Average Error: 10.9 → 10.9
Time: 14.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\right)\right)\]
\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 r87520 = 1.0;
        double r87521 = Om;
        double r87522 = Omc;
        double r87523 = r87521 / r87522;
        double r87524 = 2.0;
        double r87525 = pow(r87523, r87524);
        double r87526 = r87520 - r87525;
        double r87527 = t;
        double r87528 = l;
        double r87529 = r87527 / r87528;
        double r87530 = pow(r87529, r87524);
        double r87531 = r87524 * r87530;
        double r87532 = r87520 + r87531;
        double r87533 = r87526 / r87532;
        double r87534 = sqrt(r87533);
        double r87535 = asin(r87534);
        return r87535;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r87536 = 1.0;
        double r87537 = 3.0;
        double r87538 = pow(r87536, r87537);
        double r87539 = Om;
        double r87540 = Omc;
        double r87541 = r87539 / r87540;
        double r87542 = 2.0;
        double r87543 = pow(r87541, r87542);
        double r87544 = pow(r87543, r87537);
        double r87545 = r87538 - r87544;
        double r87546 = r87543 + r87536;
        double r87547 = r87543 * r87546;
        double r87548 = r87536 * r87536;
        double r87549 = r87547 + r87548;
        double r87550 = t;
        double r87551 = l;
        double r87552 = r87550 / r87551;
        double r87553 = pow(r87552, r87542);
        double r87554 = r87542 * r87553;
        double r87555 = r87536 + r87554;
        double r87556 = r87549 * r87555;
        double r87557 = r87545 / r87556;
        double r87558 = sqrt(r87557);
        double r87559 = asin(r87558);
        double r87560 = log1p(r87559);
        double r87561 = expm1(r87560);
        return r87561;
}

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 10.9

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

  3. Applied flip3--10.9

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

  4. Applied associate-/l/10.9

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

  5. Simplified10.9

    \[\leadsto \sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\color{blue}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}}\right)\]
  6. Using strategy rm

  7. Applied expm1-log1p-u10.9

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

  8. Final simplification10.9

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

Reproduce

herbie shell --seed 2020057 +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)))))))