Average Error: 10.4 → 10.5
Time: 16.1s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}} \cdot \frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)\]
\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{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}} \cdot \frac{\sqrt[3]{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)
double f(double t, double l, double Om, double Omc) {
        double r51723 = 1.0;
        double r51724 = Om;
        double r51725 = Omc;
        double r51726 = r51724 / r51725;
        double r51727 = 2.0;
        double r51728 = pow(r51726, r51727);
        double r51729 = r51723 - r51728;
        double r51730 = t;
        double r51731 = l;
        double r51732 = r51730 / r51731;
        double r51733 = pow(r51732, r51727);
        double r51734 = r51727 * r51733;
        double r51735 = r51723 + r51734;
        double r51736 = r51729 / r51735;
        double r51737 = sqrt(r51736);
        double r51738 = asin(r51737);
        return r51738;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r51739 = 1.0;
        double r51740 = Om;
        double r51741 = Omc;
        double r51742 = r51740 / r51741;
        double r51743 = 2.0;
        double r51744 = pow(r51742, r51743);
        double r51745 = r51739 - r51744;
        double r51746 = cbrt(r51745);
        double r51747 = r51746 * r51746;
        double r51748 = t;
        double r51749 = l;
        double r51750 = r51748 / r51749;
        double r51751 = pow(r51750, r51743);
        double r51752 = fma(r51743, r51751, r51739);
        double r51753 = sqrt(r51752);
        double r51754 = r51747 / r51753;
        double r51755 = r51746 / r51753;
        double r51756 = r51754 * r51755;
        double r51757 = sqrt(r51756);
        double r51758 = asin(r51757);
        return r51758;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 10.4

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

  2. Simplified10.4

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

  4. Applied add-sqr-sqrt10.5

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

  5. Applied add-cube-cbrt10.5

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

  6. Applied times-frac10.5

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

  7. Final simplification10.5

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

Reproduce

herbie shell --seed 2019350 +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)))))))