Average Error: 10.7 → 10.7
Time: 41.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} + 1}{\frac{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}{1 - \frac{Om}{Omc}}}}\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} + 1}{\frac{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2}{1 - \frac{Om}{Omc}}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r2062814 = 1.0;
        double r2062815 = Om;
        double r2062816 = Omc;
        double r2062817 = r2062815 / r2062816;
        double r2062818 = 2.0;
        double r2062819 = pow(r2062817, r2062818);
        double r2062820 = r2062814 - r2062819;
        double r2062821 = t;
        double r2062822 = l;
        double r2062823 = r2062821 / r2062822;
        double r2062824 = pow(r2062823, r2062818);
        double r2062825 = r2062818 * r2062824;
        double r2062826 = r2062814 + r2062825;
        double r2062827 = r2062820 / r2062826;
        double r2062828 = sqrt(r2062827);
        double r2062829 = asin(r2062828);
        return r2062829;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2062830 = Om;
        double r2062831 = Omc;
        double r2062832 = r2062830 / r2062831;
        double r2062833 = 1.0;
        double r2062834 = r2062832 + r2062833;
        double r2062835 = t;
        double r2062836 = l;
        double r2062837 = r2062835 / r2062836;
        double r2062838 = r2062837 * r2062837;
        double r2062839 = 2.0;
        double r2062840 = r2062838 * r2062839;
        double r2062841 = r2062833 + r2062840;
        double r2062842 = r2062833 - r2062832;
        double r2062843 = r2062841 / r2062842;
        double r2062844 = r2062834 / r2062843;
        double r2062845 = sqrt(r2062844);
        double r2062846 = asin(r2062845);
        return r2062846;
}

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 10.7

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

  2. Simplified10.7

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

  4. Applied *-un-lft-identity10.7

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

  5. Applied difference-of-squares10.7

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

  6. Applied associate-/l*10.7

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

  7. Final simplification10.7

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

Reproduce

herbie shell --seed 2019151 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))