Toniolo and Linder, Equation (2)

Average Error: 9.9 → 1.0

Time: 20.5s

Precision: binary64

Cost: 14664

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+111}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

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) -4e+111)
   (asin (/ (- (sqrt 0.5)) (/ t l)))
   (if (<= (/ t l) 2e+143)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (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) <= -4e+111) {
		tmp = asin((-sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 2e+143) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} 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) <= (-4d+111)) then
        tmp = asin((-sqrt(0.5d0) / (t / l)))
    else if ((t / l) <= 2d+143) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    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) <= -4e+111) {
		tmp = Math.asin((-Math.sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 2e+143) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} 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) <= -4e+111:
		tmp = math.asin((-math.sqrt(0.5) / (t / l)))
	elif (t / l) <= 2e+143:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	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) <= -4e+111)
		tmp = asin(Float64(Float64(-sqrt(0.5)) / Float64(t / l)));
	elseif (Float64(t / l) <= 2e+143)
		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(t / l)))))));
	else
		tmp = asin(Float64(l * Float64(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) <= -4e+111)
		tmp = asin((-sqrt(0.5) / (t / l)));
	elseif ((t / l) <= 2e+143)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	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], -4e+111], N[ArcSin[N[((-N[Sqrt[0.5], $MachinePrecision]) / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+143], 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[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999983e111

   1. Initial program 28.8

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

   2. Applied egg-rr 1.2

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

   3. Simplified 1.2

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

      [Start] 1.2	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      associate-*r/ [=>] 1.2	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      unpow2 [=>] 1.2	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [<=] 9.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 9.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{Omc \cdot Omc}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 9.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      *-rgt-identity [=>] 9.8	\[ \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 9.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 9.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [=>] 1.2	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 1.2	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   4. Applied egg-rr 1.2

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

   5. Taylor expanded in Om around 0 35.1

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

   6. Simplified 35.3

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

      [Start] 35.1	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{{\left(\sqrt{2}\right)}^{2} \cdot {t}^{2}}{{\ell}^{2}}}}\right) \]
      associate-/l* [=>] 35.1	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{{\left(\sqrt{2}\right)}^{2}}{\frac{{\ell}^{2}}{{t}^{2}}}}}}\right) \]
      associate-/r/ [=>] 35.4	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \color{blue}{\frac{{\left(\sqrt{2}\right)}^{2}}{{\ell}^{2}} \cdot {t}^{2}}}}\right) \]
      unpow2 [=>] 35.4	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{{\ell}^{2}} \cdot {t}^{2}}}\right) \]
      rem-square-sqrt [=>] 35.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{\color{blue}{2}}{{\ell}^{2}} \cdot {t}^{2}}}\right) \]
      unpow2 [=>] 35.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\color{blue}{\ell \cdot \ell}} \cdot {t}^{2}}}\right) \]
      unpow2 [=>] 35.3	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell \cdot \ell} \cdot \color{blue}{\left(t \cdot t\right)}}}\right) \]

   7. Taylor expanded in t around -inf 0.8

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

   8. Simplified 1.6

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

      [Start] 0.8	\[ \sin^{-1} \left(-1 \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      mul-1-neg [=>] 0.8	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
      associate-/l* [=>] 1.6	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   if -3.99999999999999983e111 < (/.f64 t l) < 2e143

   1. Initial program 0.9

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

   2. Applied egg-rr 2.8

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

   3. Applied egg-rr 0.9

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

   4. Applied egg-rr 0.9

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

   if 2e143 < (/.f64 t l)

   1. Initial program 29.8

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

   2. Taylor expanded in t around -inf 35.8

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

   3. Simplified 32.5

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

      [Start] 35.8	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 35.8	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 35.8	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 35.8	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 35.8	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 32.5	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [<=] 32.5	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 32.5	\[ \sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]

   4. Taylor expanded in Om around 0 32.5

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

   5. Applied egg-rr 0.6

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

   6. Simplified 0.6

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

      [Start] 0.6	\[ 0 + \sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \]
      +-lft-identity [=>] 0.6	\[ \color{blue}{\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+111}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

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	1.7
Cost	19712

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

Alternative 3
Error	2.1
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	12.9
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+206} \lor \neg \left(\frac{t}{\ell} \leq 10^{-8}\right):\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \]

Alternative 5
Error	12.9
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	2.2
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	31.3
Cost	7104

\[\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right) \]

Alternative 8
Error	31.6
Cost	6464

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

Reproduce

herbie shell --seed 2023237 
(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)))))))