Toniolo and Linder, Equation (2)

Average Error: 10.2 → 0.8

Time: 16.7s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\sqrt{0.5}}{t}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\left(t_1 \cdot \left(-\ell\right)\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot t_1\right)\\ \end{array} \]

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 (/ (sqrt 0.5) t)))
   (if (<= (/ t l) -2e+154)
     (asin (* (* t_1 (- l)) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
     (if (<= (/ t l) 5e+153)
       (asin
        (sqrt
         (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
       (asin (* 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 = sqrt(0.5) / t;
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = asin(((t_1 * -l) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else if ((t / l) <= 5e+153) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((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) :: tmp
    t_1 = sqrt(0.5d0) / t
    if ((t / l) <= (-2d+154)) then
        tmp = asin(((t_1 * -l) * sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    else if ((t / l) <= 5d+153) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((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 = Math.sqrt(0.5) / t;
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = Math.asin(((t_1 * -l) * Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else if ((t / l) <= 5e+153) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((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 = math.sqrt(0.5) / t
	tmp = 0
	if (t / l) <= -2e+154:
		tmp = math.asin(((t_1 * -l) * math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	elif (t / l) <= 5e+153:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((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(sqrt(0.5) / t)
	tmp = 0.0
	if (Float64(t / l) <= -2e+154)
		tmp = asin(Float64(Float64(t_1 * Float64(-l)) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	elseif (Float64(t / l) <= 5e+153)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(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 = sqrt(0.5) / t;
	tmp = 0.0;
	if ((t / l) <= -2e+154)
		tmp = asin(((t_1 * -l) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	elseif ((t / l) <= 5e+153)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((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[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+154], N[ArcSin[N[(N[(t$95$1 * (-l)), $MachinePrecision] * 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], 5e+153], 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[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2.00000000000000007e154

   1. Initial program 33.9

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

   2. Taylor expanded in t around -inf 7.5

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

   3. Simplified 0.2

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 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) (neg.f64 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) (neg.f64 l))): 25 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) (neg.f64 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) (neg.f64 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<= distribute-rgt-neg-in_binary64 (neg.f64 (*.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)))) (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)))): 36 points increase in error, 38 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (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
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.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 t around 0 7.5

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

   5. Simplified 0.2

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)} \]
      Proof
      (*.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l)) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l))) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 39 points increase in error, 41 points decrease in error
      (*.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc)))))): 25 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) 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 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) 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
      (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.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

   if -2.00000000000000007e154 < (/.f64 t l) < 5.00000000000000018e153

   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 0.9

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

   if 5.00000000000000018e153 < (/.f64 t l)

   1. Initial program 34.2

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

   2. Taylor expanded in t around inf 37.2

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

   3. Simplified 34.0

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

   4. Taylor expanded in Om around 0 0.5

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

   5. Simplified 0.5

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)} \]
      Proof
      (*.f64 l (/.f64 (sqrt.f64 1/2) t)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)): 40 points increase in error, 40 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	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	1.1
Cost	26624

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

Alternative 3
Error	11.1
Cost	20616

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{if}\;\ell \leq -1.3188001006228094 \cdot 10^{-226}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.6726243458196315 \cdot 10^{-207}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.2
Cost	20168

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\frac{\ell \cdot \ell}{t}}, 1\right)}}\right)\\ \mathbf{if}\;\ell \leq -2.5794476924957625 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 8.73942859047012 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	25.7
Cost	14300

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ t_2 := \sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{if}\;t \leq -1.2957272213448302 \cdot 10^{-45}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq 9.773266327473208 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq 8.012953079407455 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3452831232783784 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq 1.1628429289754013 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.778282437993681 \cdot 10^{+179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.1141642451405887 \cdot 10^{+217}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	25.4
Cost	14300

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ t_2 := \sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{if}\;t \leq -1.2957272213448302 \cdot 10^{-45}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq 9.773266327473208 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{elif}\;t \leq 8.012953079407455 \cdot 10^{+69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.3452831232783784 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq 1.1628429289754013 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.778282437993681 \cdot 10^{+179}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.1141642451405887 \cdot 10^{+217}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	17.0
Cost	14160

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell \cdot \ell} \cdot \left(t \cdot t\right)}}\right)\\ \mathbf{if}\;t \leq -7.979594363944894 \cdot 10^{+175}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;t \leq -8.970261192921055 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.492190603879028 \cdot 10^{-154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{elif}\;t \leq 2.822620350509152 \cdot 10^{+169}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.1141642451405887 \cdot 10^{+217}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)}\right)\\ \end{array} \]

Alternative 8
Error	25.8
Cost	13648

\[\begin{array}{l} t_1 := \sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{if}\;t \leq -1.2957272213448302 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.773266327473208 \cdot 10^{-13}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;t \leq 1.778282437993681 \cdot 10^{+179}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;t \leq 1.1141642451405887 \cdot 10^{+217}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	21.7
Cost	13644

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.5088045830432677 \cdot 10^{-85}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.1283752222269853 \cdot 10^{-207}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq 1.1087125981518063 \cdot 10^{-105}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \sqrt{\frac{0.5}{t \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 10
Error	23.4
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.5088045830432677 \cdot 10^{-85}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 8.73942859047012 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 11
Error	31.1
Cost	6464

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

Reproduce

herbie shell --seed 2022294 
(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)))))))