Average Error: 10.3 → 10.3
Time: 25.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\]
\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{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)
double f(double t, double l, double Om, double Omc) {
        double r2559318 = 1.0;
        double r2559319 = Om;
        double r2559320 = Omc;
        double r2559321 = r2559319 / r2559320;
        double r2559322 = 2.0;
        double r2559323 = pow(r2559321, r2559322);
        double r2559324 = r2559318 - r2559323;
        double r2559325 = t;
        double r2559326 = l;
        double r2559327 = r2559325 / r2559326;
        double r2559328 = pow(r2559327, r2559322);
        double r2559329 = r2559322 * r2559328;
        double r2559330 = r2559318 + r2559329;
        double r2559331 = r2559324 / r2559330;
        double r2559332 = sqrt(r2559331);
        double r2559333 = asin(r2559332);
        return r2559333;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2559334 = 1.0;
        double r2559335 = Om;
        double r2559336 = Omc;
        double r2559337 = r2559335 / r2559336;
        double r2559338 = r2559337 * r2559337;
        double r2559339 = r2559334 - r2559338;
        double r2559340 = t;
        double r2559341 = l;
        double r2559342 = r2559340 / r2559341;
        double r2559343 = r2559342 * r2559342;
        double r2559344 = r2559343 + r2559343;
        double r2559345 = r2559334 + r2559344;
        double r2559346 = r2559339 / r2559345;
        double r2559347 = sqrt(r2559346);
        double r2559348 = asin(r2559347);
        return r2559348;
}

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. Simplified10.3

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

  3. Final simplification10.3

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

Reproduce

herbie shell --seed 2019162 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))