Average Error: 10.3 → 10.3
Time: 24.4s
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 r54293 = 1.0;
        double r54294 = Om;
        double r54295 = Omc;
        double r54296 = r54294 / r54295;
        double r54297 = 2.0;
        double r54298 = pow(r54296, r54297);
        double r54299 = r54293 - r54298;
        double r54300 = t;
        double r54301 = l;
        double r54302 = r54300 / r54301;
        double r54303 = pow(r54302, r54297);
        double r54304 = r54297 * r54303;
        double r54305 = r54293 + r54304;
        double r54306 = r54299 / r54305;
        double r54307 = sqrt(r54306);
        double r54308 = asin(r54307);
        return r54308;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r54309 = 1.0;
        double r54310 = Om;
        double r54311 = Omc;
        double r54312 = r54310 / r54311;
        double r54313 = 2.0;
        double r54314 = pow(r54312, r54313);
        double r54315 = r54309 - r54314;
        double r54316 = t;
        double r54317 = l;
        double r54318 = r54316 / r54317;
        double r54319 = pow(r54318, r54313);
        double r54320 = r54313 * r54319;
        double r54321 = r54309 + r54320;
        double r54322 = r54315 / r54321;
        double r54323 = sqrt(r54322);
        double r54324 = asin(r54323);
        return r54324;
}

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)))))))