Toniolo and Linder, Equation (2)

Average Error: 9.6 → 0.8

Time: 15.2s

Precision: binary64

Cost: 26952

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+123}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \left(\frac{\ell}{t} \cdot {2}^{-0.5}\right)\right)\\ \end{array} \]

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 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t l) -1e+123)
     (asin (/ (* l (- (sqrt 0.5))) t))
     (if (<= (/ t l) 10000000.0)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (* (sqrt t_1) (* (/ l t) (pow 2.0 -0.5))))))))

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 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+123) {
		tmp = asin(((l * -sqrt(0.5)) / t));
	} else if ((t / l) <= 10000000.0) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt(t_1) * ((l / t) * pow(2.0, -0.5))));
	}
	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) ** 2.0d0)
    if ((t / l) <= (-1d+123)) then
        tmp = asin(((l * -sqrt(0.5d0)) / t))
    else if ((t / l) <= 10000000.0d0) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt(t_1) * ((l / t) * (2.0d0 ** (-0.5d0)))))
    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 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+123) {
		tmp = Math.asin(((l * -Math.sqrt(0.5)) / t));
	} else if ((t / l) <= 10000000.0) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt(t_1) * ((l / t) * Math.pow(2.0, -0.5))));
	}
	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 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -1e+123:
		tmp = math.asin(((l * -math.sqrt(0.5)) / t))
	elif (t / l) <= 10000000.0:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt(t_1) * ((l / t) * math.pow(2.0, -0.5))))
	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(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t / l) <= -1e+123)
		tmp = asin(Float64(Float64(l * Float64(-sqrt(0.5))) / t));
	elseif (Float64(t / l) <= 10000000.0)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(sqrt(t_1) * Float64(Float64(l / t) * (2.0 ^ -0.5))));
	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) ^ 2.0);
	tmp = 0.0;
	if ((t / l) <= -1e+123)
		tmp = asin(((l * -sqrt(0.5)) / t));
	elseif ((t / l) <= 10000000.0)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt(t_1) * ((l / t) * (2.0 ^ -0.5))));
	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[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+123], N[ArcSin[N[(N[(l * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 10000000.0], 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[t$95$1], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -9.99999999999999978e122

   1. Initial program 29.6

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

   2. Taylor expanded in t around -inf 9.2

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

   3. Simplified 9.8

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

      [Start] 9.2	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 9.2	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 9.2	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 9.2	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 9.2	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 9.8	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Applied egg-rr 36.8

      \[\leadsto \sin^{-1} \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} - 1\right)}\right) \]

   5. Simplified 0.3

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

      [Start] 36.8	\[ \sin^{-1} \left(-\left(e^{\mathsf{log1p}\left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} - 1\right)\right) \]
      expm1-def [=>] 0.3	\[ \sin^{-1} \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)}\right) \]
      expm1-log1p [=>] 0.3	\[ \sin^{-1} \left(-\color{blue}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(-\color{blue}{\frac{\ell}{t} \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(-\frac{\ell}{t} \cdot \sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right) \]

   6. Taylor expanded in Om around 0 0.8

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

   if -9.99999999999999978e122 < (/.f64 t l) < 1e7

   1. Initial program 0.9

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

   2. Applied egg-rr 0.9

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

   if 1e7 < (/.f64 t l)

   1. Initial program 18.3

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

   2. Applied egg-rr 0.9

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

   3. Taylor expanded in t around inf 7.9

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

   4. Simplified 0.3

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

      [Start] 7.9	\[ \sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      *-commutative [=>] 7.9	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)} \]
      unpow2 [=>] 7.9	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [=>] 7.9	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      times-frac [=>] 0.4	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [<=] 0.4	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      associate-/l/ [<=] 0.3	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\frac{\frac{\ell}{t}}{\sqrt{2}}}\right) \]

   5. Applied egg-rr 0.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+123}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\frac{\ell}{t} \cdot {2}^{-0.5}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
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
Error	0.8
Cost	20872

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{0.5 \cdot t_1}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+123}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	20680

\[\begin{array}{l} t_1 := \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 5
Error	1.5
Cost	20356

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 6
Error	1.5
Cost	14084

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right) \cdot 0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 7
Error	1.6
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 8
Error	13.2
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 9
Error	13.2
Cost	13704

\[\begin{array}{l} t_1 := -\sqrt{0.5}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot t_1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+195}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{t_1}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 10
Error	1.7
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 11
Error	23.4
Cost	13449

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{-132} \lor \neg \left(\ell \leq 2.8 \cdot 10^{-133}\right):\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \end{array} \]

Alternative 12
Error	23.4
Cost	13449

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.36 \cdot 10^{-132} \lor \neg \left(\ell \leq 4.2 \cdot 10^{-133}\right):\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 13
Error	31.5
Cost	7104

\[\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right) \]

Alternative 14
Error	31.6
Cost	6464

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

Reproduce

herbie shell --seed 2023002 
(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)))))))