Average Error: 10.2 → 5.4
Time: 16.0s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9934801738260375 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2 \cdot \frac{t}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{t}{\ell} \le 1.9934801738260375 \cdot 10^{+150}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2 \cdot \frac{t}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\

\end{array}
double f(double t, double l, double Om, double Omc) {
        double r937641 = 1.0;
        double r937642 = Om;
        double r937643 = Omc;
        double r937644 = r937642 / r937643;
        double r937645 = 2.0;
        double r937646 = pow(r937644, r937645);
        double r937647 = r937641 - r937646;
        double r937648 = t;
        double r937649 = l;
        double r937650 = r937648 / r937649;
        double r937651 = pow(r937650, r937645);
        double r937652 = r937645 * r937651;
        double r937653 = r937641 + r937652;
        double r937654 = r937647 / r937653;
        double r937655 = sqrt(r937654);
        double r937656 = asin(r937655);
        return r937656;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r937657 = t;
        double r937658 = l;
        double r937659 = r937657 / r937658;
        double r937660 = 1.9934801738260375e+150;
        bool r937661 = r937659 <= r937660;
        double r937662 = 1.0;
        double r937663 = Om;
        double r937664 = Omc;
        double r937665 = r937663 / r937664;
        double r937666 = r937665 * r937665;
        double r937667 = r937662 - r937666;
        double r937668 = 2.0;
        double r937669 = r937668 * r937659;
        double r937670 = fma(r937669, r937659, r937662);
        double r937671 = r937667 / r937670;
        double r937672 = sqrt(r937671);
        double r937673 = asin(r937672);
        double r937674 = sqrt(r937667);
        double r937675 = sqrt(r937668);
        double r937676 = r937657 * r937675;
        double r937677 = r937676 / r937658;
        double r937678 = r937674 / r937677;
        double r937679 = asin(r937678);
        double r937680 = r937661 ? r937673 : r937679;
        return r937680;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 1.9934801738260375e+150

    1. Initial program 6.1

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

    2. Simplified6.1

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

    if 1.9934801738260375e+150 < (/ t l)

    1. Initial program 33.7

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

    2. Simplified33.7

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

    4. Applied sqrt-div33.7

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

    5. Taylor expanded around inf 1.2

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.9934801738260375 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2 \cdot \frac{t}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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)))))))