732 + r1662743;
        double r1662745 = r1662738 / r1662744;
        double r1662746 = sqrt(r1662745);
        double r1662747 = asin(r1662746);
        return r1662747;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1662748 = t;
        double r1662749 = l;
        double r1662750 = r1662748 / r1662749;
        double r1662751 = 5.092158378202942e+135;
        bool r1662752 = r1662750 <= r1662751;
        double r1662753 = 1.0;
        double r1662754 = Om;
        double r1662755 = Omc;
        double r1662756 = r1662754 / r1662755;
        double r1662757 = r1662756 * r1662756;
        double r1662758 = r1662753 - r1662757;
        double r1662759 = 2.0;
        double r1662760 = r1662759 * r1662750;
        double r1662761 = r1662750 * r1662760;
        double r1662762 = r1662761 + r1662753;
        double r1662763 = r1662758 / r1662762;
        double r1662764 = sqrt(r1662763);
        double r1662765 = asin(r1662764);
        double r1662766 = sqrt(r1662758);
        double r1662767 = sqrt(r1662759);
        double r1662768 = r1662748 * r1662767;
        double r1662769 = r1662768 / r1662749;
        double r1662770 = r1662766 / r1662769;
        double r1662771 = asin(r1662770);
        double r1662772 = r1662752 ? r1662765 : r1662771;
        return r1662772;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 5.092158378202942e+135

    1. Initial program 6.3

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

    2. Simplified6.3

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

    if 5.092158378202942e+135 < (/ t l)

    1. Initial program 31.7

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

    2. Simplified31.7

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

    4. Applied sqrt-div31.7

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

    5. Taylor expanded around inf 1.4

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.092158378202942 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{t}{\ell} \cdot \left(2 \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))