Average Error: 9.9 → 10.0
Time: 12.5s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)}\right)\right)}\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\sin^{-1} \left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)}\right)\right)}\right)
double f(double t, double l, double Om, double Omc) {
        double r84721 = 1.0;
        double r84722 = Om;
        double r84723 = Omc;
        double r84724 = r84722 / r84723;
        double r84725 = 2.0;
        double r84726 = pow(r84724, r84725);
        double r84727 = r84721 - r84726;
        double r84728 = t;
        double r84729 = l;
        double r84730 = r84728 / r84729;
        double r84731 = pow(r84730, r84725);
        double r84732 = r84725 * r84731;
        double r84733 = r84721 + r84732;
        double r84734 = r84727 / r84733;
        double r84735 = sqrt(r84734);
        double r84736 = asin(r84735);
        return r84736;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r84737 = 1.0;
        double r84738 = Om;
        double r84739 = Omc;
        double r84740 = r84738 / r84739;
        double r84741 = 2.0;
        double r84742 = pow(r84740, r84741);
        double r84743 = r84737 - r84742;
        double r84744 = sqrt(r84743);
        double r84745 = t;
        double r84746 = l;
        double r84747 = r84745 / r84746;
        double r84748 = pow(r84747, r84741);
        double r84749 = fma(r84748, r84741, r84737);
        double r84750 = r84744 / r84749;
        double r84751 = r84744 * r84750;
        double r84752 = log1p(r84751);
        double r84753 = expm1(r84752);
        double r84754 = sqrt(r84753);
        double r84755 = asin(r84754);
        return r84755;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

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 expm1-log1p-u10.0

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

  5. Applied *-un-lft-identity10.0

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

  6. Applied add-sqr-sqrt10.0

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

  7. Applied times-frac10.0

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

  8. Simplified10.0

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

  9. Simplified10.0

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

  10. Final simplification10.0

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

Reproduce

herbie shell --seed 2020083 +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)))))))