Toniolo and Linder, Equation (2)

Average Error: 10.5 → 0.8

Time: 17.7s

Precision: binary64

Cost: 20680

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100000000000:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \left(-1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot 0.5\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+117}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

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) -100000000000.0)
   (asin (* (* l (/ (sqrt 0.5) t)) (+ -1.0 (* (pow (/ Om Omc) 2.0) 0.5))))
   (if (<= (/ t l) 1e+117)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin
      (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ l (* t (sqrt 2.0))))))))

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) <= -100000000000.0) {
		tmp = asin(((l * (sqrt(0.5) / t)) * (-1.0 + (pow((Om / Omc), 2.0) * 0.5))));
	} else if ((t / l) <= 1e+117) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (l / (t * sqrt(2.0)))));
	}
	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) <= (-100000000000.0d0)) then
        tmp = asin(((l * (sqrt(0.5d0) / t)) * ((-1.0d0) + (((om / omc) ** 2.0d0) * 0.5d0))))
    else if ((t / l) <= 1d+117) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) * (om / omc)))) * (l / (t * sqrt(2.0d0)))))
    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) <= -100000000000.0) {
		tmp = Math.asin(((l * (Math.sqrt(0.5) / t)) * (-1.0 + (Math.pow((Om / Omc), 2.0) * 0.5))));
	} else if ((t / l) <= 1e+117) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (l / (t * Math.sqrt(2.0)))));
	}
	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) <= -100000000000.0:
		tmp = math.asin(((l * (math.sqrt(0.5) / t)) * (-1.0 + (math.pow((Om / Omc), 2.0) * 0.5))))
	elif (t / l) <= 1e+117:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (l / (t * math.sqrt(2.0)))))
	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) <= -100000000000.0)
		tmp = asin(Float64(Float64(l * Float64(sqrt(0.5) / t)) * Float64(-1.0 + Float64((Float64(Om / Omc) ^ 2.0) * 0.5))));
	elseif (Float64(t / l) <= 1e+117)
		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(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(l / Float64(t * sqrt(2.0)))));
	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) <= -100000000000.0)
		tmp = asin(((l * (sqrt(0.5) / t)) * (-1.0 + (((Om / Omc) ^ 2.0) * 0.5))));
	elseif ((t / l) <= 1e+117)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (l / (t * sqrt(2.0)))));
	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], -100000000000.0], N[ArcSin[N[(N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+117], 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[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e11

   1. Initial program 20.6

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

   2. Taylor expanded in Om around 0 38.2

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

   3. Simplified 31.0

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(-0.5 \cdot {\left(\frac{Om}{Omc}\right)}^{2} + 1\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\frac{\ell \cdot \ell}{t}}, 1\right)}}\right)} \]
      Proof
      (*.f64 (+.f64 (*.f64 -1/2 (pow.f64 (/.f64 Om Omc) 2)) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 t (/.f64 (*.f64 l l) t)) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc)))) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 t (/.f64 (*.f64 l l) t)) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc)))) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 t (/.f64 (*.f64 l l) t)) 1)))): 30 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc))) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 t (/.f64 (*.f64 l l) t)) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2)))) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 t (/.f64 (*.f64 l l) t)) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 t (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) t)) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) (pow.f64 l 2))) 1)))): 39 points increase in error, 2 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))) 1) (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (pow.f64 l 2)) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))) 1) (sqrt.f64 (/.f64 1 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2))) 1))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 (*.f64 -1/2 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))) 1) (sqrt.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2)))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2)))))) (*.f64 (*.f64 -1/2 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))) (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2)))))) (Rewrite<= associate-*r*_binary64 (*.f64 -1/2 (*.f64 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)) (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2)))))))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2)))))) (*.f64 -1/2 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 (/.f64 1 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2)))))) (/.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 -inf 0.6

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

   5. Simplified 0.6

      \[\leadsto \sin^{-1} \left(\left(-0.5 \cdot {\left(\frac{Om}{Omc}\right)}^{2} + 1\right) \cdot \color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)}\right) \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l)): 0 points increase in error, 0 points decrease in error
      (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
      (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 36 points increase in error, 30 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 0 points increase in error, 0 points decrease in error

   if -1e11 < (/.f64 t l) < 1.00000000000000005e117

   1. Initial program 1.0

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

   2. Applied egg-rr 1.0

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

   3. Applied egg-rr 1.0

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

   if 1.00000000000000005e117 < (/.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. Applied egg-rr 1.2

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

   3. Applied egg-rr 1.2

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

   4. Taylor expanded in t around inf 8.1

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

   5. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc)))) (/.f64 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))): 26 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
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -100000000000:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \left(-1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot 0.5\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+117}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

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	5.7
Cost	20420

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+117}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 3
Error	9.3
Cost	14408

\[\begin{array}{l} t_1 := \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -40000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{\frac{t \cdot \left(t \cdot 2\right)}{\ell}}{\ell}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + t_1 \cdot -0.5\right) \cdot \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 4
Error	5.7
Cost	14404

\[\begin{array}{l} t_1 := \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - t_1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + t_1 \cdot -0.5\right) \cdot \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\right)\\ \end{array} \]

Alternative 5
Error	17.1
Cost	13764

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.021524521674337 \cdot 10^{+122}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{\frac{t \cdot \left(t \cdot 2\right)}{\ell}}{\ell}}}\right)\\ \end{array} \]

Alternative 6
Error	31.3
Cost	13376

\[\sin^{-1} \left(\mathsf{fma}\left(\frac{-0.5}{Omc}, \frac{Om}{\frac{Omc}{Om}}, 1\right)\right) \]

Alternative 7
Error	31.2
Cost	13376

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

Alternative 8
Error	31.5
Cost	6464

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

Reproduce

herbie shell --seed 2022308 
(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)))))))