Average Error: 10.6 → 5.9
Time: 1.5m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9494802667865075 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.9494802667865075 \cdot 10^{+147}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{2 \cdot \left(\frac{t}{\ell} \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 r3709866 = 1.0;
        double r3709867 = Om;
        double r3709868 = Omc;
        double r3709869 = r3709867 / r3709868;
        double r3709870 = 2.0;
        double r3709871 = pow(r3709869, r3709870);
        double r3709872 = r3709866 - r3709871;
        double r3709873 = t;
        double r3709874 = l;
        double r3709875 = r3709873 / r3709874;
        double r3709876 = pow(r3709875, r3709870);
        double r3709877 = r3709870 * r3709876;
        double r3709878 = r3709866 + r3709877;
        double r3709879 = r3709872 / r3709878;
        double r3709880 = sqrt(r3709879);
        double r3709881 = asin(r3709880);
        return r3709881;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r3709882 = t;
        double r3709883 = l;
        double r3709884 = r3709882 / r3709883;
        double r3709885 = 1.9494802667865075e+147;
        bool r3709886 = r3709884 <= r3709885;
        double r3709887 = 1.0;
        double r3709888 = Om;
        double r3709889 = Omc;
        double r3709890 = r3709888 / r3709889;
        double r3709891 = r3709890 * r3709890;
        double r3709892 = r3709887 - r3709891;
        double r3709893 = 2.0;
        double r3709894 = r3709884 * r3709884;
        double r3709895 = r3709893 * r3709894;
        double r3709896 = r3709895 + r3709887;
        double r3709897 = r3709892 / r3709896;
        double r3709898 = sqrt(r3709897);
        double r3709899 = asin(r3709898);
        double r3709900 = sqrt(r3709892);
        double r3709901 = sqrt(r3709893);
        double r3709902 = r3709882 * r3709901;
        double r3709903 = r3709902 / r3709883;
        double r3709904 = r3709900 / r3709903;
        double r3709905 = asin(r3709904);
        double r3709906 = r3709886 ? r3709899 : r3709905;
        return r3709906;
}

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) < 1.9494802667865075e+147

    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

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

    3. Taylor expanded around 0 19.4

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

    4. Simplified6.7

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

    if 1.9494802667865075e+147 < (/ t l)

    1. Initial program 33.5

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

    2. Simplified33.5

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

    3. Taylor expanded around 0 34.6

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

    4. Simplified33.5

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

    6. Applied sqrt-div33.5

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

    7. Taylor expanded around inf 1.0

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9494802667865075 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))