Average Error: 9.5 → 5.3
Time: 27.3s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.572137631982805 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]
\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 7.572137631982805 \cdot 10^{+147}:\\
\;\;\;\;\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)\\

\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 r1607578 = 1.0;
        double r1607579 = Om;
        double r1607580 = Omc;
        double r1607581 = r1607579 / r1607580;
        double r1607582 = 2.0;
        double r1607583 = pow(r1607581, r1607582);
        double r1607584 = r1607578 - r1607583;
        double r1607585 = t;
        double r1607586 = l;
        double r1607587 = r1607585 / r1607586;
        double r1607588 = pow(r1607587, r1607582);
        double r1607589 = r1607582 * r1607588;
        double r1607590 = r1607578 + r1607589;
        double r1607591 = r1607584 / r1607590;
        double r1607592 = sqrt(r1607591);
        double r1607593 = asin(r1607592);
        return r1607593;
}
↓
double f(double t, double l, double Om, double Omc) {
        double r1607594 = t;
        double r1607595 = l;
        double r1607596 = r1607594 / r1607595;
        double r1607597 = 7.572137631982805e+147;
        bool r1607598 = r1607596 <= r1607597;
        double r1607599 = 1.0;
        double r1607600 = Om;
        double r1607601 = Omc;
        double r1607602 = r1607600 / r1607601;
        double r1607603 = r1607602 * r1607602;
        double r1607604 = r1607599 - r1607603;
        double r1607605 = r1607596 * r1607596;
        double r1607606 = 2.0;
        double r1607607 = r1607605 * r1607606;
        double r1607608 = r1607607 + r1607599;
        double r1607609 = r1607604 / r1607608;
        double r1607610 = sqrt(r1607609);
        double r1607611 = asin(r1607610);
        double r1607612 = sqrt(r1607604);
        double r1607613 = sqrt(r1607606);
        double r1607614 = r1607594 * r1607613;
        double r1607615 = r1607614 / r1607595;
        double r1607616 = r1607612 / r1607615;
        double r1607617 = asin(r1607616);
        double r1607618 = r1607598 ? r1607611 : r1607617;
        return r1607618;
}

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) < 7.572137631982805e+147

    1. Initial program 6.0

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

    2. Simplified6.0

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

    4. Applied sqrt-div6.0

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

    5. Taylor expanded around -inf 23.8

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

    6. Simplified6.0

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

    if 7.572137631982805e+147 < (/ 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

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

    4. Applied sqrt-div31.7

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

    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 simplification5.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 7.572137631982805 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))