Toniolo and Linder, Equation (2)

Average Accuracy: 84.3% → 98.7%

Time: 23.1s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \left(\ell \cdot t_1\right) \cdot -0.5\right)\right)\\ \end{array} \]

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) -2e+69)
     (asin (- (* (hypot 1.0 (/ Om Omc)) (/ l (* t (sqrt 2.0))))))
     (if (<= (/ t l) 1e+126)
       (asin (sqrt (/ (- 1.0 t_1) (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (* (/ (sqrt 0.5) t) (+ l (* (* l t_1) -0.5))))))))

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) <= -2e+69) {
		tmp = asin(-(hypot(1.0, (Om / Omc)) * (l / (t * sqrt(2.0)))));
	} else if ((t / l) <= 1e+126) {
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin(((sqrt(0.5) / t) * (l + ((l * t_1) * -0.5))));
	}
	return tmp;
}

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) <= -2e+69) {
		tmp = Math.asin(-(Math.hypot(1.0, (Om / Omc)) * (l / (t * Math.sqrt(2.0)))));
	} else if ((t / l) <= 1e+126) {
		tmp = Math.asin(Math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin(((Math.sqrt(0.5) / t) * (l + ((l * t_1) * -0.5))));
	}
	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) <= -2e+69:
		tmp = math.asin(-(math.hypot(1.0, (Om / Omc)) * (l / (t * math.sqrt(2.0)))))
	elif (t / l) <= 1e+126:
		tmp = math.asin(math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin(((math.sqrt(0.5) / t) * (l + ((l * t_1) * -0.5))))
	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) <= -2e+69)
		tmp = asin(Float64(-Float64(hypot(1.0, Float64(Om / Omc)) * Float64(l / Float64(t * sqrt(2.0))))));
	elseif (Float64(t / l) <= 1e+126)
		tmp = asin(sqrt(Float64(Float64(1.0 - t_1) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) / t) * Float64(l + Float64(Float64(l * t_1) * -0.5))));
	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) <= -2e+69)
		tmp = asin(-(hypot(1.0, (Om / Omc)) * (l / (t * sqrt(2.0)))));
	elseif ((t / l) <= 1e+126)
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin(((sqrt(0.5) / t) * (l + ((l * t_1) * -0.5))));
	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], -2e+69], N[ArcSin[(-N[(N[Sqrt[1.0 ^ 2 + N[(Om / Omc), $MachinePrecision] ^ 2], $MachinePrecision] * N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+126], N[ArcSin[N[Sqrt[N[(N[(1.0 - t$95$1), $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[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] * N[(l + N[(N[(l * t$95$1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2.0000000000000001e69

   1. Initial program 70.1%

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

   2. Applied egg-rr 67.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\left(\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) \cdot 0.5}\right)} \]
      Step-by-step derivation

      [Start] 70.1	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      pow1/2 [=>] 70.1	\[ \sin^{-1} \color{blue}{\left({\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}^{0.5}\right)} \]
      pow-to-exp [=>] 67.1	\[ \sin^{-1} \color{blue}{\left(e^{\log \left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right) \cdot 0.5}\right)} \]
      log-div [=>] 67.1	\[ \sin^{-1} \left(e^{\color{blue}{\left(\log \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) - \log \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)} \cdot 0.5}\right) \]
      sub-neg [=>] 67.1	\[ \sin^{-1} \left(e^{\left(\log \color{blue}{\left(1 + \left(-{\left(\frac{Om}{Omc}\right)}^{2}\right)\right)} - \log \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) \cdot 0.5}\right) \]
      log1p-def [=>] 67.1	\[ \sin^{-1} \left(e^{\left(\color{blue}{\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right)} - \log \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) \cdot 0.5}\right) \]
      log1p-udef [<=] 67.1	\[ \sin^{-1} \left(e^{\left(\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \color{blue}{\mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right) \cdot 0.5}\right) \]

   3. Simplified 67.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{e^{\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)} \]
      Step-by-step derivation

      [Start] 67.1	\[ \sin^{-1} \left(e^{\left(\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) \cdot 0.5}\right) \]
      exp-prod [=>] 67.1	\[ \sin^{-1} \color{blue}{\left({\left(e^{\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}^{0.5}\right)} \]
      unpow1/2 [=>] 67.1	\[ \sin^{-1} \color{blue}{\left(\sqrt{e^{\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)} \]

   4. Applied egg-rr 35.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} - 1\right)} \]
      Step-by-step derivation

      [Start] 67.1	\[ \sin^{-1} \left(\sqrt{e^{\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right) \]
      expm1-log1p-u [=>] 67.1	\[ \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{e^{\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\right)\right)} \]
      expm1-udef [=>] 35.8	\[ \sin^{-1} \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{e^{\mathsf{log1p}\left(-{\left(\frac{Om}{Omc}\right)}^{2}\right) - \mathsf{log1p}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)} - 1\right)} \]

   5. Simplified 98.0%

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

      [Start] 35.8	\[ \sin^{-1} \left(e^{\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} - 1\right) \]
      expm1-def [=>] 98.0	\[ \sin^{-1} \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right)} \]
      expm1-log1p [=>] 98.0	\[ \sin^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]

   6. Taylor expanded in t around -inf 83.6%

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

   7. Simplified 99.6%

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

      [Start] 83.6	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 + \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      associate-*r* [=>] 83.6	\[ \sin^{-1} \color{blue}{\left(\left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \cdot \sqrt{1 + \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 83.6	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 + \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right)} \]
      unpow2 [=>] 83.6	\[ \sin^{-1} \left(\sqrt{1 + \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      unpow2 [=>] 83.6	\[ \sin^{-1} \left(\sqrt{1 + \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      times-frac [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 + \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      hypot-1-def [=>] 99.6	\[ \sin^{-1} \left(\color{blue}{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)} \cdot \left(-1 \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)\right) \]
      associate-*r/ [=>] 99.6	\[ \sin^{-1} \left(\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \color{blue}{\frac{-1 \cdot \ell}{\sqrt{2} \cdot t}}\right) \]
      mul-1-neg [=>] 99.6	\[ \sin^{-1} \left(\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\color{blue}{-\ell}}{\sqrt{2} \cdot t}\right) \]

   if -2.0000000000000001e69 < (/.f64 t l) < 9.99999999999999925e125

   1. Initial program 98.6%

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

   2. Applied egg-rr 98.7%

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

      [Start] 98.6	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      unpow2 [=>] 98.6	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]
      clear-num [=>] 98.6	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right)}}\right) \]
      un-div-inv [=>] 98.7	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}}\right) \]

   if 9.99999999999999925e125 < (/.f64 t l)

   1. Initial program 47.3%

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

   2. Applied egg-rr 47.3%

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

      [Start] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      unpow2 [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]
      clear-num [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right)}}\right) \]
      clear-num [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{1}{\frac{\ell}{t}} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right)}}\right) \]
      frac-times [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{1 \cdot 1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}}\right) \]
      metadata-eval [=>] 47.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\color{blue}{1}}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right) \]

   3. Taylor expanded in l around 0 83.5%

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

   4. Simplified 99.7%

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

      [Start] 83.5	\[ \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      associate-/l* [=>] 79.1	\[ \sin^{-1} \left(\color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      associate-/r/ [=>] 83.5	\[ \sin^{-1} \left(\color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      unpow2 [=>] 83.5	\[ \sin^{-1} \left(\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right) \cdot \sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]
      unpow2 [=>] 83.5	\[ \sin^{-1} \left(\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right) \cdot \sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]
      times-frac [=>] 99.7	\[ \sin^{-1} \left(\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right) \cdot \sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]
      unpow2 [<=] 99.7	\[ \sin^{-1} \left(\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right) \cdot \sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

   5. Taylor expanded in Om around 0 83.5%

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

   6. Simplified 99.7%

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

      [Start] 83.5	\[ \sin^{-1} \left(-0.5 \cdot \frac{\sqrt{0.5} \cdot \left({Om}^{2} \cdot \ell\right)}{{Omc}^{2} \cdot t} + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 83.5	\[ \sin^{-1} \left(\color{blue}{\frac{\sqrt{0.5} \cdot \left({Om}^{2} \cdot \ell\right)}{{Omc}^{2} \cdot t} \cdot -0.5} + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 83.5	\[ \sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left({Om}^{2} \cdot \ell\right)}{\color{blue}{t \cdot {Omc}^{2}}} \cdot -0.5 + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 83.5	\[ \sin^{-1} \left(\color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \frac{{Om}^{2} \cdot \ell}{{Omc}^{2}}\right)} \cdot -0.5 + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*l* [=>] 83.5	\[ \sin^{-1} \left(\color{blue}{\frac{\sqrt{0.5}}{t} \cdot \left(\frac{{Om}^{2} \cdot \ell}{{Omc}^{2}} \cdot -0.5\right)} + \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-*l/ [<=] 83.5	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\frac{{Om}^{2} \cdot \ell}{{Omc}^{2}} \cdot -0.5\right) + \color{blue}{\frac{\sqrt{0.5}}{t} \cdot \ell}\right) \]
      distribute-lft-out [=>] 83.5	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \left(\frac{{Om}^{2} \cdot \ell}{{Omc}^{2}} \cdot -0.5 + \ell\right)\right)} \]
      associate-/l* [=>] 83.5	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\color{blue}{\frac{{Om}^{2}}{\frac{{Omc}^{2}}{\ell}}} \cdot -0.5 + \ell\right)\right) \]
      associate-/r/ [=>] 83.5	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\color{blue}{\left(\frac{{Om}^{2}}{{Omc}^{2}} \cdot \ell\right)} \cdot -0.5 + \ell\right)\right) \]
      unpow2 [=>] 83.5	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\left(\frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}} \cdot \ell\right) \cdot -0.5 + \ell\right)\right) \]
      unpow2 [=>] 83.5	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\left(\frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}} \cdot \ell\right) \cdot -0.5 + \ell\right)\right) \]
      times-frac [=>] 99.7	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\left(\color{blue}{\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)} \cdot \ell\right) \cdot -0.5 + \ell\right)\right) \]
      unpow2 [<=] 99.7	\[ \sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\left(\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} \cdot \ell\right) \cdot -0.5 + \ell\right)\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \left(\ell \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.4%
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
Accuracy	98.6%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+45}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	98.4%
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 200000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	98.1%
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+25}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5
Accuracy	98.0%
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 200000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	98.0%
Cost	20616

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 200000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \left(\ell \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	97.9%
Cost	20360

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 8
Accuracy	97.9%
Cost	20360

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(-\mathsf{hypot}\left(1, \frac{Om}{Omc}\right) \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 9
Accuracy	75.5%
Cost	14160

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}}}\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+186}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-270}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\ell \leq 3.1 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 10
Accuracy	63.5%
Cost	13905

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-54}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\ell \leq -7.6 \cdot 10^{-165} \lor \neg \left(\ell \leq 5.4 \cdot 10^{-64}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 11
Accuracy	63.7%
Cost	13712

\[\begin{array}{l} t_1 := \sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{-151}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-64}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	63.6%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \cdot 10^{-22} \lor \neg \left(\ell \leq 2.9 \cdot 10^{-36}\right):\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 13
Accuracy	54.9%
Cost	7369

\[\begin{array}{l} \mathbf{if}\;\ell \leq -6.9 \cdot 10^{-165} \lor \neg \left(\ell \leq 7 \cdot 10^{-182}\right):\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(-0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}\right)\\ \end{array} \]

Alternative 14
Accuracy	54.5%
Cost	7240

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{-165}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{-178}:\\ \;\;\;\;\sin^{-1} \left(-0.5 \cdot \frac{Om \cdot Om}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 15
Accuracy	51.1%
Cost	6464

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

Reproduce

herbie shell --seed 2023157 
(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)))))))