Toniolo and Linder, Equation (2)

Average Accuracy: 83.6% → 98.7%

Time: 28.4s

Precision: binary64

Cost: 26756

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\frac{-\ell}{t}}{{0.5}^{-0.5}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{t_1 \cdot 0.5}}}\right)\\ \end{array} \]

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) -1e+157)
     (asin (* (sqrt t_1) (/ (/ (- l) t) (pow 0.5 -0.5))))
     (if (<= (/ t l) 5e+136)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ 1.0 (* (/ l t) (/ l t))))))))
       (asin (/ l (/ t (sqrt (* 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 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+157) {
		tmp = asin((sqrt(t_1) * ((-l / t) / pow(0.5, -0.5))));
	} else if ((t / l) <= 5e+136) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} else {
		tmp = asin((l / (t / sqrt((t_1 * 0.5)))));
	}
	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) <= (-1d+157)) then
        tmp = asin((sqrt(t_1) * ((-l / t) / (0.5d0 ** (-0.5d0)))))
    else if ((t / l) <= 5d+136) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (1.0d0 / ((l / t) * (l / t))))))))
    else
        tmp = asin((l / (t / sqrt((t_1 * 0.5d0)))))
    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) <= -1e+157) {
		tmp = Math.asin((Math.sqrt(t_1) * ((-l / t) / Math.pow(0.5, -0.5))));
	} else if ((t / l) <= 5e+136) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	} else {
		tmp = Math.asin((l / (t / Math.sqrt((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 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -1e+157:
		tmp = math.asin((math.sqrt(t_1) * ((-l / t) / math.pow(0.5, -0.5))))
	elif (t / l) <= 5e+136:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))))
	else:
		tmp = math.asin((l / (t / math.sqrt((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(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t / l) <= -1e+157)
		tmp = asin(Float64(sqrt(t_1) * Float64(Float64(Float64(-l) / t) / (0.5 ^ -0.5))));
	elseif (Float64(t / l) <= 5e+136)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(1.0 / Float64(Float64(l / t) * Float64(l / t))))))));
	else
		tmp = asin(Float64(l / Float64(t / sqrt(Float64(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 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t / l) <= -1e+157)
		tmp = asin((sqrt(t_1) * ((-l / t) / (0.5 ^ -0.5))));
	elseif ((t / l) <= 5e+136)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((l / t) * (l / t))))))));
	else
		tmp = asin((l / (t / sqrt((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[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+157], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[((-l) / t), $MachinePrecision] / N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+136], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(1.0 / N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -9.99999999999999983e156

   1. Initial program 45.8%

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

   2. Applied egg-rr 45.8%

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

      [Start] 45.8	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      unpow2 [=>] 45.8	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]
      div-inv [=>] 45.8	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\left(t \cdot \frac{1}{\ell}\right)} \cdot \frac{t}{\ell}\right)}}\right) \]
      associate-*l* [=>] 45.8	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(t \cdot \left(\frac{1}{\ell} \cdot \frac{t}{\ell}\right)\right)}}}\right) \]
      inv-pow [=>] 45.8	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(t \cdot \left(\color{blue}{{\ell}^{-1}} \cdot \frac{t}{\ell}\right)\right)}}\right) \]

   3. Taylor expanded in t around -inf 88.2%

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

   4. Simplified 99.5%

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

      [Start] 88.2	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 88.2	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 88.2	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      distribute-rgt-neg-in [=>] 88.2	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right)} \]
      unpow2 [=>] 88.2	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      unpow2 [=>] 88.2	\[ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      times-frac [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      unpow2 [<=] 99.6	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \left(-\frac{\sqrt{0.5} \cdot \ell}{t}\right)\right) \]
      *-commutative [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right)\right) \]
      associate-/l* [=>] 99.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\ell}{\frac{t}{\sqrt{0.5}}}}\right)\right) \]

   5. Applied egg-rr 48.6%

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

      [Start] 99.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\right) \]
      add-sqr-sqrt [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}{\frac{t}{\sqrt{0.5}}}\right)\right) \]
      div-inv [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{\ell} \cdot \sqrt{\ell}}{\color{blue}{t \cdot \frac{1}{\sqrt{0.5}}}}\right)\right) \]
      times-frac [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\sqrt{\ell}}{t} \cdot \frac{\sqrt{\ell}}{\frac{1}{\sqrt{0.5}}}}\right)\right) \]
      pow1/2 [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{\ell}}{t} \cdot \frac{\sqrt{\ell}}{\frac{1}{\color{blue}{{0.5}^{0.5}}}}\right)\right) \]
      pow-flip [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{\ell}}{t} \cdot \frac{\sqrt{\ell}}{\color{blue}{{0.5}^{\left(-0.5\right)}}}\right)\right) \]
      metadata-eval [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{\ell}}{t} \cdot \frac{\sqrt{\ell}}{{0.5}^{\color{blue}{-0.5}}}\right)\right) \]

   6. Simplified 99.6%

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

      [Start] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\sqrt{\ell}}{t} \cdot \frac{\sqrt{\ell}}{{0.5}^{-0.5}}\right)\right) \]
      associate-*r/ [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\color{blue}{\frac{\frac{\sqrt{\ell}}{t} \cdot \sqrt{\ell}}{{0.5}^{-0.5}}}\right)\right) \]
      associate-*l/ [=>] 48.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\color{blue}{\frac{\sqrt{\ell} \cdot \sqrt{\ell}}{t}}}{{0.5}^{-0.5}}\right)\right) \]
      rem-square-sqrt [=>] 99.6	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\frac{\color{blue}{\ell}}{t}}{{0.5}^{-0.5}}\right)\right) \]

   if -9.99999999999999983e156 < (/.f64 t l) < 5.0000000000000002e136

   1. Initial program 98.3%

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

   2. Applied egg-rr 98.3%

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

      [Start] 98.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      unpow2 [=>] 98.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}}\right) \]
      clear-num [=>] 98.4	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\color{blue}{\frac{1}{\frac{\ell}{t}}} \cdot \frac{t}{\ell}\right)}}\right) \]
      clear-num [=>] 98.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{1}{\frac{\ell}{t}} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}\right)}}\right) \]
      frac-times [=>] 98.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \color{blue}{\frac{1 \cdot 1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}}\right) \]
      metadata-eval [=>] 98.3	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\color{blue}{1}}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right) \]

   if 5.0000000000000002e136 < (/.f64 t l)

   1. Initial program 51.5%

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

   2. Taylor expanded in t around -inf 38.7%

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

   3. Simplified 38.7%

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

      [Start] 38.7	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 38.7	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 38.7	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 38.7	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 38.7	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      associate-/l* [=>] 38.7	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right) \]

   4. Applied egg-rr 43.7%

      \[\leadsto \sin^{-1} \left(-\color{blue}{\left(0 + \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)}\right) \]
      Proof

      [Start] 38.7	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right) \]
      add-log-exp [=>] 39.5	\[ \sin^{-1} \left(-\color{blue}{\log \left(e^{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)}\right) \]
      *-un-lft-identity [=>] 39.5	\[ \sin^{-1} \left(-\log \color{blue}{\left(1 \cdot e^{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)}\right) \]
      log-prod [=>] 39.5	\[ \sin^{-1} \left(-\color{blue}{\left(\log 1 + \log \left(e^{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)\right)}\right) \]
      metadata-eval [=>] 39.5	\[ \sin^{-1} \left(-\left(\color{blue}{0} + \log \left(e^{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)\right)\right) \]
      add-log-exp [<=] 38.7	\[ \sin^{-1} \left(-\left(0 + \color{blue}{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}}\right)\right) \]
      associate-*r/ [=>] 38.7	\[ \sin^{-1} \left(-\left(0 + \color{blue}{\frac{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}}\right)\right) \]
      div-inv [=>] 38.7	\[ \sin^{-1} \left(-\left(0 + \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \sqrt{0.5}\right) \cdot \frac{1}{\frac{t}{\ell}}}\right)\right) \]
      sqrt-unprod [=>] 38.7	\[ \sin^{-1} \left(-\left(0 + \color{blue}{\sqrt{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot 0.5}} \cdot \frac{1}{\frac{t}{\ell}}\right)\right) \]
      times-frac [=>] 43.8	\[ \sin^{-1} \left(-\left(0 + \sqrt{\left(1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right) \cdot 0.5} \cdot \frac{1}{\frac{t}{\ell}}\right)\right) \]
      pow2 [=>] 43.8	\[ \sin^{-1} \left(-\left(0 + \sqrt{\left(1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}\right) \cdot 0.5} \cdot \frac{1}{\frac{t}{\ell}}\right)\right) \]
      clear-num [<=] 43.7	\[ \sin^{-1} \left(-\left(0 + \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \color{blue}{\frac{\ell}{t}}\right)\right) \]

   5. Simplified 43.7%

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

      [Start] 43.7	\[ \sin^{-1} \left(-\left(0 + \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\right) \]
      +-lft-identity [=>] 43.7	\[ \sin^{-1} \left(-\color{blue}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \frac{\ell}{t}}\right) \]
      associate-*r/ [=>] 43.7	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5} \cdot \ell}{t}}\right) \]
      associate-*l/ [<=] 43.7	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t} \cdot \ell}\right) \]
      *-commutative [=>] 43.7	\[ \sin^{-1} \left(-\color{blue}{\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}}\right) \]
      *-commutative [=>] 43.7	\[ \sin^{-1} \left(-\ell \cdot \frac{\sqrt{\color{blue}{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}{t}\right) \]

   6. Applied egg-rr 99.6%

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

      [Start] 43.7	\[ \sin^{-1} \left(-\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right) \]
      add-sqr-sqrt [=>] 43.7	\[ \sin^{-1} \left(-\color{blue}{\sqrt{\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}} \cdot \sqrt{\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}}}\right) \]
      sqrt-unprod [=>] 44.5	\[ \sin^{-1} \left(-\color{blue}{\sqrt{\left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right) \cdot \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right)}}\right) \]
      sqr-neg [<=] 44.5	\[ \sin^{-1} \left(-\sqrt{\color{blue}{\left(-\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right) \cdot \left(-\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right)}}\right) \]
      sqrt-unprod [<=] 38.6	\[ \sin^{-1} \left(-\color{blue}{\sqrt{-\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}} \cdot \sqrt{-\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}}}\right) \]
      add-sqr-sqrt [<=] 99.6	\[ \sin^{-1} \left(-\color{blue}{\left(-\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}\right)}\right) \]
      distribute-lft-neg-in [=>] 99.6	\[ \sin^{-1} \left(-\color{blue}{\left(-\ell\right) \cdot \frac{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{t}}\right) \]
      clear-num [=>] 99.6	\[ \sin^{-1} \left(-\left(-\ell\right) \cdot \color{blue}{\frac{1}{\frac{t}{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}}\right) \]
      un-div-inv [=>] 99.6	\[ \sin^{-1} \left(-\color{blue}{\frac{-\ell}{\frac{t}{\sqrt{0.5 \cdot \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\frac{-\ell}{t}}{{0.5}^{-0.5}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}\right)\\ \end{array} \]

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	21000

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+136}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{t_1 \cdot 0.5}}}\right)\\ \end{array} \]

Alternative 3
Accuracy	98.5%
Cost	20872

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+16}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+62}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{t_1 \cdot 0.5}}}\right)\\ \end{array} \]

Alternative 4
Accuracy	97.7%
Cost	20872

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+96}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{t_1 \cdot 0.5}}}\right)\\ \end{array} \]

Alternative 5
Accuracy	97.1%
Cost	20488

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]

Alternative 6
Accuracy	97.1%
Cost	20488

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}\right)\\ \end{array} \]

Alternative 7
Accuracy	97.1%
Cost	20488

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}}\right)\\ \end{array} \]

Alternative 8
Accuracy	84.0%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-29}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)\\ \end{array} \]

Alternative 9
Accuracy	86.1%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-29}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell} \cdot \frac{t \cdot t}{\ell}}}\right)\\ \end{array} \]

Alternative 10
Accuracy	80.5%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+202}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 11
Accuracy	80.2%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\left(-\ell\right) \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+202}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 12
Accuracy	80.2%
Cost	13704

\[\begin{array}{l} t_1 := -\sqrt{0.5}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot t_1}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+202}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{t_1}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 13
Accuracy	80.2%
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+202}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 14
Accuracy	63.8%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{-88}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 2.46 \cdot 10^{-85}:\\ \;\;\;\;\sin^{-1} \left(\left(-\ell\right) \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \end{array} \]

Alternative 15
Accuracy	50.5%
Cost	7104

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

Alternative 16
Accuracy	50.1%
Cost	6464

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

Reproduce

herbie shell --seed 2023143 
(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)))))))