Average Error: 10.5 → 5.7
Time: 1.6m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.528201376140965 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 7.528201376140965 \cdot 10^{+69}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\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 r3826923 = 1.0;
        double r3826924 = Om;
        double r3826925 = Omc;
        double r3826926 = r3826924 / r3826925;
        double r3826927 = 2.0;
        double r3826928 = pow(r3826926, r3826927);
        double r3826929 = r3826923 - r3826928;
        double r3826930 = t;
        double r3826931 = l;
        double r3826932 = r3826930 / r3826931;
        double r3826933 = pow(r3826932, r3826927);
        double r3826934 = r3826927 * r3826933;
        double r3826935 = r3826923 + r3826934;
        double r3826936 = r3826929 / r3826935;
        double r3826937 = sqrt(r3826936);
        double r3826938 = asin(r3826937);
        return r3826938;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r3826939 = t;
        double r3826940 = l;
        double r3826941 = r3826939 / r3826940;
        double r3826942 = 7.528201376140965e+69;
        bool r3826943 = r3826941 <= r3826942;
        double r3826944 = 1.0;
        double r3826945 = Om;
        double r3826946 = Omc;
        double r3826947 = r3826945 / r3826946;
        double r3826948 = r3826947 * r3826947;
        double r3826949 = r3826944 - r3826948;
        double r3826950 = 2.0;
        double r3826951 = r3826940 / r3826939;
        double r3826952 = r3826951 * r3826951;
        double r3826953 = r3826950 / r3826952;
        double r3826954 = r3826944 + r3826953;
        double r3826955 = r3826949 / r3826954;
        double r3826956 = sqrt(r3826955);
        double r3826957 = asin(r3826956);
        double r3826958 = sqrt(r3826949);
        double r3826959 = sqrt(r3826950);
        double r3826960 = r3826939 * r3826959;
        double r3826961 = r3826960 / r3826940;
        double r3826962 = r3826958 / r3826961;
        double r3826963 = asin(r3826962);
        double r3826964 = r3826943 ? r3826957 : r3826963;
        return r3826964;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 7.528201376140965e+69

    1. Initial program 6.8

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

    2. Simplified6.8

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

    3. Taylor expanded around 0 23.0

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

    4. Simplified6.8

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

    if 7.528201376140965e+69 < (/ t l)

    1. Initial program 25.8

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

    2. Simplified25.8

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

    3. Taylor expanded around 0 30.3

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

    4. Simplified25.8

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

    6. Applied sqrt-div25.8

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

    7. 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 simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.528201376140965 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))