Toniolo and Linder, Equation (2)

Average Accuracy: 84.3% → 98.8%

Time: 20.7s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \left(\ell \cdot t_1\right) \cdot -0.5\right)\right)\\ \end{array} \]

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) -5e+135)
     (asin (- (/ (/ 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) <= -5e+135) {
		tmp = asin(-((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;
}

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) <= (-5d+135)) then
        tmp = asin(-((l / t) / sqrt(2.0d0)))
    else if ((t / l) <= 1d+126) then
        tmp = asin(sqrt(((1.0d0 - t_1) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin(((sqrt(0.5d0) / t) * (l + ((l * 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 = Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -5e+135) {
		tmp = Math.asin(-((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) <= -5e+135:
		tmp = math.asin(-((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) <= -5e+135)
		tmp = asin(Float64(-Float64(Float64(l / 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) <= -5e+135)
		tmp = asin(-((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], -5e+135], N[ArcSin[(-N[(N[(l / t), $MachinePrecision] / N[Sqrt[2.0], $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) < -5.00000000000000029e135

   1. Initial program 56.4%

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

   2. Applied egg-rr 97.5%

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

      [Start] 56.4	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      sqrt-div [=>] 56.4	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      div-inv [=>] 56.4	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      add-sqr-sqrt [=>] 56.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]
      hypot-1-def [=>] 56.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]
      *-commutative [=>] 56.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]
      sqrt-prod [=>] 56.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]
      unpow2 [=>] 56.4	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]
      sqrt-prod [=>] 0.0	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)} \cdot \sqrt{2}\right)}\right) \]
      add-sqr-sqrt [<=] 97.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Simplified 97.5%

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

      [Start] 97.5	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 97.5	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [<=] 79.8	\[ \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 79.8	\[ \sin^{-1} \left(\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{Omc \cdot Omc}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 79.8	\[ \sin^{-1} \left(\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      associate-*r/ [=>] 79.8	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      *-rgt-identity [=>] 79.8	\[ \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 79.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 79.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [=>] 97.5	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 97.5	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]

   4. Taylor expanded in t around -inf 82.0%

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

   5. Simplified 82.1%

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

      [Start] 82.0	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 82.0	\[ \sin^{-1} \color{blue}{\left(-\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 82.0	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}}\right) \]
      unpow2 [=>] 82.0	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [=>] 82.0	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      associate-/l/ [<=] 82.1	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \color{blue}{\frac{\frac{\ell}{t}}{\sqrt{2}}}\right) \]

   6. Taylor expanded in Om around 0 99.7%

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

   if -5.00000000000000029e135 < (/.f64 t l) < 9.99999999999999925e125

   1. Initial program 98.7%

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

   2. Applied egg-rr 98.7%

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

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

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

      [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%

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

      [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%

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

      [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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+126}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \left(\ell \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot -0.5\right)\right)\\ \end{array} \]

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

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

Alternative 3
Accuracy	98.4%
Cost	20872

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

Alternative 4
Accuracy	98.1%
Cost	20680

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\sqrt{2}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(-t_1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+25}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 5
Accuracy	98.1%
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 200000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	98.0%
Cost	20616

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(-\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 200000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + \left(\ell \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	97.9%
Cost	20360

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\sqrt{2}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+135}:\\ \;\;\;\;\sin^{-1} \left(-t_1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} t_1\\ \end{array} \]

Alternative 8
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 9
Accuracy	63.5%
Cost	13905

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\sqrt{2}}\\ t_2 := \sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -2.9 \cdot 10^{-54}:\\ \;\;\;\;\sin^{-1} t_1\\ \mathbf{elif}\;\ell \leq -7.6 \cdot 10^{-165} \lor \neg \left(\ell \leq 5.4 \cdot 10^{-64}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(-t_1\right)\\ \end{array} \]

Alternative 10
Accuracy	63.7%
Cost	13712

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\sqrt{2}}\\ t_2 := \sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -1.7 \cdot 10^{-151}:\\ \;\;\;\;\sin^{-1} t_1\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{-164}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-64}:\\ \;\;\;\;\sin^{-1} \left(-t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
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 12
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 13
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 14
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)))))))