Average Error: 10.3 → 10.3
Time: 27.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{t}\right)\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) + \frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)\]
\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(\left(\sqrt[3]{\frac{t}{\ell}} \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{t}\right)\right) \cdot \left(\sqrt[3]{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right) + \frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1}}\right)
double f(double t, double l, double Om, double Omc) {
        double r2560428 = 1.0;
        double r2560429 = Om;
        double r2560430 = Omc;
        double r2560431 = r2560429 / r2560430;
        double r2560432 = 2.0;
        double r2560433 = pow(r2560431, r2560432);
        double r2560434 = r2560428 - r2560433;
        double r2560435 = t;
        double r2560436 = l;
        double r2560437 = r2560435 / r2560436;
        double r2560438 = pow(r2560437, r2560432);
        double r2560439 = r2560432 * r2560438;
        double r2560440 = r2560428 + r2560439;
        double r2560441 = r2560434 / r2560440;
        double r2560442 = sqrt(r2560441);
        double r2560443 = asin(r2560442);
        return r2560443;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2560444 = 1.0;
        double r2560445 = Om;
        double r2560446 = Omc;
        double r2560447 = r2560445 / r2560446;
        double r2560448 = r2560447 * r2560447;
        double r2560449 = r2560444 - r2560448;
        double r2560450 = t;
        double r2560451 = l;
        double r2560452 = r2560450 / r2560451;
        double r2560453 = cbrt(r2560452);
        double r2560454 = r2560444 / r2560451;
        double r2560455 = cbrt(r2560454);
        double r2560456 = cbrt(r2560450);
        double r2560457 = r2560455 * r2560456;
        double r2560458 = r2560453 * r2560457;
        double r2560459 = r2560453 * r2560452;
        double r2560460 = r2560458 * r2560459;
        double r2560461 = r2560452 * r2560452;
        double r2560462 = r2560460 + r2560461;
        double r2560463 = r2560462 + r2560444;
        double r2560464 = r2560449 / r2560463;
        double r2560465 = sqrt(r2560464);
        double r2560466 = asin(r2560465);
        return r2560466;
}

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. Using strategy rm

  4. Applied add-cube-cbrt10.3

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

  5. Applied associate-*l*10.3

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

  7. Applied div-inv10.3

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

  8. Applied cbrt-prod10.3

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

  9. Final simplification10.3

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

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)))))))