Average Error: 10.3 → 10.3
Time: 23.5s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
\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 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r54256 = 1.0;
        double r54257 = Om;
        double r54258 = Omc;
        double r54259 = r54257 / r54258;
        double r54260 = 2.0;
        double r54261 = pow(r54259, r54260);
        double r54262 = r54256 - r54261;
        double r54263 = t;
        double r54264 = l;
        double r54265 = r54263 / r54264;
        double r54266 = pow(r54265, r54260);
        double r54267 = r54260 * r54266;
        double r54268 = r54256 + r54267;
        double r54269 = r54262 / r54268;
        double r54270 = sqrt(r54269);
        double r54271 = asin(r54270);
        return r54271;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r54272 = 1.0;
        double r54273 = Om;
        double r54274 = Omc;
        double r54275 = r54273 / r54274;
        double r54276 = 2.0;
        double r54277 = pow(r54275, r54276);
        double r54278 = r54272 - r54277;
        double r54279 = t;
        double r54280 = l;
        double r54281 = r54279 / r54280;
        double r54282 = pow(r54281, r54276);
        double r54283 = r54276 * r54282;
        double r54284 = r54272 + r54283;
        double r54285 = r54278 / r54284;
        double r54286 = sqrt(r54285);
        double r54287 = asin(r54286);
        return r54287;
}

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 10.3

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

  2. Final simplification10.3

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

Reproduce

herbie shell --seed 2019326 
(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)))))))