Average Error: 10.1 → 5.4
Time: 24.0s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.2007149219171108 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} + \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.2007149219171108 \cdot 10^{+153}:\\
\;\;\;\;\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 r887622 = 1.0;
        double r887623 = Om;
        double r887624 = Omc;
        double r887625 = r887623 / r887624;
        double r887626 = 2.0;
        double r887627 = pow(r887625, r887626);
        double r887628 = r887622 - r887627;
        double r887629 = t;
        double r887630 = l;
        double r887631 = r887629 / r887630;
        double r887632 = pow(r887631, r887626);
        double r887633 = r887626 * r887632;
        double r887634 = r887622 + r887633;
        double r887635 = r887628 / r887634;
        double r887636 = sqrt(r887635);
        double r887637 = asin(r887636);
        return r887637;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r887638 = t;
        double r887639 = l;
        double r887640 = r887638 / r887639;
        double r887641 = 1.2007149219171108e+153;
        bool r887642 = r887640 <= r887641;
        double r887643 = 1.0;
        double r887644 = Om;
        double r887645 = Omc;
        double r887646 = r887644 / r887645;
        double r887647 = r887646 * r887646;
        double r887648 = r887643 - r887647;
        double r887649 = sqrt(r887648);
        double r887650 = r887640 + r887640;
        double r887651 = r887640 * r887650;
        double r887652 = r887651 + r887643;
        double r887653 = sqrt(r887652);
        double r887654 = r887649 / r887653;
        double r887655 = asin(r887654);
        double r887656 = 2.0;
        double r887657 = sqrt(r887656);
        double r887658 = r887638 * r887657;
        double r887659 = r887658 / r887639;
        double r887660 = r887649 / r887659;
        double r887661 = asin(r887660);
        double r887662 = r887642 ? r887655 : r887661;
        return r887662;
}

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) < 1.2007149219171108e+153

    1. Initial program 6.0

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

    2. Simplified6.0

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

    4. Applied add-log-exp6.0

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

    6. Applied sqrt-div6.1

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

    7. Simplified6.1

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

    if 1.2007149219171108e+153 < (/ t l)

    1. Initial program 34.9

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

    2. Simplified34.9

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

    4. Applied add-log-exp34.9

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

    6. Applied sqrt-div34.9

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

    7. Simplified34.9

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

    8. Taylor expanded around inf 1.4

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

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.2007149219171108 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} + \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))