Average Error: 10.1 → 10.1
Time: 9.0s
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 r64909 = 1.0;
        double r64910 = Om;
        double r64911 = Omc;
        double r64912 = r64910 / r64911;
        double r64913 = 2.0;
        double r64914 = pow(r64912, r64913);
        double r64915 = r64909 - r64914;
        double r64916 = t;
        double r64917 = l;
        double r64918 = r64916 / r64917;
        double r64919 = pow(r64918, r64913);
        double r64920 = r64913 * r64919;
        double r64921 = r64909 + r64920;
        double r64922 = r64915 / r64921;
        double r64923 = sqrt(r64922);
        double r64924 = asin(r64923);
        return r64924;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r64925 = 1.0;
        double r64926 = Om;
        double r64927 = Omc;
        double r64928 = r64926 / r64927;
        double r64929 = 2.0;
        double r64930 = pow(r64928, r64929);
        double r64931 = r64925 - r64930;
        double r64932 = t;
        double r64933 = l;
        double r64934 = r64932 / r64933;
        double r64935 = pow(r64934, r64929);
        double r64936 = r64929 * r64935;
        double r64937 = r64925 + r64936;
        double r64938 = r64931 / r64937;
        double r64939 = sqrt(r64938);
        double r64940 = asin(r64939);
        double r64941 = log1p(r64940);
        double r64942 = expm1(r64941);
        return r64942;
}

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 10.1

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

  3. Applied expm1-log1p-u10.1

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

  4. Final simplification10.1

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

Reproduce

herbie shell --seed 2020089 +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)))))))