Average Error: 10.6 → 5.5
Time: 26.2s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.708797556308962 \cdot 10^{+114}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 2.708797556308962 \cdot 10^{+114}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\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 r1576381 = 1.0;
        double r1576382 = Om;
        double r1576383 = Omc;
        double r1576384 = r1576382 / r1576383;
        double r1576385 = 2.0;
        double r1576386 = pow(r1576384, r1576385);
        double r1576387 = r1576381 - r1576386;
        double r1576388 = t;
        double r1576389 = l;
        double r1576390 = r1576388 / r1576389;
        double r1576391 = pow(r1576390, r1576385);
        double r1576392 = r1576385 * r1576391;
        double r1576393 = r1576381 + r1576392;
        double r1576394 = r1576387 / r1576393;
        double r1576395 = sqrt(r1576394);
        double r1576396 = asin(r1576395);
        return r1576396;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1576397 = t;
        double r1576398 = l;
        double r1576399 = r1576397 / r1576398;
        double r1576400 = 2.708797556308962e+114;
        bool r1576401 = r1576399 <= r1576400;
        double r1576402 = 1.0;
        double r1576403 = Om;
        double r1576404 = Omc;
        double r1576405 = r1576403 / r1576404;
        double r1576406 = r1576405 * r1576405;
        double r1576407 = r1576402 - r1576406;
        double r1576408 = 2.0;
        double r1576409 = r1576398 / r1576397;
        double r1576410 = r1576409 * r1576409;
        double r1576411 = r1576408 / r1576410;
        double r1576412 = r1576402 + r1576411;
        double r1576413 = r1576407 / r1576412;
        double r1576414 = sqrt(r1576413);
        double r1576415 = asin(r1576414);
        double r1576416 = sqrt(r1576407);
        double r1576417 = sqrt(r1576408);
        double r1576418 = r1576397 * r1576417;
        double r1576419 = r1576418 / r1576398;
        double r1576420 = r1576416 / r1576419;
        double r1576421 = asin(r1576420);
        double r1576422 = r1576401 ? r1576415 : r1576421;
        return r1576422;
}

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) < 2.708797556308962e+114

    1. Initial program 6.4

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

    2. Simplified6.4

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

    if 2.708797556308962e+114 < (/ t l)

    1. Initial program 29.8

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

    2. Simplified29.8

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

    3. Taylor expanded around inf 39.1

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

    4. Simplified29.8

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

    6. Applied sqrt-div29.8

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

    7. Taylor expanded around inf 1.0

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

  4. Final simplification5.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 2.708797556308962 \cdot 10^{+114}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))