Average Error: 10.3 → 5.5
Time: 23.6s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 8.36205026708708 \cdot 10^{+127}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 8.36205026708708 \cdot 10^{+127}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \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 r2202566 = 1.0;
        double r2202567 = Om;
        double r2202568 = Omc;
        double r2202569 = r2202567 / r2202568;
        double r2202570 = 2.0;
        double r2202571 = pow(r2202569, r2202570);
        double r2202572 = r2202566 - r2202571;
        double r2202573 = t;
        double r2202574 = l;
        double r2202575 = r2202573 / r2202574;
        double r2202576 = pow(r2202575, r2202570);
        double r2202577 = r2202570 * r2202576;
        double r2202578 = r2202566 + r2202577;
        double r2202579 = r2202572 / r2202578;
        double r2202580 = sqrt(r2202579);
        double r2202581 = asin(r2202580);
        return r2202581;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2202582 = t;
        double r2202583 = l;
        double r2202584 = r2202582 / r2202583;
        double r2202585 = 8.36205026708708e+127;
        bool r2202586 = r2202584 <= r2202585;
        double r2202587 = 1.0;
        double r2202588 = Om;
        double r2202589 = Omc;
        double r2202590 = r2202588 / r2202589;
        double r2202591 = r2202590 * r2202590;
        double r2202592 = r2202587 - r2202591;
        double r2202593 = 2.0;
        double r2202594 = r2202584 * r2202584;
        double r2202595 = fma(r2202593, r2202594, r2202587);
        double r2202596 = r2202592 / r2202595;
        double r2202597 = sqrt(r2202596);
        double r2202598 = asin(r2202597);
        double r2202599 = sqrt(r2202592);
        double r2202600 = sqrt(r2202593);
        double r2202601 = r2202582 * r2202600;
        double r2202602 = r2202601 / r2202583;
        double r2202603 = r2202599 / r2202602;
        double r2202604 = asin(r2202603);
        double r2202605 = r2202586 ? r2202598 : r2202604;
        return r2202605;
}

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) < 8.36205026708708e+127

    1. Initial program 6.4

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

    2. Simplified6.4

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

    if 8.36205026708708e+127 < (/ t l)

    1. Initial program 30.0

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

    2. Simplified30.0

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

    4. Applied sqrt-div30.0

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

    5. Taylor expanded around inf 1.3

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 8.36205026708708 \cdot 10^{+127}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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)))))))