Average Error: 10.1 → 5.7
Time: 24.4s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.3085858754480278 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.3085858754480278 \cdot 10^{+135}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\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 r1533607 = 1.0;
        double r1533608 = Om;
        double r1533609 = Omc;
        double r1533610 = r1533608 / r1533609;
        double r1533611 = 2.0;
        double r1533612 = pow(r1533610, r1533611);
        double r1533613 = r1533607 - r1533612;
        double r1533614 = t;
        double r1533615 = l;
        double r1533616 = r1533614 / r1533615;
        double r1533617 = pow(r1533616, r1533611);
        double r1533618 = r1533611 * r1533617;
        double r1533619 = r1533607 + r1533618;
        double r1533620 = r1533613 / r1533619;
        double r1533621 = sqrt(r1533620);
        double r1533622 = asin(r1533621);
        return r1533622;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1533623 = t;
        double r1533624 = l;
        double r1533625 = r1533623 / r1533624;
        double r1533626 = 1.3085858754480278e+135;
        bool r1533627 = r1533625 <= r1533626;
        double r1533628 = 1.0;
        double r1533629 = Om;
        double r1533630 = Omc;
        double r1533631 = r1533629 / r1533630;
        double r1533632 = r1533631 * r1533631;
        double r1533633 = r1533628 - r1533632;
        double r1533634 = 2.0;
        double r1533635 = r1533624 / r1533623;
        double r1533636 = r1533635 * r1533635;
        double r1533637 = r1533634 / r1533636;
        double r1533638 = r1533628 + r1533637;
        double r1533639 = r1533633 / r1533638;
        double r1533640 = sqrt(r1533639);
        double r1533641 = asin(r1533640);
        double r1533642 = sqrt(r1533633);
        double r1533643 = sqrt(r1533634);
        double r1533644 = r1533623 * r1533643;
        double r1533645 = r1533644 / r1533624;
        double r1533646 = r1533642 / r1533645;
        double r1533647 = asin(r1533646);
        double r1533648 = r1533627 ? r1533641 : r1533647;
        return r1533648;
}

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) < 1.3085858754480278e+135

    1. Initial program 6.4

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

    2. Simplified6.4

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

    4. Applied add-cube-cbrt6.5

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

    5. Applied add-cube-cbrt6.5

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

    6. Applied times-frac6.5

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

    7. Applied associate-*l*6.5

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

    8. Taylor expanded around 0 24.0

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

    9. Simplified6.5

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

    if 1.3085858754480278e+135 < (/ t l)

    1. Initial program 30.9

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

    2. Simplified30.9

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

    4. Applied sqrt-div30.9

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

    5. Taylor expanded around inf 1.3

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.3085858754480278 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))