Average Error: 10.4 → 10.4
Time: 10.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)\right)
double f(double t, double l, double Om, double Omc) {
        double r77845 = 1.0;
        double r77846 = Om;
        double r77847 = Omc;
        double r77848 = r77846 / r77847;
        double r77849 = 2.0;
        double r77850 = pow(r77848, r77849);
        double r77851 = r77845 - r77850;
        double r77852 = t;
        double r77853 = l;
        double r77854 = r77852 / r77853;
        double r77855 = pow(r77854, r77849);
        double r77856 = r77849 * r77855;
        double r77857 = r77845 + r77856;
        double r77858 = r77851 / r77857;
        double r77859 = sqrt(r77858);
        double r77860 = asin(r77859);
        return r77860;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r77861 = 1.0;
        double r77862 = Om;
        double r77863 = Omc;
        double r77864 = r77862 / r77863;
        double r77865 = 2.0;
        double r77866 = pow(r77864, r77865);
        double r77867 = r77861 - r77866;
        double r77868 = t;
        double r77869 = l;
        double r77870 = r77868 / r77869;
        double r77871 = pow(r77870, r77865);
        double r77872 = r77865 * r77871;
        double r77873 = r77861 + r77872;
        double r77874 = r77867 / r77873;
        double r77875 = sqrt(r77874);
        double r77876 = asin(r77875);
        double r77877 = log1p(r77876);
        double r77878 = expm1(r77877);
        return r77878;
}

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. Initial program 10.4

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

  3. Applied expm1-log1p-u10.4

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)\right)}\]

  4. Final simplification10.4

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))