Toniolo and Linder, Equation (2)

Average Error: 10.4 → 0.8

Time: 14.3s

Precision: binary64

Cost: 26888

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{\sqrt{2}} \cdot \sqrt{t_1}}{t}\right)\\ \end{array} \]

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
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t l) -4e+157)
     (asin
      (* (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))) (* (/ (sqrt 0.5) t) (- l))))
     (if (<= (/ t l) 1e+142)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (/ (* (/ l (sqrt 2.0)) (sqrt t_1)) 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 t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -4e+157) {
		tmp = asin((sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * ((sqrt(0.5) / t) * -l)));
	} else if ((t / l) <= 1e+142) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((((l / sqrt(2.0)) * sqrt(t_1)) / 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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t / l) <= (-4d+157)) then
        tmp = asin((sqrt((1.0d0 - ((om / omc) / (omc / om)))) * ((sqrt(0.5d0) / t) * -l)))
    else if ((t / l) <= 1d+142) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((((l / sqrt(2.0d0)) * sqrt(t_1)) / 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 t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -4e+157) {
		tmp = Math.asin((Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * ((Math.sqrt(0.5) / t) * -l)));
	} else if ((t / l) <= 1e+142) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((((l / Math.sqrt(2.0)) * Math.sqrt(t_1)) / 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):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -4e+157:
		tmp = math.asin((math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * ((math.sqrt(0.5) / t) * -l)))
	elif (t / l) <= 1e+142:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((((l / math.sqrt(2.0)) * math.sqrt(t_1)) / 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)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t / l) <= -4e+157)
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))) * Float64(Float64(sqrt(0.5) / t) * Float64(-l))));
	elseif (Float64(t / l) <= 1e+142)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(Float64(l / sqrt(2.0)) * sqrt(t_1)) / 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)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t / l) <= -4e+157)
		tmp = asin((sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) * ((sqrt(0.5) / t) * -l)));
	elseif ((t / l) <= 1e+142)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((((l / sqrt(2.0)) * sqrt(t_1)) / 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_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -4e+157], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+142], N[ArcSin[N[Sqrt[N[(t$95$1 / 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[(N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.99999999999999993e157

   1. Initial program 33.8

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

   2. Taylor expanded in t around -inf 8.1

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

   3. Simplified 0.2

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)} \]
      Proof
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 0 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 10 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc))))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 10 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 0 points increase in error, 2 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 2 points increase in error, 8 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (sqrt.f64 1/2) (/.f64 t l)))))): 10 points increase in error, 0 points decrease in error
      (asin.f64 (*.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))))): 0 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))))): 0 points increase in error, 10 points decrease in error
      (asin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 8 points increase in error, 2 points decrease in error
      (asin.f64 (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)))))))): 2 points increase in error, 8 points decrease in error

   4. Applied egg-rr 0.2

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

   if -3.99999999999999993e157 < (/.f64 t l) < 1.00000000000000005e142

   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

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

   if 1.00000000000000005e142 < (/.f64 t l)

   1. Initial program 32.9

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

   2. Applied egg-rr 1.5

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

   3. Simplified 1.5

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      Proof
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 0 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 10 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc))))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 10 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 0 points increase in error, 2 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))))) (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 2 points increase in error, 8 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (sqrt.f64 1/2) (/.f64 t l)))))): 10 points increase in error, 0 points decrease in error
      (asin.f64 (*.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))))): 0 points increase in error, 0 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))))): 0 points increase in error, 10 points decrease in error
      (asin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 8 points increase in error, 2 points decrease in error
      (asin.f64 (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)))))))): 2 points increase in error, 8 points decrease in error

   4. Taylor expanded in t around inf 7.8

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

   5. Simplified 7.8

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)} \]
      Proof
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc)))) (/.f64 l (*.f64 (sqrt.f64 2) t)))): 0 points increase in error, 0 points decrease in error
      (asin.f64 (*.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, 4 points decrease in error
      (asin.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))))) (/.f64 l (*.f64 (sqrt.f64 2) t)))): 4 points increase in error, 0 points decrease in error
      (asin.f64 (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

   6. Applied egg-rr 0.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{\sqrt{2}} \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t}\right)\\ \end{array} \]

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	0.8
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+129}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}} \cdot \sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}\right)\\ \end{array} \]

Alternative 3
Error	1.4
Cost	20680

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{Om}{Omc \cdot Omc}} \cdot \frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1} \cdot \sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	20680

\[\begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}} \cdot \sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}\right)\\ \end{array} \]

Alternative 5
Error	5.7
Cost	20420

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}} \cdot \sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}\right)\\ \end{array} \]

Alternative 6
Error	5.8
Cost	14532

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 7
Error	18.1
Cost	14028

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot t\right)}{\ell \cdot \ell}}}\right)\\ \mathbf{if}\;\ell \leq -2.2 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{-162}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 8
Error	23.7
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-59}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{\left(Om \cdot \frac{Om}{Omc}\right) \cdot -0.5}{Omc}\right)\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 9
Error	23.8
Cost	13385

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-58} \lor \neg \left(\ell \leq 2.45 \cdot 10^{-73}\right):\\ \;\;\;\;\sin^{-1} \left(1 + \frac{\left(Om \cdot \frac{Om}{Omc}\right) \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 10
Error	31.9
Cost	7104

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

Alternative 11
Error	32.1
Cost	6464

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

Reproduce

herbie shell --seed 2022343 
(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)))))))