Average Error: 10.3 → 1.0
Time: 17.4s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -1.6785619795363593 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{elif}\;\frac{t}{\ell} \le 6.140685024486043 \cdot 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\mathsf{fma}\left(\frac{t}{\ell} \cdot 2, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \end{array}\]
\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 -1.6785619795363593 \cdot 10^{+152}:\\
\;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\

\mathbf{elif}\;\frac{t}{\ell} \le 6.140685024486043 \cdot 10^{+118}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\mathsf{fma}\left(\frac{t}{\ell} \cdot 2, \frac{t}{\ell}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\

\end{array}
double f(double t, double l, double Om, double Omc) {
        double r931783 = 1.0;
        double r931784 = Om;
        double r931785 = Omc;
        double r931786 = r931784 / r931785;
        double r931787 = 2.0;
        double r931788 = pow(r931786, r931787);
        double r931789 = r931783 - r931788;
        double r931790 = t;
        double r931791 = l;
        double r931792 = r931790 / r931791;
        double r931793 = pow(r931792, r931787);
        double r931794 = r931787 * r931793;
        double r931795 = r931783 + r931794;
        double r931796 = r931789 / r931795;
        double r931797 = sqrt(r931796);
        double r931798 = asin(r931797);
        return r931798;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r931799 = t;
        double r931800 = l;
        double r931801 = r931799 / r931800;
        double r931802 = -1.6785619795363593e+152;
        bool r931803 = r931801 <= r931802;
        double r931804 = 1.0;
        double r931805 = Om;
        double r931806 = Omc;
        double r931807 = r931805 / r931806;
        double r931808 = r931807 * r931807;
        double r931809 = exp(r931808);
        double r931810 = log(r931809);
        double r931811 = r931804 - r931810;
        double r931812 = sqrt(r931811);
        double r931813 = 2.0;
        double r931814 = sqrt(r931813);
        double r931815 = r931814 * r931799;
        double r931816 = r931815 / r931800;
        double r931817 = r931812 / r931816;
        double r931818 = fabs(r931817);
        double r931819 = asin(r931818);
        double r931820 = 6.140685024486043e+118;
        bool r931821 = r931801 <= r931820;
        double r931822 = r931801 * r931813;
        double r931823 = fma(r931822, r931801, r931804);
        double r931824 = r931811 / r931823;
        double r931825 = sqrt(r931824);
        double r931826 = asin(r931825);
        double r931827 = r931821 ? r931826 : r931819;
        double r931828 = r931803 ? r931819 : r931827;
        return r931828;
}

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) < -1.6785619795363593e+152 or 6.140685024486043e+118 < (/ 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

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

    4. Applied add-log-exp32.2

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

    6. Applied add-sqr-sqrt32.2

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

    7. Applied add-sqr-sqrt32.2

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

    8. Applied times-frac32.2

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

    9. Applied rem-sqrt-square32.2

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

    10. Taylor expanded around inf 1.2

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

    if -1.6785619795363593e+152 < (/ t l) < 6.140685024486043e+118

    1. Initial program 1.0

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

    2. Simplified1.0

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

    4. Applied add-log-exp1.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -1.6785619795363593 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{elif}\;\frac{t}{\ell} \le 6.140685024486043 \cdot 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{\mathsf{fma}\left(\frac{t}{\ell} \cdot 2, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +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)))))))