Toniolo and Linder, Equation (2)

Average Error: 10.5 → 0.9

Time: 20.3s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{0.5}}{t}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \left(-\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_2}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \sqrt{t_2}\right)\\ \end{array} \]

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 (/ (* l (sqrt 0.5)) t)) (t_2 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t l) -5e+158)
     (asin (* t_1 (- (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))))
     (if (<= (/ t l) 4e+94)
       (asin (sqrt (/ t_2 (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
       (asin (* t_1 (sqrt 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 = (l * sqrt(0.5)) / t;
	double t_2 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -5e+158) {
		tmp = asin((t_1 * -sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else if ((t / l) <= 4e+94) {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin((t_1 * sqrt(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 = (l * sqrt(0.5d0)) / t
    t_2 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t / l) <= (-5d+158)) then
        tmp = asin((t_1 * -sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    else if ((t / l) <= 4d+94) then
        tmp = asin(sqrt((t_2 / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin((t_1 * sqrt(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 = (l * Math.sqrt(0.5)) / t;
	double t_2 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -5e+158) {
		tmp = Math.asin((t_1 * -Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else if ((t / l) <= 4e+94) {
		tmp = Math.asin(Math.sqrt((t_2 / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin((t_1 * Math.sqrt(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 = (l * math.sqrt(0.5)) / t
	t_2 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -5e+158:
		tmp = math.asin((t_1 * -math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	elif (t / l) <= 4e+94:
		tmp = math.asin(math.sqrt((t_2 / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin((t_1 * math.sqrt(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(Float64(l * sqrt(0.5)) / t)
	t_2 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t / l) <= -5e+158)
		tmp = asin(Float64(t_1 * Float64(-sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))))));
	elseif (Float64(t / l) <= 4e+94)
		tmp = asin(sqrt(Float64(t_2 / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(t_1 * sqrt(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 = (l * sqrt(0.5)) / t;
	t_2 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t / l) <= -5e+158)
		tmp = asin((t_1 * -sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	elseif ((t / l) <= 4e+94)
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin((t_1 * sqrt(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[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5e+158], N[ArcSin[N[(t$95$1 * (-N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 4e+94], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.9999999999999996e158

   1. Initial program 35.2

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

   2. Taylor expanded in t around -inf 7.4

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

   3. Applied egg-rr 0.3

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

   if -4.9999999999999996e158 < (/.f64 t l) < 4.0000000000000001e94

   1. Initial program 1.1

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

   if 4.0000000000000001e94 < (/.f64 t l)

   1. Initial program 26.3

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

   2. Taylor expanded in t around inf 6.8

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

   3. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (*.f64 (/.f64 (sqrt.f64 1/2) t) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))) (*.f64 (/.f64 (sqrt.f64 1/2) t) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc))))) (*.f64 (/.f64 (sqrt.f64 1/2) t) l)): 17 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)))) (*.f64 (/.f64 (sqrt.f64 1/2) t) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))))) (*.f64 (/.f64 (sqrt.f64 1/2) t) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 23 points increase in error, 37 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in Om around 0 0.3

      \[\leadsto \color{blue}{\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \left(-\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	26888

\[\begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{0.5}}{t}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \left(-\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\frac{\ell}{t} \cdot \left(\ell \cdot 0.5\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \end{array} \]

Alternative 2
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 3
Error	1.6
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\frac{\ell}{t} \cdot \left(\ell \cdot 0.5\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 4
Error	1.3
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \frac{\frac{-\ell}{\sqrt{2}}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Error	1.3
Cost	20680

\[\begin{array}{l} t_1 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 6
Error	1.5
Cost	20680

\[\begin{array}{l} t_1 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\frac{\ell}{t} \cdot \left(\ell \cdot 0.5\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 7
Error	24.1
Cost	14564

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ t_2 := \sin^{-1} \left(\sqrt{\frac{\ell \cdot \ell}{\frac{t \cdot t}{0.5}}}\right)\\ \mathbf{if}\;\ell \leq -6.368342407717429 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -5.170541273801775 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.05781555034143 \cdot 10^{-88}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -4.480292111669076 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7.416680697680594 \cdot 10^{-221}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\ell \leq 9.027944660380968 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.536596423874 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5.489600907821648 \cdot 10^{+21}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.167422908707455 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 8
Error	24.2
Cost	14308

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ t_2 := \sin^{-1} \left(\sqrt{\frac{\ell \cdot \ell}{\frac{t \cdot t}{0.5}}}\right)\\ \mathbf{if}\;\ell \leq -6.368342407717429 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -5.170541273801775 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.05781555034143 \cdot 10^{-88}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -4.480292111669076 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7.416680697680594 \cdot 10^{-221}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\ell \leq 9.027944660380968 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 6.536596423874 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 5.489600907821648 \cdot 10^{+21}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.167422908707455 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 9
Error	23.5
Cost	14176

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ t_2 := \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\\ \mathbf{if}\;\ell \leq -6.368342407717429 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -5.170541273801775 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.05781555034143 \cdot 10^{-88}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -4.480292111669076 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7.416680697680594 \cdot 10^{-221}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 4.13764240951604 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.9340668135933346 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.968636984140756 \cdot 10^{-66}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 10
Error	23.5
Cost	14176

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ t_2 := \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\\ \mathbf{if}\;\ell \leq -6.368342407717429 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -5.170541273801775 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.05781555034143 \cdot 10^{-88}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -4.480292111669076 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 7.416680697680594 \cdot 10^{-221}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\ell \leq 4.13764240951604 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 4.9340668135933346 \cdot 10^{-73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.968636984140756 \cdot 10^{-66}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 11
Error	14.7
Cost	14160

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t \cdot \frac{t}{\ell}}{\ell \cdot 0.5}}}\right)\\ \mathbf{if}\;\ell \leq -1.05781555034143 \cdot 10^{-88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -4.480292111669076 \cdot 10^{-143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq -1.8478182051827075 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.682205398163397 \cdot 10^{-233}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	1.5
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\frac{\ell}{t} \cdot \left(\ell \cdot 0.5\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 13
Error	24.3
Cost	13384

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;t \leq -7.003874561217993 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.739279599132316 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	24.3
Cost	13384

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{if}\;t \leq -7.003874561217993 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.739279599132316 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	31.3
Cost	6464

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

Reproduce

herbie shell --seed 2022315 
(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)))))))