Average Error: 10.0 → 10.0
Time: 11.7s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 \cdot 1 - {\left(\frac{Om}{Omc}\right)}^{2} \cdot {\left(\frac{Om}{Omc}\right)}^{2}}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right)\]
\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{1 \cdot 1 - {\left(\frac{Om}{Omc}\right)}^{2} \cdot {\left(\frac{Om}{Omc}\right)}^{2}}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right) \cdot \left(1 + {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right)
double f(double t, double l, double Om, double Omc) {
        double r78118 = 1.0;
        double r78119 = Om;
        double r78120 = Omc;
        double r78121 = r78119 / r78120;
        double r78122 = 2.0;
        double r78123 = pow(r78121, r78122);
        double r78124 = r78118 - r78123;
        double r78125 = t;
        double r78126 = l;
        double r78127 = r78125 / r78126;
        double r78128 = pow(r78127, r78122);
        double r78129 = r78122 * r78128;
        double r78130 = r78118 + r78129;
        double r78131 = r78124 / r78130;
        double r78132 = sqrt(r78131);
        double r78133 = asin(r78132);
        return r78133;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r78134 = 1.0;
        double r78135 = r78134 * r78134;
        double r78136 = Om;
        double r78137 = Omc;
        double r78138 = r78136 / r78137;
        double r78139 = 2.0;
        double r78140 = pow(r78138, r78139);
        double r78141 = r78140 * r78140;
        double r78142 = r78135 - r78141;
        double r78143 = t;
        double r78144 = l;
        double r78145 = r78143 / r78144;
        double r78146 = pow(r78145, r78139);
        double r78147 = r78139 * r78146;
        double r78148 = r78134 + r78147;
        double r78149 = r78134 + r78140;
        double r78150 = r78148 * r78149;
        double r78151 = r78142 / r78150;
        double r78152 = sqrt(r78151);
        double r78153 = asin(r78152);
        return r78153;
}

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 10.0

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

  3. Applied flip--10.0

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

  4. Applied associate-/l/10.0

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

  5. Final simplification10.0

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

Reproduce

herbie shell --seed 2019352 
(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)))))))