Average Error: 10.6 → 5.9
Time: 1.2m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9494802667865075 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.9494802667865075 \cdot 10^{+147}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\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 r3305772 = 1.0;
        double r3305773 = Om;
        double r3305774 = Omc;
        double r3305775 = r3305773 / r3305774;
        double r3305776 = 2.0;
        double r3305777 = pow(r3305775, r3305776);
        double r3305778 = r3305772 - r3305777;
        double r3305779 = t;
        double r3305780 = l;
        double r3305781 = r3305779 / r3305780;
        double r3305782 = pow(r3305781, r3305776);
        double r3305783 = r3305776 * r3305782;
        double r3305784 = r3305772 + r3305783;
        double r3305785 = r3305778 / r3305784;
        double r3305786 = sqrt(r3305785);
        double r3305787 = asin(r3305786);
        return r3305787;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r3305788 = t;
        double r3305789 = l;
        double r3305790 = r3305788 / r3305789;
        double r3305791 = 1.9494802667865075e+147;
        bool r3305792 = r3305790 <= r3305791;
        double r3305793 = 1.0;
        double r3305794 = Om;
        double r3305795 = Omc;
        double r3305796 = r3305794 / r3305795;
        double r3305797 = r3305796 * r3305796;
        double r3305798 = r3305793 - r3305797;
        double r3305799 = sqrt(r3305798);
        double r3305800 = r3305790 * r3305790;
        double r3305801 = 2.0;
        double r3305802 = fma(r3305800, r3305801, r3305793);
        double r3305803 = sqrt(r3305802);
        double r3305804 = r3305799 / r3305803;
        double r3305805 = asin(r3305804);
        double r3305806 = sqrt(r3305801);
        double r3305807 = r3305788 * r3305806;
        double r3305808 = r3305807 / r3305789;
        double r3305809 = r3305799 / r3305808;
        double r3305810 = asin(r3305809);
        double r3305811 = r3305792 ? r3305805 : r3305810;
        return r3305811;
}

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) < 1.9494802667865075e+147

    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

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

    4. Applied sqrt-div6.7

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

    if 1.9494802667865075e+147 < (/ t l)

    1. Initial program 33.5

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

    2. Simplified33.5

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

    4. Applied sqrt-div33.5

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

    5. Taylor expanded around -inf 1.0

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9494802667865075 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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)))))))