Toniolo and Linder, Equation (2)

Average Error: 10.6 → 0.9

Time: 19.7s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{t_1 \cdot 0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

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 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t l) -1e+154)
     (- (asin (/ l (/ t (sqrt (* t_1 0.5))))))
     (if (<= (/ t l) 5e+71)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
       (asin (/ l (* (sqrt 2.0) 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 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+154) {
		tmp = -asin((l / (t / sqrt((t_1 * 0.5)))));
	} else if ((t / l) <= 5e+71) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((l / (sqrt(2.0) * 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 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t / l) <= (-1d+154)) then
        tmp = -asin((l / (t / sqrt((t_1 * 0.5d0)))))
    else if ((t / l) <= 5d+71) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((l / (sqrt(2.0d0) * 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 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+154) {
		tmp = -Math.asin((l / (t / Math.sqrt((t_1 * 0.5)))));
	} else if ((t / l) <= 5e+71) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((l / (Math.sqrt(2.0) * 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 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -1e+154:
		tmp = -math.asin((l / (t / math.sqrt((t_1 * 0.5)))))
	elif (t / l) <= 5e+71:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((l / (math.sqrt(2.0) * 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 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t / l) <= -1e+154)
		tmp = Float64(-asin(Float64(l / Float64(t / sqrt(Float64(t_1 * 0.5))))));
	elseif (Float64(t / l) <= 5e+71)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(l / Float64(sqrt(2.0) * 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 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t / l) <= -1e+154)
		tmp = -asin((l / (t / sqrt((t_1 * 0.5)))));
	elseif ((t / l) <= 5e+71)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((l / (sqrt(2.0) * 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[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+154], (-N[ArcSin[N[(l / N[(t / N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 5e+71], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 35.5

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

   2. Taylor expanded in t around -inf 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 35.5

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

   5. Simplified 0.3

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

      [Start] 35.5	\[ \left(0 - e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)}\right) + 1 \]
      associate-+l- [=>] 35.5	\[ \color{blue}{0 - \left(e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)} - 1\right)} \]
      expm1-def [=>] 0.3	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right)\right)} \]
      expm1-log1p [=>] 0.3	\[ 0 - \color{blue}{\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} \]
      sub0-neg [=>] 0.3	\[ \color{blue}{-\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)} \]
      associate-*r/ [=>] 0.3	\[ -\sin^{-1} \color{blue}{\left(\frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell}{t}\right)} \]
      associate-*l/ [<=] 0.3	\[ -\sin^{-1} \color{blue}{\left(\frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t} \cdot \ell\right)} \]
      *-commutative [=>] 0.3	\[ -\sin^{-1} \color{blue}{\left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)} \]
      *-commutative [=>] 0.3	\[ -\sin^{-1} \left(\ell \cdot \frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{t}\right) \]

   6. Applied egg-rr 0.4

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

   if -1.00000000000000004e154 < (/.f64 t l) < 4.99999999999999972e71

   1. Initial program 1.0

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

   2. Applied egg-rr 1.0

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

   if 4.99999999999999972e71 < (/.f64 t l)

   1. Initial program 25.3

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

   2. Applied egg-rr 1.3

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

   3. Simplified 1.3

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

      [Start] 1.3	\[ \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      associate-*l/ [=>] 1.3	\[ \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}\right)}\right) \]

   4. Taylor expanded in t around inf 8.6

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

   5. Simplified 8.6

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

      [Start] 8.6	\[ \sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      *-commutative [=>] 8.6	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)} \]
      unpow2 [=>] 8.6	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [=>] 8.6	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]

   6. Taylor expanded in Om around 0 0.8

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+154}:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	45632

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{\sqrt{2}}{\ell} \cdot t\right)}\right)\right)\right) \]

Alternative 2
Error	1.1
Cost	32832

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

Alternative 3
Error	1.1
Cost	26624

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

Alternative 4
Error	1.5
Cost	20488

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right) \cdot 0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	1.5
Cost	20488

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right) \cdot 0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 6
Error	1.5
Cost	20488

\[\begin{array}{l} t_1 := \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\\ \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{\frac{t}{t_1}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 7
Error	1.6
Cost	14084

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right) \cdot 0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 8
Error	1.7
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 9
Error	32.8
Cost	13444

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 10
Error	32.8
Cost	13444

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 11
Error	32.8
Cost	13444

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;-\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 12
Error	44.3
Cost	13120

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

Alternative 13
Error	44.3
Cost	13120

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

Reproduce

herbie shell --seed 2022354 
(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)))))))