Average Error: 10.4 → 10.5
Time: 23.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} + 1}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}} \cdot \frac{1 - \frac{Om}{Omc}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right)\]
\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}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}} \cdot \frac{1 - \frac{Om}{Omc}}{\sqrt{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r2179695 = 1.0;
        double r2179696 = Om;
        double r2179697 = Omc;
        double r2179698 = r2179696 / r2179697;
        double r2179699 = 2.0;
        double r2179700 = pow(r2179698, r2179699);
        double r2179701 = r2179695 - r2179700;
        double r2179702 = t;
        double r2179703 = l;
        double r2179704 = r2179702 / r2179703;
        double r2179705 = pow(r2179704, r2179699);
        double r2179706 = r2179699 * r2179705;
        double r2179707 = r2179695 + r2179706;
        double r2179708 = r2179701 / r2179707;
        double r2179709 = sqrt(r2179708);
        double r2179710 = asin(r2179709);
        return r2179710;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2179711 = Om;
        double r2179712 = Omc;
        double r2179713 = r2179711 / r2179712;
        double r2179714 = 1.0;
        double r2179715 = r2179713 + r2179714;
        double r2179716 = t;
        double r2179717 = l;
        double r2179718 = r2179716 / r2179717;
        double r2179719 = r2179718 * r2179718;
        double r2179720 = r2179719 + r2179719;
        double r2179721 = r2179714 + r2179720;
        double r2179722 = sqrt(r2179721);
        double r2179723 = r2179715 / r2179722;
        double r2179724 = r2179714 - r2179713;
        double r2179725 = r2179724 / r2179722;
        double r2179726 = r2179723 * r2179725;
        double r2179727 = sqrt(r2179726);
        double r2179728 = asin(r2179727);
        return r2179728;
}

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. Simplified10.4

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

  4. Applied add-sqr-sqrt10.4

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

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

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

  6. Applied difference-of-squares10.5

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

  7. Applied times-frac10.5

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

  8. Final simplification10.5

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

Reproduce

herbie shell --seed 2019163 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))