Average Error: 10.7 → 10.7
Time: 22.9s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{Omc}}{\mathsf{fma}\left(2 \cdot \frac{t}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\right)\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{Omc}}{\mathsf{fma}\left(2 \cdot \frac{t}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\right)\right)
double f(double t, double l, double Om, double Omc) {
        double r1005745 = 1.0;
        double r1005746 = Om;
        double r1005747 = Omc;
        double r1005748 = r1005746 / r1005747;
        double r1005749 = 2.0;
        double r1005750 = pow(r1005748, r1005749);
        double r1005751 = r1005745 - r1005750;
        double r1005752 = t;
        double r1005753 = l;
        double r1005754 = r1005752 / r1005753;
        double r1005755 = pow(r1005754, r1005749);
        double r1005756 = r1005749 * r1005755;
        double r1005757 = r1005745 + r1005756;
        double r1005758 = r1005751 / r1005757;
        double r1005759 = sqrt(r1005758);
        double r1005760 = asin(r1005759);
        return r1005760;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1005761 = 1.0;
        double r1005762 = Om;
        double r1005763 = Omc;
        double r1005764 = r1005762 / r1005763;
        double r1005765 = r1005761 + r1005764;
        double r1005766 = r1005761 - r1005764;
        double r1005767 = 2.0;
        double r1005768 = t;
        double r1005769 = l;
        double r1005770 = r1005768 / r1005769;
        double r1005771 = r1005767 * r1005770;
        double r1005772 = fma(r1005771, r1005770, r1005761);
        double r1005773 = r1005766 / r1005772;
        double r1005774 = r1005765 * r1005773;
        double r1005775 = sqrt(r1005774);
        double r1005776 = asin(r1005775);
        double r1005777 = log1p(r1005776);
        double r1005778 = expm1(r1005777);
        return r1005778;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 10.7

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

  2. Simplified10.7

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

  4. Applied expm1-log1p-u10.7

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

  6. Applied *-un-lft-identity10.7

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

  7. Applied *-un-lft-identity10.7

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

  8. Applied difference-of-squares10.7

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

  9. Applied times-frac10.7

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

  10. Simplified10.7

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

  11. Final simplification10.7

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))