Average Error: 9.9 → 10.1
Time: 44.9s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\left(\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right) \cdot \frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot \left({\left(\sqrt{\frac{t}{\ell}}\right)}^{2} \cdot {\left(\sqrt{\frac{t}{\ell}}\right)}^{2}\right)}}\right)\]
\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{\left(\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right) \cdot \frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot \left({\left(\sqrt{\frac{t}{\ell}}\right)}^{2} \cdot {\left(\sqrt{\frac{t}{\ell}}\right)}^{2}\right)}}\right)
double f(double t, double l, double Om, double Omc) {
        double r72629 = 1.0;
        double r72630 = Om;
        double r72631 = Omc;
        double r72632 = r72630 / r72631;
        double r72633 = 2.0;
        double r72634 = pow(r72632, r72633);
        double r72635 = r72629 - r72634;
        double r72636 = t;
        double r72637 = l;
        double r72638 = r72636 / r72637;
        double r72639 = pow(r72638, r72633);
        double r72640 = r72633 * r72639;
        double r72641 = r72629 + r72640;
        double r72642 = r72635 / r72641;
        double r72643 = sqrt(r72642);
        double r72644 = asin(r72643);
        return r72644;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r72645 = 1.0;
        double r72646 = Om;
        double r72647 = Omc;
        double r72648 = r72646 / r72647;
        double r72649 = 2.0;
        double r72650 = pow(r72648, r72649);
        double r72651 = r72645 - r72650;
        double r72652 = cbrt(r72651);
        double r72653 = r72652 * r72652;
        double r72654 = t;
        double r72655 = l;
        double r72656 = r72654 / r72655;
        double r72657 = sqrt(r72656);
        double r72658 = pow(r72657, r72649);
        double r72659 = r72658 * r72658;
        double r72660 = r72649 * r72659;
        double r72661 = r72645 + r72660;
        double r72662 = r72652 / r72661;
        double r72663 = r72653 * r72662;
        double r72664 = sqrt(r72663);
        double r72665 = asin(r72664);
        return r72665;
}

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 9.9

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

  3. Applied *-un-lft-identity9.9

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

  4. Applied add-cube-cbrt10.0

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

  5. Applied times-frac10.0

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

  6. Simplified10.0

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

  8. Applied add-sqr-sqrt10.1

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

  9. Applied unpow-prod-down10.1

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

  10. Final simplification10.1

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

Reproduce

herbie shell --seed 2020046 +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)))))))