Average Error: 10.1 → 10.2
Time: 24.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{\sqrt[3]{t}}{\sqrt[3]{\ell}} \cdot \frac{\frac{t}{\ell}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}\right) + 1}}\right)\]
\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{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{\sqrt[3]{t}}{\sqrt[3]{\ell}} \cdot \frac{\frac{t}{\ell}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}\right) + 1}}\right)
double f(double t, double l, double Om, double Omc) {
        double r2247761 = 1.0;
        double r2247762 = Om;
        double r2247763 = Omc;
        double r2247764 = r2247762 / r2247763;
        double r2247765 = 2.0;
        double r2247766 = pow(r2247764, r2247765);
        double r2247767 = r2247761 - r2247766;
        double r2247768 = t;
        double r2247769 = l;
        double r2247770 = r2247768 / r2247769;
        double r2247771 = pow(r2247770, r2247765);
        double r2247772 = r2247765 * r2247771;
        double r2247773 = r2247761 + r2247772;
        double r2247774 = r2247767 / r2247773;
        double r2247775 = sqrt(r2247774);
        double r2247776 = asin(r2247775);
        return r2247776;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2247777 = 1.0;
        double r2247778 = Om;
        double r2247779 = Omc;
        double r2247780 = r2247778 / r2247779;
        double r2247781 = r2247780 * r2247780;
        double r2247782 = r2247777 - r2247781;
        double r2247783 = t;
        double r2247784 = l;
        double r2247785 = r2247783 / r2247784;
        double r2247786 = r2247785 * r2247785;
        double r2247787 = cbrt(r2247783);
        double r2247788 = cbrt(r2247784);
        double r2247789 = r2247787 / r2247788;
        double r2247790 = r2247788 / r2247787;
        double r2247791 = r2247790 * r2247790;
        double r2247792 = r2247785 / r2247791;
        double r2247793 = r2247789 * r2247792;
        double r2247794 = r2247786 + r2247793;
        double r2247795 = r2247794 + r2247777;
        double r2247796 = r2247782 / r2247795;
        double r2247797 = sqrt(r2247796);
        double r2247798 = asin(r2247797);
        return r2247798;
}

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. Simplified10.1

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

  4. Applied add-cube-cbrt10.2

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

  5. Applied add-cube-cbrt10.2

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

  6. Applied times-frac10.2

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

  7. Applied associate-*r*10.2

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

  8. Simplified10.2

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

  9. Final simplification10.2

    \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{\sqrt[3]{t}}{\sqrt[3]{\ell}} \cdot \frac{\frac{t}{\ell}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{t}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{t}}}\right) + 1}}\right)\]

Reproduce

herbie shell --seed 2019164 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))