Average Error: 10.1 → 1.3
Time: 22.7s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -4.0820958753315573 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{elif}\;\frac{t}{\ell} \le 1.504437824641995 \cdot 10^{+83}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \end{array}\]
\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 -4.0820958753315573 \cdot 10^{+161}:\\
\;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\

\mathbf{elif}\;\frac{t}{\ell} \le 1.504437824641995 \cdot 10^{+83}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\

\end{array}
double f(double t, double l, double Om, double Omc) {
        double r2210233 = 1.0;
        double r2210234 = Om;
        double r2210235 = Omc;
        double r2210236 = r2210234 / r2210235;
        double r2210237 = 2.0;
        double r2210238 = pow(r2210236, r2210237);
        double r2210239 = r2210233 - r2210238;
        double r2210240 = t;
        double r2210241 = l;
        double r2210242 = r2210240 / r2210241;
        double r2210243 = pow(r2210242, r2210237);
        double r2210244 = r2210237 * r2210243;
        double r2210245 = r2210233 + r2210244;
        double r2210246 = r2210239 / r2210245;
        double r2210247 = sqrt(r2210246);
        double r2210248 = asin(r2210247);
        return r2210248;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r2210249 = t;
        double r2210250 = l;
        double r2210251 = r2210249 / r2210250;
        double r2210252 = -4.0820958753315573e+161;
        bool r2210253 = r2210251 <= r2210252;
        double r2210254 = 1.0;
        double r2210255 = Om;
        double r2210256 = Omc;
        double r2210257 = r2210255 / r2210256;
        double r2210258 = r2210257 * r2210257;
        double r2210259 = exp(r2210258);
        double r2210260 = log(r2210259);
        double r2210261 = r2210254 - r2210260;
        double r2210262 = sqrt(r2210261);
        double r2210263 = 2.0;
        double r2210264 = sqrt(r2210263);
        double r2210265 = r2210264 * r2210249;
        double r2210266 = r2210265 / r2210250;
        double r2210267 = r2210262 / r2210266;
        double r2210268 = fabs(r2210267);
        double r2210269 = asin(r2210268);
        double r2210270 = 1.504437824641995e+83;
        bool r2210271 = r2210251 <= r2210270;
        double r2210272 = r2210251 * r2210251;
        double r2210273 = r2210272 + r2210272;
        double r2210274 = r2210254 + r2210273;
        double r2210275 = r2210261 / r2210274;
        double r2210276 = sqrt(r2210275);
        double r2210277 = asin(r2210276);
        double r2210278 = r2210271 ? r2210277 : r2210269;
        double r2210279 = r2210253 ? r2210269 : r2210278;
        return r2210279;
}

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) < -4.0820958753315573e+161 or 1.504437824641995e+83 < (/ t l)

    1. Initial program 28.7

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

    2. Simplified28.7

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

    4. Applied add-log-exp28.7

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

    6. Applied add-sqr-sqrt28.8

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

    7. Applied add-sqr-sqrt28.8

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

    8. Applied times-frac28.7

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

    9. Applied rem-sqrt-square28.7

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

    10. Taylor expanded around inf 1.1

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

    if -4.0820958753315573e+161 < (/ t l) < 1.504437824641995e+83

    1. Initial program 1.3

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

    2. Simplified1.3

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

    4. Applied add-log-exp1.3

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -4.0820958753315573 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{elif}\;\frac{t}{\ell} \le 1.504437824641995 \cdot 10^{+83}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}{1 + \left(\frac{t}{\ell} \cdot \frac{t}{\ell} + \frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \log \left(e^{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))