Average Error: 10.1 → 10.1
Time: 13.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\right)\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\right)\right)
double f(double t, double l, double Om, double Omc) {
        double r74431 = 1.0;
        double r74432 = Om;
        double r74433 = Omc;
        double r74434 = r74432 / r74433;
        double r74435 = 2.0;
        double r74436 = pow(r74434, r74435);
        double r74437 = r74431 - r74436;
        double r74438 = t;
        double r74439 = l;
        double r74440 = r74438 / r74439;
        double r74441 = pow(r74440, r74435);
        double r74442 = r74435 * r74441;
        double r74443 = r74431 + r74442;
        double r74444 = r74437 / r74443;
        double r74445 = sqrt(r74444);
        double r74446 = asin(r74445);
        return r74446;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r74447 = 1.0;
        double r74448 = 3.0;
        double r74449 = pow(r74447, r74448);
        double r74450 = Om;
        double r74451 = Omc;
        double r74452 = r74450 / r74451;
        double r74453 = 2.0;
        double r74454 = pow(r74452, r74453);
        double r74455 = pow(r74454, r74448);
        double r74456 = r74449 - r74455;
        double r74457 = r74454 + r74447;
        double r74458 = r74454 * r74457;
        double r74459 = r74447 * r74447;
        double r74460 = r74458 + r74459;
        double r74461 = t;
        double r74462 = l;
        double r74463 = r74461 / r74462;
        double r74464 = pow(r74463, r74453);
        double r74465 = r74453 * r74464;
        double r74466 = r74447 + r74465;
        double r74467 = r74460 * r74466;
        double r74468 = r74456 / r74467;
        double r74469 = sqrt(r74468);
        double r74470 = asin(r74469);
        double r74471 = log1p(r74470);
        double r74472 = expm1(r74471);
        return r74472;
}

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 10.1

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

  3. Applied expm1-log1p-u10.1

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)\right)}\]
  4. Using strategy rm

  5. Applied flip3--10.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{1 \cdot 1 + \left({\left(\frac{Om}{Omc}\right)}^{2} \cdot {\left(\frac{Om}{Omc}\right)}^{2} + 1 \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right)\right)\]

  6. Applied associate-/l/10.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\color{blue}{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right) \cdot \left(1 \cdot 1 + \left({\left(\frac{Om}{Omc}\right)}^{2} \cdot {\left(\frac{Om}{Omc}\right)}^{2} + 1 \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}}}\right)\right)\right)\]

  7. Simplified10.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\color{blue}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}}\right)\right)\right)\]

  8. Final simplification10.1

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\right)\right)\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(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)))))))