Toniolo and Linder, Equation (2)

Average Error: 10.6 → 0.9

Time: 15.2s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{-t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \ell\right)\right)\\ \end{array} \]

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
 (if (<= (/ t l) -4e+156)
   (asin (/ l (/ (- t) (sqrt 0.5))))
   (if (<= (/ t l) 5e+124)
     (asin
      (sqrt
       (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin
      (* (/ (sqrt 0.5) t) (* (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))) l))))))

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 tmp;
	if ((t / l) <= -4e+156) {
		tmp = asin((l / (-t / sqrt(0.5))));
	} else if ((t / l) <= 5e+124) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin(((sqrt(0.5) / t) * (sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * l)));
	}
	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) :: tmp
    if ((t / l) <= (-4d+156)) then
        tmp = asin((l / (-t / sqrt(0.5d0))))
    else if ((t / l) <= 5d+124) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin(((sqrt(0.5d0) / t) * (sqrt((1.0d0 - ((om / omc) / (omc / om)))) * l)))
    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 tmp;
	if ((t / l) <= -4e+156) {
		tmp = Math.asin((l / (-t / Math.sqrt(0.5))));
	} else if ((t / l) <= 5e+124) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin(((Math.sqrt(0.5) / t) * (Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * l)));
	}
	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):
	tmp = 0
	if (t / l) <= -4e+156:
		tmp = math.asin((l / (-t / math.sqrt(0.5))))
	elif (t / l) <= 5e+124:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin(((math.sqrt(0.5) / t) * (math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * l)))
	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)
	tmp = 0.0
	if (Float64(t / l) <= -4e+156)
		tmp = asin(Float64(l / Float64(Float64(-t) / sqrt(0.5))));
	elseif (Float64(t / l) <= 5e+124)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) / t) * Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))) * l)));
	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)
	tmp = 0.0;
	if ((t / l) <= -4e+156)
		tmp = asin((l / (-t / sqrt(0.5))));
	elseif ((t / l) <= 5e+124)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin(((sqrt(0.5) / t) * (sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * l)));
	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_] := If[LessEqual[N[(t / l), $MachinePrecision], -4e+156], N[ArcSin[N[(l / N[((-t) / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+124], 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[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] * N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.9999999999999999e156

   1. Initial program 34.0

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

   2. Taylor expanded in t around -inf 7.7

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

   3. Simplified 0.2

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

      [Start] 7.7	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      associate-*r* [=>] 7.7	\[ \sin^{-1} \color{blue}{\left(\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 7.7	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right)} \]
      unpow2 [=>] 7.7	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      unpow2 [=>] 7.7	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      times-frac [=>] 0.3	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      unpow2 [<=] 0.3	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      mul-1-neg [=>] 0.3	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)}\right) \]
      associate-/l* [=>] 1.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)\right) \]
      associate-/r/ [=>] 0.2	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{t} \cdot \ell}\right)\right) \]

   4. Taylor expanded in Om around 0 0.6

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

   5. Simplified 0.6

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

      [Start] 0.6	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*r/ [=>] 0.6	\[ \sin^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\sqrt{0.5} \cdot \ell\right)}{t}\right)} \]
      mul-1-neg [=>] 0.6	\[ \sin^{-1} \left(\frac{\color{blue}{-\sqrt{0.5} \cdot \ell}}{t}\right) \]
      distribute-rgt-neg-in [=>] 0.6	\[ \sin^{-1} \left(\frac{\color{blue}{\sqrt{0.5} \cdot \left(-\ell\right)}}{t}\right) \]
      *-commutative [=>] 0.6	\[ \sin^{-1} \left(\frac{\color{blue}{\left(-\ell\right) \cdot \sqrt{0.5}}}{t}\right) \]
      associate-*l/ [<=] 0.6	\[ \sin^{-1} \color{blue}{\left(\frac{-\ell}{t} \cdot \sqrt{0.5}\right)} \]
      *-commutative [=>] 0.6	\[ \sin^{-1} \color{blue}{\left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)} \]

   6. Applied egg-rr 0.6

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

   if -3.9999999999999999e156 < (/.f64 t l) < 4.9999999999999996e124

   1. Initial program 1.1

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

   2. Applied egg-rr 1.0

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

   if 4.9999999999999996e124 < (/.f64 t l)

   1. Initial program 31.5

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

   2. Taylor expanded in t around -inf 42.7

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

   3. Simplified 39.4

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

      [Start] 42.7	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      associate-*r* [=>] 42.7	\[ \sin^{-1} \color{blue}{\left(\left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 42.7	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right)} \]
      unpow2 [=>] 42.7	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      unpow2 [=>] 42.7	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      times-frac [=>] 39.4	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      unpow2 [<=] 39.4	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      mul-1-neg [=>] 39.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)}\right) \]
      associate-/l* [=>] 39.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)\right) \]
      associate-/r/ [=>] 39.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{0.5}}{t} \cdot \ell}\right)\right) \]

   4. Applied egg-rr 0.3

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

   5. Simplified 0.3

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

      [Start] 0.3	\[ \sin^{-1} \left(0 + \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right) \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right) \]
      +-lft-identity [=>] 0.3	\[ \sin^{-1} \color{blue}{\left(\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right) \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]
      associate-*l* [=>] 0.3	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\right)} \]

   6. Applied egg-rr 0.3

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{-t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \ell\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
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	0.9
Cost	20680

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{-t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\sqrt{t_1} \cdot \ell\right)\right)\\ \end{array} \]

Alternative 3
Error	1.0
Cost	14792

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{-t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+99}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \left(\frac{Om}{Omc} \cdot \frac{Om \cdot \ell}{Omc}\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4
Error	1.8
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{-t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	32.9
Cost	13444

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	32.9
Cost	13444

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{-t}{\sqrt{0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	44.1
Cost	13120

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

Reproduce

herbie shell --seed 2023056 
(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)))))))