Toniolo and Linder, Equation (2)

Average Error: 9.9 → 1.4

Time: 15.0s

Precision: binary64

Cost: 21000

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ t_2 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \frac{\ell}{t_1}\right)\\ \end{array} \]

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 (* t (sqrt 2.0))) (t_2 (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
   (if (<= (/ t l) -2e+173)
     (asin (* t_2 (/ (- l) t_1)))
     (if (<= (/ t l) 1e+120)
       (asin
        (sqrt
         (/
          (- 1.0 (pow (/ Om Omc) 2.0))
          (+ 1.0 (* 2.0 (/ 1.0 (* (/ l t) (/ l t))))))))
       (asin (* t_2 (/ l t_1)))))))

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 = t * sqrt(2.0);
	double t_2 = sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	double tmp;
	if ((t / l) <= -2e+173) {
		tmp = asin((t_2 * (-l / t_1)));
	} else if ((t / l) <= 1e+120) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} else {
		tmp = asin((t_2 * (l / t_1)));
	}
	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 = t * sqrt(2.0d0)
    t_2 = sqrt((1.0d0 - ((om / omc) * (om / omc))))
    if ((t / l) <= (-2d+173)) then
        tmp = asin((t_2 * (-l / t_1)))
    else if ((t / l) <= 1d+120) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t) * (l / t))))))))
    else
        tmp = asin((t_2 * (l / t_1)))
    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 = t * Math.sqrt(2.0);
	double t_2 = Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	double tmp;
	if ((t / l) <= -2e+173) {
		tmp = Math.asin((t_2 * (-l / t_1)));
	} else if ((t / l) <= 1e+120) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} else {
		tmp = Math.asin((t_2 * (l / t_1)));
	}
	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 = t * math.sqrt(2.0)
	t_2 = math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))
	tmp = 0
	if (t / l) <= -2e+173:
		tmp = math.asin((t_2 * (-l / t_1)))
	elif (t / l) <= 1e+120:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))))
	else:
		tmp = math.asin((t_2 * (l / t_1)))
	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(t * sqrt(2.0))
	t_2 = sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))
	tmp = 0.0
	if (Float64(t / l) <= -2e+173)
		tmp = asin(Float64(t_2 * Float64(Float64(-l) / t_1)));
	elseif (Float64(t / l) <= 1e+120)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(1.0 / Float64(Float64(l / t) * Float64(l / t))))))));
	else
		tmp = asin(Float64(t_2 * Float64(l / t_1)));
	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 = t * sqrt(2.0);
	t_2 = sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	tmp = 0.0;
	if ((t / l) <= -2e+173)
		tmp = asin((t_2 * (-l / t_1)));
	elseif ((t / l) <= 1e+120)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	else
		tmp = asin((t_2 * (l / t_1)));
	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[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+173], N[ArcSin[N[(t$95$2 * N[((-l) / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+120], 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[(1.0 / N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$2 * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e173

   1. Initial program 30.3

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

   2. Applied egg-rr 1.8

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

   3. Applied egg-rr 1.8

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

   4. Taylor expanded in t around -inf 9.0

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

   5. Simplified 0.2

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{-\ell}{\sqrt{2} \cdot t}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc)))) (/.f64 (neg.f64 l) (*.f64 (sqrt.f64 2) t))): 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 (neg.f64 l) (*.f64 (sqrt.f64 2) t))): 37 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 (neg.f64 l) (*.f64 (sqrt.f64 2) t))): 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 (neg.f64 l) (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (/.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 l)) (*.f64 (sqrt.f64 2) t))): 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-*r/_binary64 (*.f64 -1 (/.f64 l (*.f64 (sqrt.f64 2) t))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1 (/.f64 l (*.f64 (sqrt.f64 2) t))) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (/.f64 l (*.f64 (sqrt.f64 2) t)) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error

   if -2e173 < (/.f64 t l) < 9.9999999999999998e119

   1. Initial program 1.9

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

   2. Applied egg-rr 1.9

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

   if 9.9999999999999998e119 < (/.f64 t l)

   1. Initial program 29.1

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

   2. Applied egg-rr 1.4

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

   3. Taylor expanded in t around inf 8.6

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

   4. Simplified 0.3

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

   5. Taylor expanded in l around 0 8.6

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

   6. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)} \]
      Proof
      (*.f64 (/.f64 l (*.f64 (sqrt.f64 2) t)) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l (*.f64 (sqrt.f64 2) t)) (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc)))))): 37 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l (*.f64 (sqrt.f64 2) t)) (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 l (*.f64 (sqrt.f64 2) t)) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

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	1.9
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{Om}{Omc \cdot Omc}} \cdot \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 3
Error	1.4
Cost	20680

\[\begin{array}{l} t_1 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	5.4
Cost	20420

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Error	5.3
Cost	20420

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	5.4
Cost	14404

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7
Error	5.4
Cost	14404

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8
Error	12.4
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right) + -1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 9
Error	10.8
Cost	13892

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{\frac{t \cdot \left(t \cdot 2\right)}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 10
Error	13.0
Cost	13640

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -5.028329886114347 \cdot 10^{+220}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.014440800698129076:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	12.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 12
Error	12.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 13
Error	12.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right) + -1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 14
Error	31.7
Cost	6464

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

Reproduce

herbie shell --seed 2022295 
(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)))))))