Average Error: 10.1 → 5.8
Time: 20.4s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.1033967040349165 \cdot 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2 \cdot \frac{t}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 5.1033967040349165 \cdot 10^{+126}:\\
\;\;\;\;\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 r1011250 = 1.0;
        double r1011251 = Om;
        double r1011252 = Omc;
        double r1011253 = r1011251 / r1011252;
        double r1011254 = 2.0;
        double r1011255 = pow(r1011253, r1011254);
        double r1011256 = r1011250 - r1011255;
        double r1011257 = t;
        double r1011258 = l;
        double r1011259 = r1011257 / r1011258;
        double r1011260 = pow(r1011259, r1011254);
        double r1011261 = r1011254 * r1011260;
        double r1011262 = r1011250 + r1011261;
        double r1011263 = r1011256 / r1011262;
        double r1011264 = sqrt(r1011263);
        double r1011265 = asin(r1011264);
        return r1011265;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1011266 = t;
        double r1011267 = l;
        double r1011268 = r1011266 / r1011267;
        double r1011269 = 5.1033967040349165e+126;
        bool r1011270 = r1011268 <= r1011269;
        double r1011271 = 1.0;
        double r1011272 = Om;
        double r1011273 = Omc;
        double r1011274 = r1011272 / r1011273;
        double r1011275 = r1011274 * r1011274;
        double r1011276 = r1011271 - r1011275;
        double r1011277 = 2.0;
        double r1011278 = r1011277 * r1011268;
        double r1011279 = fma(r1011278, r1011268, r1011271);
        double r1011280 = r1011276 / r1011279;
        double r1011281 = sqrt(r1011280);
        double r1011282 = asin(r1011281);
        double r1011283 = sqrt(r1011276);
        double r1011284 = sqrt(r1011277);
        double r1011285 = r1011266 * r1011284;
        double r1011286 = r1011285 / r1011267;
        double r1011287 = r1011283 / r1011286;
        double r1011288 = asin(r1011287);
        double r1011289 = r1011270 ? r1011282 : r1011288;
        return r1011289;
}

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) < 5.1033967040349165e+126

    1. Initial program 6.5

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

    2. Simplified6.5

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

    if 5.1033967040349165e+126 < (/ 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

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

    4. Applied sqrt-div30.0

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

    5. Taylor expanded around inf 1.6

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.1033967040349165 \cdot 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(2 \cdot \frac{t}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +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)))))))