Average Error: 10.4 → 10.5
Time: 20.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\sqrt{1} + \sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{\frac{-1}{\ell}}{-1} \cdot t\right)}^{2}, 1\right)}} \cdot \frac{\sqrt{1} - \sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}}\right)\]
\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{1} + \sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{\frac{-1}{\ell}}{-1} \cdot t\right)}^{2}, 1\right)}} \cdot \frac{\sqrt{1} - \sqrt{{\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 r56682 = 1.0;
        double r56683 = Om;
        double r56684 = Omc;
        double r56685 = r56683 / r56684;
        double r56686 = 2.0;
        double r56687 = pow(r56685, r56686);
        double r56688 = r56682 - r56687;
        double r56689 = t;
        double r56690 = l;
        double r56691 = r56689 / r56690;
        double r56692 = pow(r56691, r56686);
        double r56693 = r56686 * r56692;
        double r56694 = r56682 + r56693;
        double r56695 = r56688 / r56694;
        double r56696 = sqrt(r56695);
        double r56697 = asin(r56696);
        return r56697;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r56698 = 1.0;
        double r56699 = sqrt(r56698);
        double r56700 = Om;
        double r56701 = Omc;
        double r56702 = r56700 / r56701;
        double r56703 = 2.0;
        double r56704 = pow(r56702, r56703);
        double r56705 = sqrt(r56704);
        double r56706 = r56699 + r56705;
        double r56707 = -1.0;
        double r56708 = l;
        double r56709 = r56707 / r56708;
        double r56710 = r56709 / r56707;
        double r56711 = t;
        double r56712 = r56710 * r56711;
        double r56713 = pow(r56712, r56703);
        double r56714 = fma(r56703, r56713, r56698);
        double r56715 = sqrt(r56714);
        double r56716 = r56706 / r56715;
        double r56717 = r56699 - r56705;
        double r56718 = r56711 / r56708;
        double r56719 = pow(r56718, r56703);
        double r56720 = fma(r56703, r56719, r56698);
        double r56721 = sqrt(r56720);
        double r56722 = r56717 / r56721;
        double r56723 = r56716 * r56722;
        double r56724 = sqrt(r56723);
        double r56725 = asin(r56724);
        return r56725;
}

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-sqrt10.4

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

  5. Applied add-sqr-sqrt10.4

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

  6. Applied add-sqr-sqrt10.4

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

  7. Applied difference-of-squares10.5

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

  8. Applied times-frac10.5

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

  9. Taylor expanded around -inf 50.9

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

  10. Simplified10.5

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

  11. Final simplification10.5

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))