Toniolo and Linder, Equation (2)

Average Error: 16.8% → 1.14%

Time: 19.1s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{t_1 \cdot 0.5}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \frac{\ell}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}} \cdot -0.5\right)\right)\\ \end{array} \]

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+150)
     (asin (* (* l (sqrt (* t_1 0.5))) (/ -1.0 t)))
     (if (<= (/ t l) 5e+150)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
       (asin
        (*
         (/ (sqrt 0.5) t)
         (+ l (* (/ l (* (/ Omc Om) (/ Omc Om))) -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+150) {
		tmp = asin(((l * sqrt((t_1 * 0.5))) * (-1.0 / t)));
	} else if ((t / l) <= 5e+150) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin(((sqrt(0.5) / t) * (l + ((l / ((Omc / Om) * (Omc / Om))) * -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+150)) then
        tmp = asin(((l * sqrt((t_1 * 0.5d0))) * ((-1.0d0) / t)))
    else if ((t / l) <= 5d+150) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin(((sqrt(0.5d0) / t) * (l + ((l / ((omc / om) * (omc / om))) * (-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+150) {
		tmp = Math.asin(((l * Math.sqrt((t_1 * 0.5))) * (-1.0 / t)));
	} else if ((t / l) <= 5e+150) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin(((Math.sqrt(0.5) / t) * (l + ((l / ((Omc / Om) * (Omc / Om))) * -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+150:
		tmp = math.asin(((l * math.sqrt((t_1 * 0.5))) * (-1.0 / t)))
	elif (t / l) <= 5e+150:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin(((math.sqrt(0.5) / t) * (l + ((l / ((Omc / Om) * (Omc / Om))) * -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+150)
		tmp = asin(Float64(Float64(l * sqrt(Float64(t_1 * 0.5))) * Float64(-1.0 / t)));
	elseif (Float64(t / l) <= 5e+150)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) / t) * Float64(l + Float64(Float64(l / Float64(Float64(Omc / Om) * Float64(Omc / Om))) * -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+150)
		tmp = asin(((l * sqrt((t_1 * 0.5))) * (-1.0 / t)));
	elseif ((t / l) <= 5e+150)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin(((sqrt(0.5) / t) * (l + ((l / ((Omc / Om) * (Omc / Om))) * -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+150], N[ArcSin[N[(N[(l * N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+150], 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[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] * N[(l + N[(N[(l / N[(N[(Omc / Om), $MachinePrecision] * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -9.99999999999999981e149

   1. Initial program 54.92

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

   2. Taylor expanded in t around -inf 12.01

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

   3. Simplified 13.32

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

      [Start] 12.01	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 12.01	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 12.01	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 12.01	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 12.01	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 13.32	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Applied egg-rr 2.01

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

   5. Simplified 0.44

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

      [Start] 2.01	\[ \sin^{-1} \left(-\frac{1}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell}}\right) \]
      associate-/r/ [=>] 0.44	\[ \sin^{-1} \left(-\color{blue}{\frac{1}{t} \cdot \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell\right)}\right) \]
      *-commutative [=>] 0.44	\[ \sin^{-1} \left(-\frac{1}{t} \cdot \color{blue}{\left(\ell \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right)}\right) \]
      *-commutative [=>] 0.44	\[ \sin^{-1} \left(-\frac{1}{t} \cdot \left(\ell \cdot \sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}\right)\right) \]

   if -9.99999999999999981e149 < (/.f64 t l) < 5.00000000000000009e150

   1. Initial program 1.4

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

   if 5.00000000000000009e150 < (/.f64 t l)

   1. Initial program 55.42

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

   2. Taylor expanded in t around inf 11.35

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

   3. Taylor expanded in Om around 0 12

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

   4. Simplified 11.47

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

      [Start] 12	\[ \sin^{-1} \left(-0.5 \cdot \frac{\sqrt{0.5} \cdot \left({Om}^{2} \cdot \ell\right)}{{Omc}^{2} \cdot t} + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 12	\[ \sin^{-1} \left(\color{blue}{\frac{\sqrt{0.5} \cdot \left({Om}^{2} \cdot \ell\right)}{{Omc}^{2} \cdot t} \cdot -0.5} + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [<=] 12	\[ \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left({Om}^{2} \cdot \ell\right)}{\color{blue}{t \cdot {Omc}^{2}}} \cdot -0.5 + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 11.81	\[ \sin^{-1} \left(\color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \frac{{Om}^{2} \cdot \ell}{{Omc}^{2}}\right)} \cdot -0.5 + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*l* [=>] 11.81	\[ \sin^{-1} \left(\color{blue}{\frac{\sqrt{0.5}}{t} \cdot \left(\frac{{Om}^{2} \cdot \ell}{{Omc}^{2}} \cdot -0.5\right)} + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*l/ [<=] 11.82	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\frac{{Om}^{2} \cdot \ell}{{Omc}^{2}} \cdot -0.5\right) + \color{blue}{\frac{\sqrt{0.5}}{t} \cdot \ell}\right) \]
      distribute-lft-out [=>] 11.82	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \left(\frac{{Om}^{2} \cdot \ell}{{Omc}^{2}} \cdot -0.5 + \ell\right)\right)} \]
      *-commutative [=>] 11.82	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\frac{\color{blue}{\ell \cdot {Om}^{2}}}{{Omc}^{2}} \cdot -0.5 + \ell\right)\right) \]
      unpow2 [=>] 11.82	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\frac{\ell \cdot \color{blue}{\left(Om \cdot Om\right)}}{{Omc}^{2}} \cdot -0.5 + \ell\right)\right) \]
      unpow2 [=>] 11.82	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\frac{\ell \cdot \left(Om \cdot Om\right)}{\color{blue}{Omc \cdot Omc}} \cdot -0.5 + \ell\right)\right) \]
      associate-/l* [=>] 11.47	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\color{blue}{\frac{\ell}{\frac{Omc \cdot Omc}{Om \cdot Om}}} \cdot -0.5 + \ell\right)\right) \]

   5. Applied egg-rr 0.56

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

4. Final simplification 1.14

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \frac{\ell}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}} \cdot -0.5\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.68%
Cost	32832

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

Alternative 2
Error	1.35%
Cost	20872

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1.2 \cdot 10^{+50}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{t_1 \cdot 0.5}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 20000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 3
Error	1.13%
Cost	20872

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \sqrt{t_1 \cdot 0.5}\right) \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \frac{\ell}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 4
Error	2.37%
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\left(1 - \frac{\frac{t}{\ell}}{\frac{\ell}{t}}\right) \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 5
Error	2.47%
Cost	20292

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\left(1 - \frac{\frac{t}{\ell}}{\frac{\ell}{t}}\right) \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \frac{\ell}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 6
Error	2.7%
Cost	14536

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-1}{\frac{t}{\ell \cdot \sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\left(1 - \frac{\frac{t}{\ell}}{\frac{\ell}{t}}\right) \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \frac{\ell}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7
Error	2.8%
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \frac{\ell}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 8
Error	2.87%
Cost	14408

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-1}{\frac{t}{\ell \cdot \sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right)}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \frac{\ell}{\frac{Omc}{Om} \cdot \frac{Omc}{Om}} \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 9
Error	37.89%
Cost	13914

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{-63}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -3.7 \cdot 10^{-131} \lor \neg \left(\ell \leq -1.62 \cdot 10^{-164}\right) \land \left(\ell \leq 1.5 \cdot 10^{-123} \lor \neg \left(\ell \leq 3.5 \cdot 10^{-47}\right) \land \ell \leq 4 \cdot 10^{+23}\right):\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 10
Error	37.87%
Cost	13913

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-63}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.9 \cdot 10^{-131}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq -1.62 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{-121} \lor \neg \left(\ell \leq 2.4 \cdot 10^{-48}\right) \land \ell \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 11
Error	2.91%
Cost	13896

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	20.38%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+213}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 13
Error	3.72%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 14
Error	3.5%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 15
Error	3.5%
Cost	13640

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -200000000:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	48.9%
Cost	6464

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

Reproduce

herbie shell --seed 2023121 
(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)))))))