Average Error: 10.0 → 5.6
Time: 24.2s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.092158378202942 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 5.092158378202942 \cdot 10^{+135}:\\
\;\;\;\;\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)\\

\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 r1377297 = 1.0;
        double r1377298 = Om;
        double r1377299 = Omc;
        double r1377300 = r1377298 / r1377299;
        double r1377301 = 2.0;
        double r1377302 = pow(r1377300, r1377301);
        double r1377303 = r1377297 - r1377302;
        double r1377304 = t;
        double r1377305 = l;
        double r1377306 = r1377304 / r1377305;
        double r1377307 = pow(r1377306, r1377301);
        double r1377308 = r1377301 * r1377307;
        double r1377309 = r1377297 + r1377308;
        double r1377310 = r1377303 / r1377309;
        double r1377311 = sqrt(r1377310);
        double r1377312 = asin(r1377311);
        return r1377312;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1377313 = t;
        double r1377314 = l;
        double r1377315 = r1377313 / r1377314;
        double r1377316 = 5.092158378202942e+135;
        bool r1377317 = r1377315 <= r1377316;
        double r1377318 = 1.0;
        double r1377319 = Om;
        double r1377320 = Omc;
        double r1377321 = r1377319 / r1377320;
        double r1377322 = r1377321 * r1377321;
        double r1377323 = r1377318 - r1377322;
        double r1377324 = r1377315 * r1377315;
        double r1377325 = 2.0;
        double r1377326 = fma(r1377324, r1377325, r1377318);
        double r1377327 = r1377323 / r1377326;
        double r1377328 = sqrt(r1377327);
        double r1377329 = asin(r1377328);
        double r1377330 = sqrt(r1377323);
        double r1377331 = sqrt(r1377325);
        double r1377332 = r1377313 * r1377331;
        double r1377333 = r1377332 / r1377314;
        double r1377334 = r1377330 / r1377333;
        double r1377335 = asin(r1377334);
        double r1377336 = r1377317 ? r1377329 : r1377335;
        return r1377336;
}

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) < 5.092158378202942e+135

    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

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

    if 5.092158378202942e+135 < (/ t l)

    1. Initial program 31.7

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

    2. Simplified31.7

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

    4. Applied sqrt-div31.7

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

    5. Taylor expanded around inf 1.4

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.092158378202942 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +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)))))))