Average Error: 10.2 → 5.6
Time: 31.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.410996405697987 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 2.410996405697987 \cdot 10^{+143}:\\
\;\;\;\;\sin^{-1} \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)\\

\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 r1532564 = 1.0;
        double r1532565 = Om;
        double r1532566 = Omc;
        double r1532567 = r1532565 / r1532566;
        double r1532568 = 2.0;
        double r1532569 = pow(r1532567, r1532568);
        double r1532570 = r1532564 - r1532569;
        double r1532571 = t;
        double r1532572 = l;
        double r1532573 = r1532571 / r1532572;
        double r1532574 = pow(r1532573, r1532568);
        double r1532575 = r1532568 * r1532574;
        double r1532576 = r1532564 + r1532575;
        double r1532577 = r1532570 / r1532576;
        double r1532578 = sqrt(r1532577);
        double r1532579 = asin(r1532578);
        return r1532579;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1532580 = t;
        double r1532581 = l;
        double r1532582 = r1532580 / r1532581;
        double r1532583 = 2.410996405697987e+143;
        bool r1532584 = r1532582 <= r1532583;
        double r1532585 = 1.0;
        double r1532586 = Om;
        double r1532587 = Omc;
        double r1532588 = r1532586 / r1532587;
        double r1532589 = r1532588 * r1532588;
        double r1532590 = r1532585 - r1532589;
        double r1532591 = sqrt(r1532590);
        double r1532592 = 2.0;
        double r1532593 = r1532582 * r1532582;
        double r1532594 = r1532592 * r1532593;
        double r1532595 = r1532585 + r1532594;
        double r1532596 = sqrt(r1532595);
        double r1532597 = r1532591 / r1532596;
        double r1532598 = asin(r1532597);
        double r1532599 = sqrt(r1532592);
        double r1532600 = r1532580 * r1532599;
        double r1532601 = r1532600 / r1532581;
        double r1532602 = r1532591 / r1532601;
        double r1532603 = asin(r1532602);
        double r1532604 = r1532584 ? r1532598 : r1532603;
        return r1532604;
}

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) < 2.410996405697987e+143

    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

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

    3. Taylor expanded around inf 23.7

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

    4. Simplified6.3

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

    6. Applied sqrt-div6.4

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

    if 2.410996405697987e+143 < (/ 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

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

    3. Taylor expanded around inf 36.6

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

    4. Simplified31.3

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

    6. Applied sqrt-div31.3

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

    7. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.410996405697987 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))