Average Error: 10.5 → 5.8
Time: 2.0m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.639427664533362 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 7.639427664533362 \cdot 10^{+141}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{2 \cdot \left(\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 r6162116 = 1.0;
        double r6162117 = Om;
        double r6162118 = Omc;
        double r6162119 = r6162117 / r6162118;
        double r6162120 = 2.0;
        double r6162121 = pow(r6162119, r6162120);
        double r6162122 = r6162116 - r6162121;
        double r6162123 = t;
        double r6162124 = l;
        double r6162125 = r6162123 / r6162124;
        double r6162126 = pow(r6162125, r6162120);
        double r6162127 = r6162120 * r6162126;
        double r6162128 = r6162116 + r6162127;
        double r6162129 = r6162122 / r6162128;
        double r6162130 = sqrt(r6162129);
        double r6162131 = asin(r6162130);
        return r6162131;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r6162132 = t;
        double r6162133 = l;
        double r6162134 = r6162132 / r6162133;
        double r6162135 = 7.639427664533362e+141;
        bool r6162136 = r6162134 <= r6162135;
        double r6162137 = 1.0;
        double r6162138 = Om;
        double r6162139 = Omc;
        double r6162140 = r6162138 / r6162139;
        double r6162141 = r6162140 * r6162140;
        double r6162142 = r6162137 - r6162141;
        double r6162143 = 2.0;
        double r6162144 = r6162134 * r6162134;
        double r6162145 = r6162143 * r6162144;
        double r6162146 = r6162145 + r6162137;
        double r6162147 = r6162142 / r6162146;
        double r6162148 = sqrt(r6162147);
        double r6162149 = asin(r6162148);
        double r6162150 = sqrt(r6162142);
        double r6162151 = sqrt(r6162143);
        double r6162152 = r6162132 * r6162151;
        double r6162153 = r6162152 / r6162133;
        double r6162154 = r6162150 / r6162153;
        double r6162155 = asin(r6162154);
        double r6162156 = r6162136 ? r6162149 : r6162155;
        return r6162156;
}

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) < 7.639427664533362e+141

    1. Initial program 6.6

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

    2. Simplified6.6

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

    if 7.639427664533362e+141 < (/ t l)

    1. Initial program 32.2

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

    2. Simplified32.2

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

    4. Applied sqrt-div32.3

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

    5. Taylor expanded around inf 1.5

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.639427664533362 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))