Average Error: 10.6 → 6.0
Time: 1.2m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 4.3448374903793984 \cdot 10^{+86}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 4.3448374903793984 \cdot 10^{+86}:\\
\;\;\;\;\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 r3964803 = 1.0;
        double r3964804 = Om;
        double r3964805 = Omc;
        double r3964806 = r3964804 / r3964805;
        double r3964807 = 2.0;
        double r3964808 = pow(r3964806, r3964807);
        double r3964809 = r3964803 - r3964808;
        double r3964810 = t;
        double r3964811 = l;
        double r3964812 = r3964810 / r3964811;
        double r3964813 = pow(r3964812, r3964807);
        double r3964814 = r3964807 * r3964813;
        double r3964815 = r3964803 + r3964814;
        double r3964816 = r3964809 / r3964815;
        double r3964817 = sqrt(r3964816);
        double r3964818 = asin(r3964817);
        return r3964818;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r3964819 = t;
        double r3964820 = l;
        double r3964821 = r3964819 / r3964820;
        double r3964822 = 4.3448374903793984e+86;
        bool r3964823 = r3964821 <= r3964822;
        double r3964824 = 1.0;
        double r3964825 = Om;
        double r3964826 = Omc;
        double r3964827 = r3964825 / r3964826;
        double r3964828 = r3964827 * r3964827;
        double r3964829 = r3964824 - r3964828;
        double r3964830 = sqrt(r3964829);
        double r3964831 = r3964821 * r3964821;
        double r3964832 = 2.0;
        double r3964833 = fma(r3964831, r3964832, r3964824);
        double r3964834 = sqrt(r3964833);
        double r3964835 = r3964830 / r3964834;
        double r3964836 = asin(r3964835);
        double r3964837 = sqrt(r3964832);
        double r3964838 = r3964819 * r3964837;
        double r3964839 = r3964838 / r3964820;
        double r3964840 = r3964830 / r3964839;
        double r3964841 = asin(r3964840);
        double r3964842 = r3964823 ? r3964836 : r3964841;
        return r3964842;
}

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) < 4.3448374903793984e+86

    1. Initial program 7.0

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

    2. Simplified7.0

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

    4. Applied sqrt-div7.0

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

    if 4.3448374903793984e+86 < (/ t l)

    1. Initial program 26.8

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

    2. Simplified26.8

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

    4. Applied sqrt-div26.8

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

    5. Taylor expanded around -inf 1.4

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

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 4.3448374903793984 \cdot 10^{+86}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 +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)))))))