Toniolo and Linder, Equation (2)

Average Error: 16.57% → 1.67%

Time: 14.8s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+162}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{\frac{t}{t_1}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\ell}{t}\right)\\ \end{array} \]

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 (sqrt (* 0.5 (- 1.0 (pow (/ Om Omc) 2.0))))))
   (if (<= (/ t l) -2e+162)
     (- (asin (/ l (/ t t_1))))
     (if (<= (/ t l) 5e+96)
       (asin
        (sqrt
         (/
          (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
          (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (* t_1 (/ l t)))))))

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 = sqrt((0.5 * (1.0 - pow((Om / Omc), 2.0))));
	double tmp;
	if ((t / l) <= -2e+162) {
		tmp = -asin((l / (t / t_1)));
	} else if ((t / l) <= 5e+96) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((t_1 * (l / t)));
	}
	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 = sqrt((0.5d0 * (1.0d0 - ((om / omc) ** 2.0d0))))
    if ((t / l) <= (-2d+162)) then
        tmp = -asin((l / (t / t_1)))
    else if ((t / l) <= 5d+96) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((t_1 * (l / t)))
    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 = Math.sqrt((0.5 * (1.0 - Math.pow((Om / Omc), 2.0))));
	double tmp;
	if ((t / l) <= -2e+162) {
		tmp = -Math.asin((l / (t / t_1)));
	} else if ((t / l) <= 5e+96) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((t_1 * (l / t)));
	}
	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 = math.sqrt((0.5 * (1.0 - math.pow((Om / Omc), 2.0))))
	tmp = 0
	if (t / l) <= -2e+162:
		tmp = -math.asin((l / (t / t_1)))
	elif (t / l) <= 5e+96:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((t_1 * (l / t)))
	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 = sqrt(Float64(0.5 * Float64(1.0 - (Float64(Om / Omc) ^ 2.0))))
	tmp = 0.0
	if (Float64(t / l) <= -2e+162)
		tmp = Float64(-asin(Float64(l / Float64(t / t_1))));
	elseif (Float64(t / l) <= 5e+96)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(t_1 * Float64(l / t)));
	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 = sqrt((0.5 * (1.0 - ((Om / Omc) ^ 2.0))));
	tmp = 0.0;
	if ((t / l) <= -2e+162)
		tmp = -asin((l / (t / t_1)));
	elseif ((t / l) <= 5e+96)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((t_1 * (l / t)));
	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[Sqrt[N[(0.5 * N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+162], (-N[ArcSin[N[(l / N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 5e+96], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.9999999999999999e162

   1. Initial program 54.55

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

   2. Applied egg-rr 54.55

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

   3. Taylor expanded in t around -inf 12.62

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

   4. Simplified 13.85

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

      [Start] 12.62	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 12.62	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 12.62	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 12.62	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 12.62	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 0.43	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [<=] 0.43	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 0.43	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [<=] 12.62	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 13.85	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   5. Applied egg-rr 54.55

      \[\leadsto \color{blue}{\left(0 - e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)}\right) + 1} \]

   6. Simplified 0.43

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

      [Start] 54.55	\[ \left(0 - e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)}\right) + 1 \]
      associate-+l- [=>] 54.55	\[ \color{blue}{0 - \left(e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)} - 1\right)} \]
      expm1-def [=>] 0.43	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)\right)} \]
      expm1-log1p [=>] 0.43	\[ 0 - \color{blue}{\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} \]
      sub0-neg [=>] 0.43	\[ \color{blue}{-\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} \]
      *-commutative [=>] 0.43	\[ -\sin^{-1} \color{blue}{\left(\frac{\ell}{t} \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right)} \]
      *-commutative [=>] 0.43	\[ -\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right) \]

   7. Applied egg-rr 0.5

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

   if -1.9999999999999999e162 < (/.f64 t l) < 5.0000000000000004e96

   1. Initial program 2.24

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

   2. Applied egg-rr 2.22

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

   3. Applied egg-rr 2.22

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

   if 5.0000000000000004e96 < (/.f64 t l)

   1. Initial program 42.51

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

   2. Applied egg-rr 42.5

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

   3. Taylor expanded in t around -inf 68.69

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

   4. Simplified 68.64

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

      [Start] 68.69	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 68.69	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 68.69	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 68.69	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 68.69	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 64.72	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [<=] 64.72	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 64.72	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [<=] 68.69	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 68.64	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   5. Applied egg-rr 0.47

      \[\leadsto \color{blue}{0 + \sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} \]

   6. Simplified 0.47

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

      [Start] 0.47	\[ 0 + \sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right) \]
      +-lft-identity [=>] 0.47	\[ \color{blue}{\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} \]
      *-commutative [=>] 0.47	\[ \sin^{-1} \color{blue}{\left(\frac{\ell}{t} \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right)} \]
      *-commutative [=>] 0.47	\[ \sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.67

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+162}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.59%
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	1.75%
Cost	20488

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+162}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+96}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 3
Error	9.95%
Cost	14404

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+51}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{t_1 \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \end{array} \]

Alternative 4
Error	9.11%
Cost	14404

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+162}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \end{array} \]

Alternative 5
Error	15.56%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \end{array} \]

Alternative 6
Error	15.36%
Cost	14152

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{t_1 \cdot 0.5}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \end{array} \]

Alternative 7
Error	20.05%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 8
Error	20.36%
Cost	13705

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -0.72 \lor \neg \left(\frac{t}{\ell} \leq 1.2 \cdot 10^{+205}\right):\\ \;\;\;\;-\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot \frac{Om}{Omc}}{\frac{Omc}{-0.5}}\right)\\ \end{array} \]

Alternative 9
Error	20.21%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -500:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+197}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot \frac{Om}{Omc}}{\frac{Omc}{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 10
Error	48.64%
Cost	7104

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

Alternative 11
Error	48.93%
Cost	6464

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

Reproduce

herbie shell --seed 2023093 
(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)))))))