Average Error: 10.4 → 5.6
Time: 31.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.0964407420254373 \cdot 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.0964407420254373 \cdot 10^{+146}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}\right)\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 r1323129 = 1.0;
        double r1323130 = Om;
        double r1323131 = Omc;
        double r1323132 = r1323130 / r1323131;
        double r1323133 = 2.0;
        double r1323134 = pow(r1323132, r1323133);
        double r1323135 = r1323129 - r1323134;
        double r1323136 = t;
        double r1323137 = l;
        double r1323138 = r1323136 / r1323137;
        double r1323139 = pow(r1323138, r1323133);
        double r1323140 = r1323133 * r1323139;
        double r1323141 = r1323129 + r1323140;
        double r1323142 = r1323135 / r1323141;
        double r1323143 = sqrt(r1323142);
        double r1323144 = asin(r1323143);
        return r1323144;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1323145 = t;
        double r1323146 = l;
        double r1323147 = r1323145 / r1323146;
        double r1323148 = 1.0964407420254373e+146;
        bool r1323149 = r1323147 <= r1323148;
        double r1323150 = 1.0;
        double r1323151 = Om;
        double r1323152 = Omc;
        double r1323153 = r1323151 / r1323152;
        double r1323154 = r1323153 * r1323153;
        double r1323155 = r1323150 - r1323154;
        double r1323156 = r1323147 * r1323147;
        double r1323157 = 2.0;
        double r1323158 = fma(r1323156, r1323157, r1323150);
        double r1323159 = r1323155 / r1323158;
        double r1323160 = expm1(r1323159);
        double r1323161 = log1p(r1323160);
        double r1323162 = sqrt(r1323161);
        double r1323163 = asin(r1323162);
        double r1323164 = sqrt(r1323155);
        double r1323165 = sqrt(r1323157);
        double r1323166 = r1323145 * r1323165;
        double r1323167 = r1323166 / r1323146;
        double r1323168 = r1323164 / r1323167;
        double r1323169 = asin(r1323168);
        double r1323170 = r1323149 ? r1323163 : r1323169;
        return r1323170;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 1.0964407420254373e+146

    1. Initial program 6.4

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

    2. Simplified6.4

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

    4. Applied log1p-expm1-u6.4

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}\right)\right)}}\right)\]

    if 1.0964407420254373e+146 < (/ t l)

    1. Initial program 34.2

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

    2. Simplified34.2

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

    4. Applied sqrt-div34.2

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

    5. 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.0964407420254373 \cdot 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))