Average Error: 10.2 → 5.4
Time: 16.4s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9934801738260375 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \left(\frac{t}{\ell} + \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.9934801738260375 \cdot 10^{+150}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \left(\frac{t}{\ell} + \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}\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 r1060408 = 1.0;
        double r1060409 = Om;
        double r1060410 = Omc;
        double r1060411 = r1060409 / r1060410;
        double r1060412 = 2.0;
        double r1060413 = pow(r1060411, r1060412);
        double r1060414 = r1060408 - r1060413;
        double r1060415 = t;
        double r1060416 = l;
        double r1060417 = r1060415 / r1060416;
        double r1060418 = pow(r1060417, r1060412);
        double r1060419 = r1060412 * r1060418;
        double r1060420 = r1060408 + r1060419;
        double r1060421 = r1060414 / r1060420;
        double r1060422 = sqrt(r1060421);
        double r1060423 = asin(r1060422);
        return r1060423;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1060424 = t;
        double r1060425 = l;
        double r1060426 = r1060424 / r1060425;
        double r1060427 = 1.9934801738260375e+150;
        bool r1060428 = r1060426 <= r1060427;
        double r1060429 = 1.0;
        double r1060430 = Om;
        double r1060431 = Omc;
        double r1060432 = r1060430 / r1060431;
        double r1060433 = r1060432 * r1060432;
        double r1060434 = r1060429 - r1060433;
        double r1060435 = r1060426 + r1060426;
        double r1060436 = r1060435 * r1060426;
        double r1060437 = r1060429 + r1060436;
        double r1060438 = r1060434 / r1060437;
        double r1060439 = sqrt(r1060438);
        double r1060440 = asin(r1060439);
        double r1060441 = sqrt(r1060434);
        double r1060442 = 2.0;
        double r1060443 = sqrt(r1060442);
        double r1060444 = r1060424 * r1060443;
        double r1060445 = r1060444 / r1060425;
        double r1060446 = r1060441 / r1060445;
        double r1060447 = asin(r1060446);
        double r1060448 = r1060428 ? r1060440 : r1060447;
        return r1060448;
}

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) < 1.9934801738260375e+150

    1. Initial program 6.1

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

    2. Simplified6.1

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

    if 1.9934801738260375e+150 < (/ t l)

    1. Initial program 33.7

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

    2. Simplified33.7

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

    4. Applied sqrt-div33.7

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

    5. Taylor expanded around inf 1.2

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9934801738260375 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \left(\frac{t}{\ell} + \frac{t}{\ell}\right) \cdot \frac{t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))