Toniolo and Linder, Equation (2)

Average Accuracy: 83.9% → 98.6%

Time: 19.5s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+137}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

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
 (if (<= (/ t l) -2e+156)
   (asin (/ (- l) (* (sqrt 2.0) t)))
   (if (<= (/ t l) 4e+137)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (/ (/ l t) (sqrt 2.0))))))

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 tmp;
	if ((t / l) <= -2e+156) {
		tmp = asin((-l / (sqrt(2.0) * t)));
	} else if ((t / l) <= 4e+137) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin(((l / t) / sqrt(2.0)));
	}
	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) :: tmp
    if ((t / l) <= (-2d+156)) then
        tmp = asin((-l / (sqrt(2.0d0) * t)))
    else if ((t / l) <= 4d+137) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin(((l / t) / sqrt(2.0d0)))
    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 tmp;
	if ((t / l) <= -2e+156) {
		tmp = Math.asin((-l / (Math.sqrt(2.0) * t)));
	} else if ((t / l) <= 4e+137) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin(((l / t) / Math.sqrt(2.0)));
	}
	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):
	tmp = 0
	if (t / l) <= -2e+156:
		tmp = math.asin((-l / (math.sqrt(2.0) * t)))
	elif (t / l) <= 4e+137:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin(((l / t) / math.sqrt(2.0)))
	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)
	tmp = 0.0
	if (Float64(t / l) <= -2e+156)
		tmp = asin(Float64(Float64(-l) / Float64(sqrt(2.0) * t)));
	elseif (Float64(t / l) <= 4e+137)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(l / t) / sqrt(2.0)));
	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)
	tmp = 0.0;
	if ((t / l) <= -2e+156)
		tmp = asin((-l / (sqrt(2.0) * t)));
	elseif ((t / l) <= 4e+137)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin(((l / t) / sqrt(2.0)));
	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_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+156], N[ArcSin[N[((-l) / N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 4e+137], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $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[(l / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 48.1%

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

   2. Applied egg-rr 97.8%

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

      [Start] 48.1	\[ \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
      sqrt-div [=>] 48.1	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      div-inv [=>] 48.1	\[ \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)} \]
      add-sqr-sqrt [=>] 48.1	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}}\right) \]
      hypot-1-def [=>] 48.1	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}}\right) \]
      *-commutative [=>] 48.1	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2}}\right)}\right) \]
      sqrt-prod [=>] 48.1	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{{\left(\frac{t}{\ell}\right)}^{2}} \cdot \sqrt{2}}\right)}\right) \]
      unpow2 [=>] 48.1	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}} \cdot \sqrt{2}\right)}\right) \]
      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 [<=] 97.8	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell}} \cdot \sqrt{2}\right)}\right) \]

   3. Simplified 97.8%

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

      [Start] 97.8	\[ \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      associate-*r/ [=>] 97.8	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
      unpow2 [=>] 97.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [<=] 86.1	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om \cdot Om}{Omc \cdot Omc}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 86.1	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{{Omc}^{2}}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 86.1	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{{Om}^{2}}}{{Omc}^{2}}} \cdot 1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      *-rgt-identity [=>] 86.1	\[ \sin^{-1} \left(\frac{\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 86.1	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [=>] 86.1	\[ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      times-frac [=>] 97.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      unpow2 [<=] 97.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right) \]
      associate-*l/ [=>] 97.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}\right)}\right) \]
      associate-/l* [=>] 97.8	\[ \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\frac{\ell}{\sqrt{2}}}}\right)}\right) \]

   4. Taylor expanded in t around -inf 87.6%

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

   5. Simplified 87.6%

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

      [Start] 87.6	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 87.6	\[ \sin^{-1} \color{blue}{\left(-\frac{\ell}{\sqrt{2} \cdot t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 87.6	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}}\right) \]
      unpow2 [=>] 87.6	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]
      unpow2 [=>] 87.6	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right) \]

   6. Taylor expanded in Om around 0 99.1%

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

   if -2e156 < (/.f64 t l) < 4.0000000000000001e137

   1. Initial program 98.4%

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

   2. Applied egg-rr 98.4%

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

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

   if 4.0000000000000001e137 < (/.f64 t l)

   1. Initial program 50.5%

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

   2. Taylor expanded in Om around 0 47.2%

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

   3. Simplified 47.2%

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

      [Start] 47.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 47.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\color{blue}{t \cdot t}}{{\ell}^{2}}}}\right) \]
      unpow2 [=>] 47.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}}}\right) \]

   4. Applied egg-rr 97.8%

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

      [Start] 47.2	\[ \sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right) \]
      sqrt-div [=>] 47.2	\[ \sin^{-1} \color{blue}{\left(\frac{\sqrt{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right)} \]
      metadata-eval [=>] 47.2	\[ \sin^{-1} \left(\frac{\color{blue}{1}}{\sqrt{1 + 2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}\right) \]
      add-sqr-sqrt [=>] 47.2	\[ \sin^{-1} \left(\frac{1}{\sqrt{1 + \color{blue}{\sqrt{2 \cdot \frac{t \cdot t}{\ell \cdot \ell}} \cdot \sqrt{2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}}}}\right) \]
      hypot-1-def [=>] 47.2	\[ \sin^{-1} \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, \sqrt{2 \cdot \frac{t \cdot t}{\ell \cdot \ell}}\right)}}\right) \]
      sqrt-prod [=>] 47.2	\[ \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \color{blue}{\sqrt{2} \cdot \sqrt{\frac{t \cdot t}{\ell \cdot \ell}}}\right)}\right) \]
      times-frac [=>] 50.5	\[ \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \sqrt{\color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}\right)}\right) \]
      sqrt-prod [=>] 97.6	\[ \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \color{blue}{\left(\sqrt{\frac{t}{\ell}} \cdot \sqrt{\frac{t}{\ell}}\right)}\right)}\right) \]
      add-sqr-sqrt [<=] 97.8	\[ \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \color{blue}{\frac{t}{\ell}}\right)}\right) \]

   5. Simplified 97.8%

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

      [Start] 97.8	\[ \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \frac{t}{\ell}\right)}\right) \]
      *-commutative [=>] 97.8	\[ \sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\ell} \cdot \sqrt{2}}\right)}\right) \]

   6. Taylor expanded in t around inf 99.0%

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

   7. Simplified 99.0%

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

      [Start] 99.0	\[ \sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right) \]
      *-commutative [=>] 99.0	\[ \sin^{-1} \left(\frac{\ell}{\color{blue}{t \cdot \sqrt{2}}}\right) \]
      associate-/r* [=>] 99.0	\[ \sin^{-1} \color{blue}{\left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+137}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	45632

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

Alternative 2
Accuracy	98.3%
Cost	32832

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

Alternative 3
Accuracy	98.3%
Cost	26624

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

Alternative 4
Accuracy	97.3%
Cost	20360

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\left(1 + \sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{\sqrt{2}}{\frac{\ell}{t}}\right)}\right)\\ \end{array} \]

Alternative 5
Accuracy	97.3%
Cost	20232

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \frac{t}{\ell}\right)}\right)\\ \end{array} \]

Alternative 6
Accuracy	97.3%
Cost	20232

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{\sqrt{2}}{\frac{\ell}{t}}\right)}\right)\\ \end{array} \]

Alternative 7
Accuracy	97.8%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+131}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 8
Accuracy	97.0%
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 9
Accuracy	79.5%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{1}{\frac{\ell \cdot \ell}{t \cdot t}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 10
Accuracy	79.5%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{1}{\frac{\ell \cdot \ell}{t \cdot t}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 11
Accuracy	96.7%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -10000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\sqrt{2} \cdot t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 12
Accuracy	59.6%
Cost	7497

\[\begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-52} \lor \neg \left(t \leq 1.72 \cdot 10^{-96}\right):\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{1}{\frac{\ell \cdot \ell}{t \cdot t}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om \cdot -0.5}{\frac{Omc}{\frac{Om}{Omc}}}\right)\\ \end{array} \]

Alternative 13
Accuracy	50.0%
Cost	7104

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

Alternative 14
Accuracy	49.7%
Cost	6464

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

Reproduce

herbie shell --seed 2023151 
(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)))))))