Average Error: 10.2 → 5.6
Time: 29.9s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.410996405697987 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 2.410996405697987 \cdot 10^{+143}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 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 r1674455 = 1.0;
        double r1674456 = Om;
        double r1674457 = Omc;
        double r1674458 = r1674456 / r1674457;
        double r1674459 = 2.0;
        double r1674460 = pow(r1674458, r1674459);
        double r1674461 = r1674455 - r1674460;
        double r1674462 = t;
        double r1674463 = l;
        double r1674464 = r1674462 / r1674463;
        double r1674465 = pow(r1674464, r1674459);
        double r1674466 = r1674459 * r1674465;
        double r1674467 = r1674455 + r1674466;
        double r1674468 = r1674461 / r1674467;
        double r1674469 = sqrt(r1674468);
        double r1674470 = asin(r1674469);
        return r1674470;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1674471 = t;
        double r1674472 = l;
        double r1674473 = r1674471 / r1674472;
        double r1674474 = 2.410996405697987e+143;
        bool r1674475 = r1674473 <= r1674474;
        double r1674476 = 1.0;
        double r1674477 = Om;
        double r1674478 = Omc;
        double r1674479 = r1674477 / r1674478;
        double r1674480 = r1674479 * r1674479;
        double r1674481 = r1674476 - r1674480;
        double r1674482 = sqrt(r1674481);
        double r1674483 = r1674473 * r1674473;
        double r1674484 = 2.0;
        double r1674485 = fma(r1674483, r1674484, r1674476);
        double r1674486 = sqrt(r1674485);
        double r1674487 = r1674482 / r1674486;
        double r1674488 = asin(r1674487);
        double r1674489 = sqrt(r1674484);
        double r1674490 = r1674471 * r1674489;
        double r1674491 = r1674490 / r1674472;
        double r1674492 = r1674482 / r1674491;
        double r1674493 = asin(r1674492);
        double r1674494 = r1674475 ? r1674488 : r1674493;
        return r1674494;
}

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) < 2.410996405697987e+143

    1. Initial program 6.3

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

    2. Simplified6.3

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

    4. Applied sqrt-div6.4

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

    if 2.410996405697987e+143 < (/ t l)

    1. Initial program 31.3

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

    2. Simplified31.3

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

    4. Applied sqrt-div31.3

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

    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 simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.410996405697987 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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)))))))