Average Error: 10.3 → 0.8
Time: 41.5s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4.162493349765146 \cdot 10^{+79}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.617798183196793 \cdot 10^{+72}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]
\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 -4.162493349765146 \cdot 10^{+79}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 3.617798183196793 \cdot 10^{+72}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}} \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{\ell \cdot \sqrt{0.5}}{t}\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) -4.162493349765146e+79)
   (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (- (/ (* l (sqrt 0.5)) t))))
   (if (<= (/ t l) 3.617798183196793e+72)
     (asin
      (sqrt
       (*
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (- 1.0 (* 4.0 (pow (/ t l) 4.0))))
        (- 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (* l (sqrt 0.5)) t))))))
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) <= -4.162493349765146e+79) {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * -((l * sqrt(0.5)) / t));
	} else if ((t / l) <= 3.617798183196793e+72) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 - (4.0 * pow((t / l), 4.0)))) * (1.0 - (2.0 * pow((t / l), 2.0)))));
	} else {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * ((l * sqrt(0.5)) / t));
	}
	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) < -4.16249334976514604e79

    1. Initial program 25.1

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

    2. Taylor expanded around -inf 8.5

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

    3. Simplified0.3

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

    if -4.16249334976514604e79 < (/.f64 t l) < 3.6177981831967929e72

    1. Initial program 0.8

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
    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_241.1

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

    5. Simplified1.1

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\]

    if 3.6177981831967929e72 < (/.f64 t l)

    1. Initial program 25.6

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

    2. Taylor expanded around inf 8.0

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

    3. Simplified0.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4.162493349765146 \cdot 10^{+79}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.617798183196793 \cdot 10^{+72}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 4 \cdot {\left(\frac{t}{\ell}\right)}^{4}} \cdot \left(1 - 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021027 
(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)))))))