Toniolo and Linder, Equation (2)

Average Error: 10.3 → 1.0

Time: 14.5s

Precision: binary64

Cost: 27336

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+159}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+27}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 + \left(1 + \left(-1 - t_1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - t_1} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

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
 (let* ((t_1 (pow (/ Om Omc) 2.0)))
   (if (<= (/ t l) -1e+159)
     (asin
      (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (- l) (* t (sqrt 2.0)))))
     (if (<= (/ t l) 1e+27)
       (asin
        (sqrt
         (/ (+ 1.0 (+ 1.0 (- -1.0 t_1))) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
       (asin (* (sqrt (- 1.0 t_1)) (/ (* 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 t_1 = pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+159) {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (-l / (t * sqrt(2.0)))));
	} else if ((t / l) <= 1e+27) {
		tmp = asin(sqrt(((1.0 + (1.0 + (-1.0 - t_1))) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin((sqrt((1.0 - t_1)) * ((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) :: t_1
    real(8) :: tmp
    t_1 = (om / omc) ** 2.0d0
    if ((t / l) <= (-1d+159)) then
        tmp = asin((sqrt((1.0d0 - ((om / omc) * (om / omc)))) * (-l / (t * sqrt(2.0d0)))))
    else if ((t / l) <= 1d+27) then
        tmp = asin(sqrt(((1.0d0 + (1.0d0 + ((-1.0d0) - t_1))) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin((sqrt((1.0d0 - t_1)) * ((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 t_1 = Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+159) {
		tmp = Math.asin((Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (-l / (t * Math.sqrt(2.0)))));
	} else if ((t / l) <= 1e+27) {
		tmp = Math.asin(Math.sqrt(((1.0 + (1.0 + (-1.0 - t_1))) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - t_1)) * ((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):
	t_1 = math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -1e+159:
		tmp = math.asin((math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (-l / (t * math.sqrt(2.0)))))
	elif (t / l) <= 1e+27:
		tmp = math.asin(math.sqrt(((1.0 + (1.0 + (-1.0 - t_1))) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - t_1)) * ((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)
	t_1 = Float64(Om / Omc) ^ 2.0
	tmp = 0.0
	if (Float64(t / l) <= -1e+159)
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(-l) / Float64(t * sqrt(2.0)))));
	elseif (Float64(t / l) <= 1e+27)
		tmp = asin(sqrt(Float64(Float64(1.0 + Float64(1.0 + Float64(-1.0 - t_1))) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - t_1)) * Float64(Float64(l * 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)
	t_1 = (Om / Omc) ^ 2.0;
	tmp = 0.0;
	if ((t / l) <= -1e+159)
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (-l / (t * sqrt(2.0)))));
	elseif ((t / l) <= 1e+27)
		tmp = asin(sqrt(((1.0 + (1.0 + (-1.0 - t_1))) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin((sqrt((1.0 - t_1)) * ((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_] := Block[{t$95$1 = N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+159], 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], If[LessEqual[N[(t / l), $MachinePrecision], 1e+27], N[ArcSin[N[Sqrt[N[(N[(1.0 + N[(1.0 + N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -9.9999999999999993e158

   1. Initial program 33.5

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

   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 [<=] 8.0	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 8.0	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{Omc \cdot Omc}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 8.0	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      *-rgt-identity [=>] 8.0	\[ \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 8.0	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 8.0	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      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. Taylor expanded in t around -inf 7.1

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

   5. Simplified 7.1

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

      [Start] 7.1	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 7.1	\[ \sin^{-1} \color{blue}{\left(-\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 7.1	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}}\right) \]
      unpow2 [=>] 7.1	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [=>] 7.1	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]

   6. Applied egg-rr 0.2

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

   if -9.9999999999999993e158 < (/.f64 t l) < 1e27

   1. Initial program 1.3

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

   2. Applied egg-rr 1.3

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

   if 1e27 < (/.f64 t l)

   1. Initial program 20.3

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

   2. Taylor expanded in t around inf 7.3

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

   3. Simplified 0.3

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

      [Start] 7.3	\[ \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      *-commutative [=>] 7.3	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]
      unpow2 [=>] 7.3	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 7.3	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 0.3	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [<=] 0.3	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+159}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+27}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 + \left(1 + \left(-1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	45632

\[\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right) \]

Alternative 2
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 3
Error	1.0
Cost	20484

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+159}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+113}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 4
Error	0.9
Cost	14664

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{0.5} \cdot \frac{\ell}{t}\right) \cdot \left(-1 + \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+50}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5
Error	0.8
Cost	14664

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{0.5} \cdot \frac{\ell}{t}\right) \cdot \left(-1 + \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot 0.5\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+113}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Error	1.9
Cost	14148

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

Alternative 7
Error	1.9
Cost	14148

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

Alternative 8
Error	12.6
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.16 \cdot 10^{+205} \lor \neg \left(\frac{t}{\ell} \leq 0.7\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot \frac{Om}{Omc}}{\frac{Omc}{-0.5}}\right)\\ \end{array} \]

Alternative 9
Error	12.6
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot \frac{Om}{Omc}}{\frac{Omc}{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10
Error	1.9
Cost	13640

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

Alternative 11
Error	31.5
Cost	7104

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

Alternative 12
Error	31.7
Cost	6464

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

Reproduce

herbie shell --seed 2023054 
(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)))))))