Toniolo and Linder, Equation (2)

Average Error: 10.5 → 0.7

Time: 20.5s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ t_2 := \ell \cdot \sqrt{t_1 \cdot 0.5}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t} \cdot t_2\right)\\ \end{array} \]

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 (pow (/ Om Omc) 2.0))) (t_2 (* l (sqrt (* t_1 0.5)))))
   (if (<= (/ t l) -5e+155)
     (asin (* t_2 (/ -1.0 t)))
     (if (<= (/ t l) 1e+154)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
       (asin (* (/ 1.0 t) t_2))))))

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 t_2 = l * sqrt((t_1 * 0.5));
	double tmp;
	if ((t / l) <= -5e+155) {
		tmp = asin((t_2 * (-1.0 / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin(((1.0 / t) * t_2));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    t_2 = l * sqrt((t_1 * 0.5d0))
    if ((t / l) <= (-5d+155)) then
        tmp = asin((t_2 * ((-1.0d0) / t)))
    else if ((t / l) <= 1d+154) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin(((1.0d0 / t) * t_2))
    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 t_2 = l * Math.sqrt((t_1 * 0.5));
	double tmp;
	if ((t / l) <= -5e+155) {
		tmp = Math.asin((t_2 * (-1.0 / t)));
	} else if ((t / l) <= 1e+154) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin(((1.0 / t) * t_2));
	}
	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)
	t_2 = l * math.sqrt((t_1 * 0.5))
	tmp = 0
	if (t / l) <= -5e+155:
		tmp = math.asin((t_2 * (-1.0 / t)))
	elif (t / l) <= 1e+154:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin(((1.0 / t) * t_2))
	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))
	t_2 = Float64(l * sqrt(Float64(t_1 * 0.5)))
	tmp = 0.0
	if (Float64(t / l) <= -5e+155)
		tmp = asin(Float64(t_2 * Float64(-1.0 / t)));
	elseif (Float64(t / l) <= 1e+154)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(1.0 / t) * t_2));
	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);
	t_2 = l * sqrt((t_1 * 0.5));
	tmp = 0.0;
	if ((t / l) <= -5e+155)
		tmp = asin((t_2 * (-1.0 / t)));
	elseif ((t / l) <= 1e+154)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin(((1.0 / t) * t_2));
	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]}, Block[{t$95$2 = N[(l * N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5e+155], N[ArcSin[N[(t$95$2 * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+154], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 / t), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.9999999999999999e155

   1. Initial program 34.8

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

   2. Taylor expanded in t around -inf 7.8

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

   3. Simplified 8.5

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

      [Start] 7.8	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 7.8	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 7.8	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 7.8	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 7.8	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 8.5	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Applied egg-rr 1.2

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

   5. Simplified 0.3

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

      [Start] 1.2	\[ \sin^{-1} \left(-\frac{1}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell}}\right) \]
      associate-/r/ [=>] 0.3	\[ \sin^{-1} \left(-\color{blue}{\frac{1}{t} \cdot \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell\right)}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(-\frac{1}{t} \cdot \color{blue}{\left(\ell \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right)}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(-\frac{1}{t} \cdot \left(\ell \cdot \sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right)\right) \]

   if -4.9999999999999999e155 < (/.f64 t l) < 1.00000000000000004e154

   1. Initial program 0.9

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

   if 1.00000000000000004e154 < (/.f64 t l)

   1. Initial program 34.9

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

   2. Taylor expanded in t around -inf 39.3

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

   3. Simplified 39.2

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

      [Start] 39.3	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 39.3	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 39.3	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 39.3	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 39.3	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 39.2	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Applied egg-rr 35.5

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

   5. Simplified 35.5

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

      [Start] 35.5	\[ \sin^{-1} \left(-\frac{1}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell}}\right) \]
      associate-/r/ [=>] 35.5	\[ \sin^{-1} \left(-\color{blue}{\frac{1}{t} \cdot \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell\right)}\right) \]
      *-commutative [=>] 35.5	\[ \sin^{-1} \left(-\frac{1}{t} \cdot \color{blue}{\left(\ell \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right)}\right) \]
      *-commutative [=>] 35.5	\[ \sin^{-1} \left(-\frac{1}{t} \cdot \left(\ell \cdot \sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t} \cdot \left(\ell \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	32832

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

Alternative 2
Error	0.7
Cost	26888

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{t_1 \cdot 0.5}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\frac{\ell}{\sqrt{2}}}{t}\right)\\ \end{array} \]

Alternative 3
Error	0.7
Cost	20616

\[\begin{array}{l} t_1 := \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t} \cdot \left(\ell \cdot t_1\right)\right)\\ \end{array} \]

Alternative 4
Error	0.7
Cost	20616

\[\begin{array}{l} t_1 := \ell \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t} \cdot t_1\right)\\ \end{array} \]

Alternative 5
Error	0.7
Cost	20292

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5 \cdot t_1}}{t}}{\frac{1}{\ell}}\right)\\ \end{array} \]

Alternative 6
Error	0.9
Cost	14664

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5 \cdot t_1}}{t}}{\frac{1}{\ell}}\right)\\ \end{array} \]

Alternative 7
Error	1.9
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -0.5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-7}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}} \cdot \left(1 + -0.5 \cdot \frac{Om \cdot \frac{Om}{Omc}}{Omc}\right)\right)\\ \end{array} \]

Alternative 8
Error	1.8
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -0.5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-7}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}{t}}{\frac{1}{\ell}}\right)\\ \end{array} \]

Alternative 9
Error	1.9
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -0.5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-7}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10
Error	13.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+209}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-7}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 11
Error	2.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -0.5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-7}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 12
Error	23.3
Cost	13385

\[\begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+64} \lor \neg \left(t \leq 3.6 \cdot 10^{+64}\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \]

Alternative 13
Error	23.3
Cost	13385

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+65} \lor \neg \left(t \leq 3.2 \cdot 10^{+64}\right):\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \]

Alternative 14
Error	31.4
Cost	7104

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

Alternative 15
Error	31.6
Cost	6464

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

Reproduce

herbie shell --seed 2023125 
(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)))))))