Toniolo and Linder, Equation (2)

Average Error: 15.35% → 1.47%

Time: 15.3s

Precision: binary64

Cost: 14664

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

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
 (if (<= (/ t l) -5e+157)
   (asin (* (/ l t) (- (sqrt 0.5))))
   (if (<= (/ t l) 2e+95)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin (/ (* l (sqrt 0.5)) 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 tmp;
	if ((t / l) <= -5e+157) {
		tmp = asin(((l / t) * -sqrt(0.5)));
	} else if ((t / l) <= 2e+95) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin(((l * sqrt(0.5)) / 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) :: tmp
    if ((t / l) <= (-5d+157)) then
        tmp = asin(((l / t) * -sqrt(0.5d0)))
    else if ((t / l) <= 2d+95) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin(((l * sqrt(0.5d0)) / 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 tmp;
	if ((t / l) <= -5e+157) {
		tmp = Math.asin(((l / t) * -Math.sqrt(0.5)));
	} else if ((t / l) <= 2e+95) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / 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):
	tmp = 0
	if (t / l) <= -5e+157:
		tmp = math.asin(((l / t) * -math.sqrt(0.5)))
	elif (t / l) <= 2e+95:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / 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)
	tmp = 0.0
	if (Float64(t / l) <= -5e+157)
		tmp = asin(Float64(Float64(l / t) * Float64(-sqrt(0.5))));
	elseif (Float64(t / l) <= 2e+95)
		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(Float64(l * sqrt(0.5)) / 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)
	tmp = 0.0;
	if ((t / l) <= -5e+157)
		tmp = asin(((l / t) * -sqrt(0.5)));
	elseif ((t / l) <= 2e+95)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin(((l * sqrt(0.5)) / 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_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+157], N[ArcSin[N[(N[(l / t), $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+95], 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[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.99999999999999976e157

   1. Initial program 51.1

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

   2. Taylor expanded in Om around 0 51.1

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

   3. Simplified 51.1

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

      [Start] 51.1	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-*r/ [=>] 51.1	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}\right) \]
      unpow2 [=>] 51.1	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 51.1	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   4. Taylor expanded in t around -inf 0.98

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

   5. Simplified 0.98

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

      [Start] 0.98	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*r/ [=>] 0.98	\[ \sin^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\sqrt{0.5} \cdot \ell\right)}{t}\right)} \]
      *-commutative [=>] 0.98	\[ \sin^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{0.5}\right)}}{t}\right) \]
      associate-*r* [=>] 0.98	\[ \sin^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \ell\right) \cdot \sqrt{0.5}}}{t}\right) \]
      neg-mul-1 [<=] 0.98	\[ \sin^{-1} \left(\frac{\color{blue}{\left(-\ell\right)} \cdot \sqrt{0.5}}{t}\right) \]

   6. Taylor expanded in l around 0 0.98

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

   7. Simplified 1.03

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

      [Start] 0.98	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 2.43	\[ \sin^{-1} \left(-1 \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]
      associate-*r/ [=>] 2.43	\[ \sin^{-1} \color{blue}{\left(\frac{-1 \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
      mul-1-neg [=>] 2.43	\[ \sin^{-1} \left(\frac{\color{blue}{-\sqrt{0.5}}}{\frac{t}{\ell}}\right) \]
      associate-/l* [<=] 0.98	\[ \sin^{-1} \color{blue}{\left(\frac{\left(-\sqrt{0.5}\right) \cdot \ell}{t}\right)} \]
      associate-*r/ [<=] 1.03	\[ \sin^{-1} \color{blue}{\left(\left(-\sqrt{0.5}\right) \cdot \frac{\ell}{t}\right)} \]
      *-commutative [<=] 1.03	\[ \sin^{-1} \color{blue}{\left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)} \]

   if -4.99999999999999976e157 < (/.f64 t l) < 2.00000000000000004e95

   1. Initial program 1.71

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

   2. Applied egg-rr 1.71

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

   3. Applied egg-rr 1.69

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

   if 2.00000000000000004e95 < (/.f64 t l)

   1. Initial program 39.55

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

   2. Taylor expanded in Om around 0 53.29

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

   3. Simplified 53.29

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

      [Start] 53.29	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-*r/ [=>] 53.29	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{2 \cdot {t}^{2}}{{\ell}^{2}}}}}\right) \]
      unpow2 [=>] 53.29	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \color{blue}{\left(t \cdot t\right)}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 53.29	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   4. Taylor expanded in t around -inf 63.06

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

   5. Simplified 63.06

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

      [Start] 63.06	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*r/ [=>] 63.06	\[ \sin^{-1} \color{blue}{\left(\frac{-1 \cdot \left(\sqrt{0.5} \cdot \ell\right)}{t}\right)} \]
      *-commutative [=>] 63.06	\[ \sin^{-1} \left(\frac{-1 \cdot \color{blue}{\left(\ell \cdot \sqrt{0.5}\right)}}{t}\right) \]
      associate-*r* [=>] 63.06	\[ \sin^{-1} \left(\frac{\color{blue}{\left(-1 \cdot \ell\right) \cdot \sqrt{0.5}}}{t}\right) \]
      neg-mul-1 [<=] 63.06	\[ \sin^{-1} \left(\frac{\color{blue}{\left(-\ell\right)} \cdot \sqrt{0.5}}{t}\right) \]

   6. Taylor expanded in l around 0 63.06

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

   7. Simplified 63.06

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

      [Start] 63.06	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 62.99	\[ \sin^{-1} \left(-1 \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]
      associate-*r/ [=>] 62.99	\[ \sin^{-1} \color{blue}{\left(\frac{-1 \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)} \]
      mul-1-neg [=>] 62.99	\[ \sin^{-1} \left(\frac{\color{blue}{-\sqrt{0.5}}}{\frac{t}{\ell}}\right) \]
      associate-/r/ [=>] 63.06	\[ \sin^{-1} \color{blue}{\left(\frac{-\sqrt{0.5}}{t} \cdot \ell\right)} \]
      distribute-neg-frac [<=] 63.06	\[ \sin^{-1} \left(\color{blue}{\left(-\frac{\sqrt{0.5}}{t}\right)} \cdot \ell\right) \]
      *-commutative [<=] 63.06	\[ \sin^{-1} \color{blue}{\left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)} \]

   8. Applied egg-rr 0.97

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

4. Final simplification 1.47

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.69%
Cost	32832

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

Alternative 2
Error	2.58%
Cost	19712

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

Alternative 3
Error	2.17%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+116}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\ell} \cdot \frac{2 \cdot t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	19.67%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+211} \lor \neg \left(\frac{t}{\ell} \leq 0.5\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 5
Error	19.67%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.5:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 6
Error	19.66%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.5:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	3.13%
Cost	13640

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

Alternative 8
Error	2.94%
Cost	13640

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

Alternative 9
Error	2.93%
Cost	13640

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

Alternative 10
Error	48.82%
Cost	6464

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

Reproduce

herbie shell --seed 2023103 
(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)))))))