Average Error: 10.4 → 5.9
Time: 22.7s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.3226963208817615 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \frac{t}{\ell} + \left(1 + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 1.3226963208817615 \cdot 10^{+73}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \frac{t}{\ell} + \left(1 + \frac{t}{\ell} \cdot \frac{t}{\ell}\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 r930218 = 1.0;
        double r930219 = Om;
        double r930220 = Omc;
        double r930221 = r930219 / r930220;
        double r930222 = 2.0;
        double r930223 = pow(r930221, r930222);
        double r930224 = r930218 - r930223;
        double r930225 = t;
        double r930226 = l;
        double r930227 = r930225 / r930226;
        double r930228 = pow(r930227, r930222);
        double r930229 = r930222 * r930228;
        double r930230 = r930218 + r930229;
        double r930231 = r930224 / r930230;
        double r930232 = sqrt(r930231);
        double r930233 = asin(r930232);
        return r930233;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r930234 = t;
        double r930235 = l;
        double r930236 = r930234 / r930235;
        double r930237 = 1.3226963208817615e+73;
        bool r930238 = r930236 <= r930237;
        double r930239 = 1.0;
        double r930240 = Om;
        double r930241 = Omc;
        double r930242 = r930240 / r930241;
        double r930243 = r930242 * r930242;
        double r930244 = r930239 - r930243;
        double r930245 = sqrt(r930244);
        double r930246 = r930236 * r930236;
        double r930247 = r930239 + r930246;
        double r930248 = r930246 + r930247;
        double r930249 = sqrt(r930248);
        double r930250 = r930245 / r930249;
        double r930251 = asin(r930250);
        double r930252 = 2.0;
        double r930253 = sqrt(r930252);
        double r930254 = r930234 * r930253;
        double r930255 = r930254 / r930235;
        double r930256 = r930245 / r930255;
        double r930257 = asin(r930256);
        double r930258 = r930238 ? r930251 : r930257;
        return r930258;
}

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) < 1.3226963208817615e+73

    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

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

    4. Applied add-sqr-sqrt6.9

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

    6. Applied sqrt-div6.9

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

    7. Simplified6.9

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

    if 1.3226963208817615e+73 < (/ t l)

    1. Initial program 25.3

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

    2. Simplified25.3

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

    4. Applied add-sqr-sqrt25.3

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

    6. Applied sqrt-div25.3

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

    7. Simplified25.3

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

    8. 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 simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 1.3226963208817615 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\sqrt{\frac{t}{\ell} \cdot \frac{t}{\ell} + \left(1 + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))