Average Error: 10.1 → 5.6
Time: 22.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 6.668151396198947 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 6.668151396198947 \cdot 10^{+152}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \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 r2328422 = 1.0;
        double r2328423 = Om;
        double r2328424 = Omc;
        double r2328425 = r2328423 / r2328424;
        double r2328426 = 2.0;
        double r2328427 = pow(r2328425, r2328426);
        double r2328428 = r2328422 - r2328427;
        double r2328429 = t;
        double r2328430 = l;
        double r2328431 = r2328429 / r2328430;
        double r2328432 = pow(r2328431, r2328426);
        double r2328433 = r2328426 * r2328432;
        double r2328434 = r2328422 + r2328433;
        double r2328435 = r2328428 / r2328434;
        double r2328436 = sqrt(r2328435);
        double r2328437 = asin(r2328436);
        return r2328437;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2328438 = t;
        double r2328439 = l;
        double r2328440 = r2328438 / r2328439;
        double r2328441 = 6.668151396198947e+152;
        bool r2328442 = r2328440 <= r2328441;
        double r2328443 = 1.0;
        double r2328444 = Om;
        double r2328445 = Omc;
        double r2328446 = r2328444 / r2328445;
        double r2328447 = r2328446 * r2328446;
        double r2328448 = r2328443 - r2328447;
        double r2328449 = sqrt(r2328448);
        double r2328450 = r2328440 * r2328440;
        double r2328451 = r2328450 + r2328450;
        double r2328452 = r2328451 + r2328443;
        double r2328453 = sqrt(r2328452);
        double r2328454 = r2328449 / r2328453;
        double r2328455 = asin(r2328454);
        double r2328456 = 2.0;
        double r2328457 = sqrt(r2328456);
        double r2328458 = r2328438 * r2328457;
        double r2328459 = r2328458 / r2328439;
        double r2328460 = r2328449 / r2328459;
        double r2328461 = asin(r2328460);
        double r2328462 = r2328442 ? r2328455 : r2328461;
        return r2328462;
}

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) < 6.668151396198947e+152

    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

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

    4. Applied sqrt-div6.3

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

    if 6.668151396198947e+152 < (/ t l)

    1. Initial program 32.7

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

    2. Simplified32.7

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

    4. Applied sqrt-div32.7

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

    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 simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 6.668151396198947 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))