Average Error: 10.1 → 5.6
Time: 33.5s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.6736910583871654 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 5.6736910583871654 \cdot 10^{+57}:\\
\;\;\;\;\sin^{-1} \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)\\

\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 r1428809 = 1.0;
        double r1428810 = Om;
        double r1428811 = Omc;
        double r1428812 = r1428810 / r1428811;
        double r1428813 = 2.0;
        double r1428814 = pow(r1428812, r1428813);
        double r1428815 = r1428809 - r1428814;
        double r1428816 = t;
        double r1428817 = l;
        double r1428818 = r1428816 / r1428817;
        double r1428819 = pow(r1428818, r1428813);
        double r1428820 = r1428813 * r1428819;
        double r1428821 = r1428809 + r1428820;
        double r1428822 = r1428815 / r1428821;
        double r1428823 = sqrt(r1428822);
        double r1428824 = asin(r1428823);
        return r1428824;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1428825 = t;
        double r1428826 = l;
        double r1428827 = r1428825 / r1428826;
        double r1428828 = 5.6736910583871654e+57;
        bool r1428829 = r1428827 <= r1428828;
        double r1428830 = 1.0;
        double r1428831 = Om;
        double r1428832 = Omc;
        double r1428833 = r1428831 / r1428832;
        double r1428834 = r1428833 * r1428833;
        double r1428835 = r1428830 - r1428834;
        double r1428836 = sqrt(r1428835);
        double r1428837 = r1428827 * r1428827;
        double r1428838 = 2.0;
        double r1428839 = fma(r1428837, r1428838, r1428830);
        double r1428840 = sqrt(r1428839);
        double r1428841 = r1428836 / r1428840;
        double r1428842 = asin(r1428841);
        double r1428843 = sqrt(r1428838);
        double r1428844 = r1428825 * r1428843;
        double r1428845 = r1428844 / r1428826;
        double r1428846 = r1428836 / r1428845;
        double r1428847 = asin(r1428846);
        double r1428848 = r1428829 ? r1428842 : r1428847;
        return r1428848;
}

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) < 5.6736910583871654e+57

    1. Initial program 6.7

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

    2. Simplified6.7

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

    4. Applied sqrt-div6.7

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

    if 5.6736910583871654e+57 < (/ t l)

    1. Initial program 23.3

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

    2. Simplified23.3

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

    4. Applied sqrt-div23.3

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

    5. Taylor expanded around inf 0.9

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.6736910583871654 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +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)))))))