Average Error: 9.8 → 5.7
Time: 2.5m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 4.743145122836656:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}} \cdot \sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}} \cdot \sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 4.743145122836656:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}} \cdot \sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}} \cdot \sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}}\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 r7574358 = 1.0;
        double r7574359 = Om;
        double r7574360 = Omc;
        double r7574361 = r7574359 / r7574360;
        double r7574362 = 2.0;
        double r7574363 = pow(r7574361, r7574362);
        double r7574364 = r7574358 - r7574363;
        double r7574365 = t;
        double r7574366 = l;
        double r7574367 = r7574365 / r7574366;
        double r7574368 = pow(r7574367, r7574362);
        double r7574369 = r7574362 * r7574368;
        double r7574370 = r7574358 + r7574369;
        double r7574371 = r7574364 / r7574370;
        double r7574372 = sqrt(r7574371);
        double r7574373 = asin(r7574372);
        return r7574373;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r7574374 = t;
        double r7574375 = l;
        double r7574376 = r7574374 / r7574375;
        double r7574377 = 4.743145122836656;
        bool r7574378 = r7574376 <= r7574377;
        double r7574379 = 1.0;
        double r7574380 = Om;
        double r7574381 = Omc;
        double r7574382 = r7574380 / r7574381;
        double r7574383 = r7574382 * r7574382;
        double r7574384 = r7574379 - r7574383;
        double r7574385 = 2.0;
        double r7574386 = r7574375 / r7574374;
        double r7574387 = r7574386 * r7574386;
        double r7574388 = r7574385 / r7574387;
        double r7574389 = r7574379 + r7574388;
        double r7574390 = r7574384 / r7574389;
        double r7574391 = cbrt(r7574390);
        double r7574392 = r7574391 * r7574391;
        double r7574393 = sqrt(r7574392);
        double r7574394 = sqrt(r7574391);
        double r7574395 = r7574393 * r7574394;
        double r7574396 = asin(r7574395);
        double r7574397 = sqrt(r7574384);
        double r7574398 = sqrt(r7574385);
        double r7574399 = r7574374 * r7574398;
        double r7574400 = r7574399 / r7574375;
        double r7574401 = r7574397 / r7574400;
        double r7574402 = asin(r7574401);
        double r7574403 = r7574378 ? r7574396 : r7574402;
        return r7574403;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 4.743145122836656

    1. Initial program 6.9

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

    2. Simplified6.9

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

    3. Taylor expanded around inf 22.0

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

    4. Simplified6.9

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

    6. Applied add-cube-cbrt7.1

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\left(\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}} \cdot \sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}\right) \cdot \sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}}}\right)\]

    7. Applied sqrt-prod7.1

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}} \cdot \sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}} \cdot \sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}} + 1}}}\right)}\]

    if 4.743145122836656 < (/ t l)

    1. Initial program 18.8

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

    2. Simplified18.8

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

    4. Applied sqrt-div18.8

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

    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 simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 4.743145122836656:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}} \cdot \sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}} \cdot \sqrt{\sqrt[3]{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))