Average Error: 10.4 → 5.9
Time: 17.9s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.3226963208817615 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \frac{t}{\ell} + \left(1 + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.3226963208817615 \cdot 10^{+73}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \frac{t}{\ell} + \left(1 + \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 r909106 = 1.0;
        double r909107 = Om;
        double r909108 = Omc;
        double r909109 = r909107 / r909108;
        double r909110 = 2.0;
        double r909111 = pow(r909109, r909110);
        double r909112 = r909106 - r909111;
        double r909113 = t;
        double r909114 = l;
        double r909115 = r909113 / r909114;
        double r909116 = pow(r909115, r909110);
        double r909117 = r909110 * r909116;
        double r909118 = r909106 + r909117;
        double r909119 = r909112 / r909118;
        double r909120 = sqrt(r909119);
        double r909121 = asin(r909120);
        return r909121;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r909122 = t;
        double r909123 = l;
        double r909124 = r909122 / r909123;
        double r909125 = 1.3226963208817615e+73;
        bool r909126 = r909124 <= r909125;
        double r909127 = 1.0;
        double r909128 = Om;
        double r909129 = Omc;
        double r909130 = r909128 / r909129;
        double r909131 = r909130 * r909130;
        double r909132 = r909127 - r909131;
        double r909133 = sqrt(r909132);
        double r909134 = r909124 * r909124;
        double r909135 = r909127 + r909134;
        double r909136 = r909134 + r909135;
        double r909137 = sqrt(r909136);
        double r909138 = r909133 / r909137;
        double r909139 = asin(r909138);
        double r909140 = 2.0;
        double r909141 = sqrt(r909140);
        double r909142 = r909122 * r909141;
        double r909143 = r909142 / r909123;
        double r909144 = r909133 / r909143;
        double r909145 = asin(r909144);
        double r909146 = r909126 ? r909139 : r909145;
        return r909146;
}

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) < 1.3226963208817615e+73

    1. Initial program 6.9

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

    2. Simplified6.9

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

    4. Applied add-sqr-sqrt6.9

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

    6. Applied sqrt-div6.9

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

    7. Simplified6.9

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

    if 1.3226963208817615e+73 < (/ t l)

    1. Initial program 25.3

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

    2. Simplified25.3

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

    4. Applied add-sqr-sqrt25.3

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

    6. Applied sqrt-div25.3

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

    7. Simplified25.3

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

    8. 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 simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.3226963208817615 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \frac{t}{\ell} + \left(1 + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))