Average Error: 10.1 → 5.6
Time: 28.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.6736910583871654 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\sqrt{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{\sqrt{2} \cdot t}{\ell}}\right)\\ \end{array}\]
\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 5.6736910583871654 \cdot 10^{+57}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\sqrt{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}}\right)\\

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

\end{array}
double f(double t, double l, double Om, double Omc) {
        double r2197804 = 1.0;
        double r2197805 = Om;
        double r2197806 = Omc;
        double r2197807 = r2197805 / r2197806;
        double r2197808 = 2.0;
        double r2197809 = pow(r2197807, r2197808);
        double r2197810 = r2197804 - r2197809;
        double r2197811 = t;
        double r2197812 = l;
        double r2197813 = r2197811 / r2197812;
        double r2197814 = pow(r2197813, r2197808);
        double r2197815 = r2197808 * r2197814;
        double r2197816 = r2197804 + r2197815;
        double r2197817 = r2197810 / r2197816;
        double r2197818 = sqrt(r2197817);
        double r2197819 = asin(r2197818);
        return r2197819;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2197820 = t;
        double r2197821 = l;
        double r2197822 = r2197820 / r2197821;
        double r2197823 = 5.6736910583871654e+57;
        bool r2197824 = r2197822 <= r2197823;
        double r2197825 = 1.0;
        double r2197826 = Om;
        double r2197827 = Omc;
        double r2197828 = r2197826 / r2197827;
        double r2197829 = r2197828 * r2197828;
        double r2197830 = exp(r2197829);
        double r2197831 = log(r2197830);
        double r2197832 = r2197825 - r2197831;
        double r2197833 = sqrt(r2197832);
        double r2197834 = r2197822 * r2197822;
        double r2197835 = 2.0;
        double r2197836 = r2197834 * r2197835;
        double r2197837 = r2197836 + r2197825;
        double r2197838 = sqrt(r2197837);
        double r2197839 = r2197833 / r2197838;
        double r2197840 = asin(r2197839);
        double r2197841 = r2197825 - r2197829;
        double r2197842 = sqrt(r2197841);
        double r2197843 = sqrt(r2197835);
        double r2197844 = r2197843 * r2197820;
        double r2197845 = r2197844 / r2197821;
        double r2197846 = r2197842 / r2197845;
        double r2197847 = asin(r2197846);
        double r2197848 = r2197824 ? r2197840 : r2197847;
        return r2197848;
}

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) < 5.6736910583871654e+57

    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. Using strategy rm

    4. Applied sqrt-div6.7

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

    6. Applied add-log-exp6.7

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

    if 5.6736910583871654e+57 < (/ t l)

    1. Initial program 23.3

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

    2. Simplified23.3

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

    4. Applied sqrt-div23.3

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

    5. Taylor expanded around -inf 0.9

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.6736910583871654 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\sqrt{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{\sqrt{2} \cdot t}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))