Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.9%

Time: 19.8s

Precision: binary64

Cost: 26888

Math

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

↓

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

FPCore

(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
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -1e+77)
     (asin (* (sqrt t_1) (/ (- l) (* t (sqrt 2.0)))))
     (if (<= (/ t l) 5e+126)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin
        (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (/ l t) (sqrt 2.0))))))))

C

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 t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -1e+77) {
		tmp = asin((sqrt(t_1) * (-l / (t * sqrt(2.0)))));
	} else if ((t / l) <= 5e+126) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * ((l / t) / sqrt(2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
end function

↓

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t / l) <= (-1d+77)) then
        tmp = asin((sqrt(t_1) * (-l / (t * sqrt(2.0d0)))))
    else if ((t / l) <= 5d+126) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) ** 2.0d0))) * ((l / t) / sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
}

↓

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -1e+77) {
		tmp = Math.asin((Math.sqrt(t_1) * (-l / (t * Math.sqrt(2.0)))));
	} else if ((t / l) <= 5e+126) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) * ((l / t) / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	return math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))

↓

def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -1e+77:
		tmp = math.asin((math.sqrt(t_1) * (-l / (t * math.sqrt(2.0)))))
	elif (t / l) <= 5e+126:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) * ((l / t) / math.sqrt(2.0))))
	return tmp

Julia

function code(t, l, Om, Omc)
	return asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))))
end

↓

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -1e+77)
		tmp = asin(Float64(sqrt(t_1) * Float64(Float64(-l) / Float64(t * sqrt(2.0)))));
	elseif (Float64(t / l) <= 5e+126)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * Float64(Float64(l / t) / sqrt(2.0))));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
end

↓

function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -1e+77)
		tmp = asin((sqrt(t_1) * (-l / (t * sqrt(2.0)))));
	elseif ((t / l) <= 5e+126)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) * ((l / t) / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

↓

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+77], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+126], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -9.99999999999999983e76

   1. Initial program 36.5%

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

   2. Applied egg-rr 98.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      Step-by-step derivation

      [Start] 36.5	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      sqrt-div [=>] 36.5	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      div-inv [=>] 36.5	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      add-sqr-sqrt [=>] 36.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]
      hypot-1-def [=>] 36.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]
      *-commutative [=>] 36.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]
      sqrt-prod [=>] 36.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]
      unpow2 [=>] 36.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]
      sqrt-prod [=>] 0.0	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]
      add-sqr-sqrt [<=] 98.7	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Simplified 98.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      Step-by-step derivation

      [Start] 98.7	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 98.7	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [<=] 94.1	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 94.1	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{Omc \cdot Omc}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 94.1	\[ \sin^{-1} \left(\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      associate-*r/ [=>] 94.1	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      *-rgt-identity [=>] 94.1	\[ \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 94.1	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 94.1	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [=>] 98.7	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 98.7	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   4. Taylor expanded in t around -inf 94.2%

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

   5. Simplified 99.6%

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

      [Start] 94.2	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      associate-*r* [=>] 94.2	\[ \sin^{-1} \color{blue}{\left(\left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 94.2	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right)} \]
      unpow2 [=>] 94.2	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      unpow2 [=>] 94.2	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      times-frac [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      unpow2 [<=] 99.6	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      *-commutative [<=] 99.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-1 \cdot \frac{\ell}{\color{blue}{t \cdot \sqrt{2}}}\right)\right) \]
      associate-*r/ [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{-1 \cdot \ell}{t \cdot \sqrt{2}}}\right) \]
      neg-mul-1 [<=] 99.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\color{blue}{-\ell}}{t \cdot \sqrt{2}}\right) \]
      *-commutative [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{-\ell}{\color{blue}{\sqrt{2} \cdot t}}\right) \]

   6. Applied egg-rr 99.6%

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

      [Start] 99.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]
      clear-num [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]
      un-div-inv [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]

   if -9.99999999999999983e76 < (/.f64 t l) < 4.99999999999999977e126

   1. Initial program 98.2%

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

   2. Applied egg-rr 96.5%

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

      [Start] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      unpow2 [=>] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]
      div-inv [=>] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\left(t \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)}}\right) \]
      associate-*l* [=>] 96.5	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(t \cdot \left(\frac{1}{\ell} \cdot \frac{t}{\ell}\right)\right)}}}\right) \]
      inv-pow [=>] 96.5	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(t \cdot \left(\color{blue}{{\ell}^{-1}} \cdot \frac{t}{\ell}\right)\right)}}\right) \]

   3. Applied egg-rr 98.2%

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

      [Start] 96.5	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(t \cdot \left({\ell}^{-1} \cdot \frac{t}{\ell}\right)\right)}}\right) \]
      associate-*r* [=>] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\left(t \cdot {\ell}^{-1}\right) \cdot \frac{t}{\ell}\right)}}}\right) \]
      unpow-1 [=>] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\left(t \cdot \color{blue}{\frac{1}{\ell}}\right) \cdot \frac{t}{\ell}\right)}}\right) \]
      div-inv [<=] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}\right)}}\right) \]
      clear-num [=>] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right)}}\right) \]
      un-div-inv [=>] 98.2	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}}\right) \]

   4. Applied egg-rr 98.2%

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

      [Start] 9.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [=>] 9.5	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]
      clear-num [=>] 9.5	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \color{blue}{\frac{1}{\frac{Omc}{Om}}}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]
      un-div-inv [=>] 9.5	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]

   if 4.99999999999999977e126 < (/.f64 t l)

   1. Initial program 49.7%

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

   2. Applied egg-rr 96.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      Step-by-step derivation

      [Start] 49.7	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      sqrt-div [=>] 49.6	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      div-inv [=>] 49.6	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      add-sqr-sqrt [=>] 49.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]
      hypot-1-def [=>] 49.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]
      *-commutative [=>] 49.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]
      sqrt-prod [=>] 49.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]
      unpow2 [=>] 49.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]
      sqrt-prod [=>] 96.7	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]
      add-sqr-sqrt [<=] 96.8	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Simplified 96.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      Step-by-step derivation

      [Start] 96.8	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 96.8	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [<=] 82.3	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 82.3	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{Omc \cdot Omc}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 82.3	\[ \sin^{-1} \left(\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      associate-*r/ [=>] 82.3	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      *-rgt-identity [=>] 82.3	\[ \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 82.3	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 82.3	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [=>] 96.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 96.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   4. Taylor expanded in t around -inf 34.1%

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

   5. Simplified 43.5%

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

      [Start] 34.1	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      associate-*r* [=>] 34.1	\[ \sin^{-1} \color{blue}{\left(\left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 34.1	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right)} \]
      unpow2 [=>] 34.1	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      unpow2 [=>] 34.1	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      times-frac [=>] 43.5	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      unpow2 [<=] 43.5	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      *-commutative [<=] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-1 \cdot \frac{\ell}{\color{blue}{t \cdot \sqrt{2}}}\right)\right) \]
      associate-*r/ [=>] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{-1 \cdot \ell}{t \cdot \sqrt{2}}}\right) \]
      neg-mul-1 [<=] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\color{blue}{-\ell}}{t \cdot \sqrt{2}}\right) \]
      *-commutative [=>] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{-\ell}{\color{blue}{\sqrt{2} \cdot t}}\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\left(0 + \frac{\frac{\ell}{t}}{\sqrt{2}}\right)}\right) \]
      Step-by-step derivation

      [Start] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right) \]
      add-log-exp [=>] 44.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\log \left(e^{\frac{-\ell}{\sqrt{2} \cdot t}}\right)}\right) \]
      *-un-lft-identity [=>] 44.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \log \color{blue}{\left(1 \cdot e^{\frac{-\ell}{\sqrt{2} \cdot t}}\right)}\right) \]
      log-prod [=>] 44.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\left(\log 1 + \log \left(e^{\frac{-\ell}{\sqrt{2} \cdot t}}\right)\right)}\right) \]
      metadata-eval [=>] 44.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\color{blue}{0} + \log \left(e^{\frac{-\ell}{\sqrt{2} \cdot t}}\right)\right)\right) \]
      add-log-exp [<=] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \color{blue}{\frac{-\ell}{\sqrt{2} \cdot t}}\right)\right) \]
      *-commutative [=>] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \frac{-\ell}{\color{blue}{t \cdot \sqrt{2}}}\right)\right) \]
      associate-/r* [=>] 43.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \color{blue}{\frac{\frac{-\ell}{t}}{\sqrt{2}}}\right)\right) \]
      add-sqr-sqrt [=>] 15.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \frac{\frac{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}{t}}{\sqrt{2}}\right)\right) \]
      sqrt-unprod [=>] 69.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \frac{\frac{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}{t}}{\sqrt{2}}\right)\right) \]
      sqr-neg [=>] 69.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \frac{\frac{\sqrt{\color{blue}{\ell \cdot \ell}}}{t}}{\sqrt{2}}\right)\right) \]
      sqrt-unprod [<=] 63.3	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \frac{\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{t}}{\sqrt{2}}\right)\right) \]
      add-sqr-sqrt [<=] 99.7	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \frac{\frac{\color{blue}{\ell}}{t}}{\sqrt{2}}\right)\right) \]

   7. Simplified 99.7%

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

      [Start] 99.7	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(0 + \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\right) \]
      +-lft-identity [=>] 99.7	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\ell}{t}}{\sqrt{2}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	26624

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

Alternative 2
Accuracy	97.7%
Cost	20488

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1 \cdot 0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}\right)\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	20488

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}\right)\\ \end{array} \]

Alternative 4
Accuracy	97.6%
Cost	14664

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1 \cdot 0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Accuracy	97.5%
Cost	14280

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{t_1 \cdot 0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Accuracy	97.7%
Cost	14280

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ t_2 := \sqrt{t_1 \cdot 0.5}\\ \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{t_2}{t}\right)\\ \end{array} \]

Alternative 7
Accuracy	97.3%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8
Accuracy	79.7%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+228}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9
Accuracy	96.7%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10
Accuracy	96.8%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 11
Accuracy	63.7%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{-84}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{-76}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 12
Accuracy	50.2%
Cost	6464

\[\sin^{-1} 1 \]

Reproduce

herbie shell --seed 2023161 
(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)))))))