Average Error: 9.5 → 5.3
Time: 42.2s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.572137631982805 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 7.572137631982805 \cdot 10^{+147}:\\
\;\;\;\;\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)\\

\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 r2212859 = 1.0;
        double r2212860 = Om;
        double r2212861 = Omc;
        double r2212862 = r2212860 / r2212861;
        double r2212863 = 2.0;
        double r2212864 = pow(r2212862, r2212863);
        double r2212865 = r2212859 - r2212864;
        double r2212866 = t;
        double r2212867 = l;
        double r2212868 = r2212866 / r2212867;
        double r2212869 = pow(r2212868, r2212863);
        double r2212870 = r2212863 * r2212869;
        double r2212871 = r2212859 + r2212870;
        double r2212872 = r2212865 / r2212871;
        double r2212873 = sqrt(r2212872);
        double r2212874 = asin(r2212873);
        return r2212874;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2212875 = t;
        double r2212876 = l;
        double r2212877 = r2212875 / r2212876;
        double r2212878 = 7.572137631982805e+147;
        bool r2212879 = r2212877 <= r2212878;
        double r2212880 = 1.0;
        double r2212881 = Om;
        double r2212882 = Omc;
        double r2212883 = r2212881 / r2212882;
        double r2212884 = r2212883 * r2212883;
        double r2212885 = r2212880 - r2212884;
        double r2212886 = r2212877 * r2212877;
        double r2212887 = 2.0;
        double r2212888 = fma(r2212886, r2212887, r2212880);
        double r2212889 = r2212885 / r2212888;
        double r2212890 = sqrt(r2212889);
        double r2212891 = asin(r2212890);
        double r2212892 = sqrt(r2212885);
        double r2212893 = sqrt(r2212887);
        double r2212894 = r2212875 * r2212893;
        double r2212895 = r2212894 / r2212876;
        double r2212896 = r2212892 / r2212895;
        double r2212897 = asin(r2212896);
        double r2212898 = r2212879 ? r2212891 : r2212897;
        return r2212898;
}

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) < 7.572137631982805e+147

    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

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

    4. Applied sqrt-div6.0

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

    5. Taylor expanded around inf 23.8

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

    6. Simplified6.0

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

    if 7.572137631982805e+147 < (/ t l)

    1. Initial program 31.7

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

    2. Simplified31.7

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

    4. Applied sqrt-div31.7

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

    5. Taylor expanded around inf 1.4

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

  4. Final simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.572137631982805 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +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)))))))