Average Error: 10.2 → 1.1
Time: 51.2s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.5616280099698165 \cdot 10^{+78}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2.538593515635445 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - \left(2 \cdot 2\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(2 \cdot 2\right)}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]
\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} \leq -1.5616280099698165 \cdot 10^{+78}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 2.538593515635445 \cdot 10^{+60}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - \left(2 \cdot 2\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(2 \cdot 2\right)}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\


\end{array}

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))
↓
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -1.5616280099698165e+78)
   (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (- (/ (sqrt 0.5) (/ t l)))))
   (if (<= (/ t l) 2.538593515635445e+60)
     (asin
      (sqrt
       (*
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (- 1.0 (* (* 2.0 2.0) (pow (/ t l) (* 2.0 2.0)))))
        (- 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (sqrt 0.5) (/ t l)))))))
double code(double t, double l, double Om, double Omc) {
	return asin(sqrt((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0)))));
}
↓
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -1.5616280099698165e+78) {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * -(sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 2.538593515635445e+60) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 - ((2.0 * 2.0) * pow((t / l), (2.0 * 2.0))))) * (1.0 - (2.0 * pow((t / l), 2.0)))));
	} else {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * (sqrt(0.5) / (t / l)));
	}
	return tmp;
}

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 3 regimes

  2. if (/.f64 t l) < -1.56162800996981652e78

    1. Initial program 24.4

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

    2. Taylor expanded around -inf 8.3

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]

    3. Simplified1.1

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} \]

    if -1.56162800996981652e78 < (/.f64 t l) < 2.53859351563544495e60

    1. Initial program 1.0

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

    3. Applied flip-+_binary64_521.0

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

    4. Applied associate-/r/_binary64_241.0

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

    5. Simplified1.0

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

    if 2.53859351563544495e60 < (/.f64 t l)

    1. Initial program 24.5

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

    2. Taylor expanded around inf 7.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

    3. Simplified1.2

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.5616280099698165 \cdot 10^{+78}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2.538593515635445 \cdot 10^{+60}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - \left(2 \cdot 2\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(2 \cdot 2\right)}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021210 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))