Average Error: 10.4 → 10.5
Time: 26.1s
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 r2179692 = 1.0;
        double r2179693 = Om;
        double r2179694 = Omc;
        double r2179695 = r2179693 / r2179694;
        double r2179696 = 2.0;
        double r2179697 = pow(r2179695, r2179696);
        double r2179698 = r2179692 - r2179697;
        double r2179699 = t;
        double r2179700 = l;
        double r2179701 = r2179699 / r2179700;
        double r2179702 = pow(r2179701, r2179696);
        double r2179703 = r2179696 * r2179702;
        double r2179704 = r2179692 + r2179703;
        double r2179705 = r2179698 / r2179704;
        double r2179706 = sqrt(r2179705);
        double r2179707 = asin(r2179706);
        return r2179707;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2179708 = Om;
        double r2179709 = Omc;
        double r2179710 = r2179708 / r2179709;
        double r2179711 = 1.0;
        double r2179712 = r2179710 + r2179711;
        double r2179713 = t;
        double r2179714 = l;
        double r2179715 = r2179713 / r2179714;
        double r2179716 = r2179715 * r2179715;
        double r2179717 = r2179716 + r2179716;
        double r2179718 = r2179711 + r2179717;
        double r2179719 = sqrt(r2179718);
        double r2179720 = r2179712 / r2179719;
        double r2179721 = r2179711 - r2179710;
        double r2179722 = r2179721 / r2179719;
        double r2179723 = r2179720 * r2179722;
        double r2179724 = sqrt(r2179723);
        double r2179725 = asin(r2179724);
        return r2179725;
}

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)))))))