Average Error: 10.1 → 5.8
Time: 21.1s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.972769457683076 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} + \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.972769457683076 \cdot 10^{+136}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} + \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 r907483 = 1.0;
        double r907484 = Om;
        double r907485 = Omc;
        double r907486 = r907484 / r907485;
        double r907487 = 2.0;
        double r907488 = pow(r907486, r907487);
        double r907489 = r907483 - r907488;
        double r907490 = t;
        double r907491 = l;
        double r907492 = r907490 / r907491;
        double r907493 = pow(r907492, r907487);
        double r907494 = r907487 * r907493;
        double r907495 = r907483 + r907494;
        double r907496 = r907489 / r907495;
        double r907497 = sqrt(r907496);
        double r907498 = asin(r907497);
        return r907498;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r907499 = t;
        double r907500 = l;
        double r907501 = r907499 / r907500;
        double r907502 = 1.972769457683076e+136;
        bool r907503 = r907501 <= r907502;
        double r907504 = 1.0;
        double r907505 = Om;
        double r907506 = Omc;
        double r907507 = r907505 / r907506;
        double r907508 = r907507 * r907507;
        double r907509 = r907504 - r907508;
        double r907510 = sqrt(r907509);
        double r907511 = r907501 + r907501;
        double r907512 = r907501 * r907511;
        double r907513 = r907512 + r907504;
        double r907514 = sqrt(r907513);
        double r907515 = r907510 / r907514;
        double r907516 = asin(r907515);
        double r907517 = 2.0;
        double r907518 = sqrt(r907517);
        double r907519 = r907499 * r907518;
        double r907520 = r907519 / r907500;
        double r907521 = r907510 / r907520;
        double r907522 = asin(r907521);
        double r907523 = r907503 ? r907516 : r907522;
        return r907523;
}

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) < 1.972769457683076e+136

    1. Initial program 6.5

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

    2. Simplified6.5

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

    4. Applied sqrt-div6.5

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

    if 1.972769457683076e+136 < (/ t l)

    1. Initial program 31.3

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

    2. Simplified31.3

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

    4. Applied sqrt-div31.3

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

    5. Taylor expanded around inf 1.6

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.972769457683076 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} + \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))