Toniolo and Linder, Equation (2)

Average Error: 10.2 → 0.8

Time: 19.2s

Precision: binary64

Cost: 27336

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 + \left(1 + \left(-1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5}}{\frac{1}{\ell}}}{t}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))

↓

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5e+125)
   (- (asin (* (sqrt 0.5) (/ l t))))
   (if (<= (/ t l) 2e+151)
     (asin
      (sqrt
       (/
        (+ 1.0 (+ 1.0 (- -1.0 (pow (/ Om Omc) 2.0))))
        (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (/ (/ (sqrt 0.5) (/ 1.0 l)) t)))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
}

↓

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+125) {
		tmp = -asin((sqrt(0.5) * (l / t)));
	} else if ((t / l) <= 2e+151) {
		tmp = asin(sqrt(((1.0 + (1.0 + (-1.0 - pow((Om / Omc), 2.0)))) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin(((sqrt(0.5) / (1.0 / l)) / t));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
end function

↓

real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-5d+125)) then
        tmp = -asin((sqrt(0.5d0) * (l / t)))
    else if ((t / l) <= 2d+151) then
        tmp = asin(sqrt(((1.0d0 + (1.0d0 + ((-1.0d0) - ((om / omc) ** 2.0d0)))) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin(((sqrt(0.5d0) / (1.0d0 / l)) / t))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
}

↓

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+125) {
		tmp = -Math.asin((Math.sqrt(0.5) * (l / t)));
	} else if ((t / l) <= 2e+151) {
		tmp = Math.asin(Math.sqrt(((1.0 + (1.0 + (-1.0 - Math.pow((Om / Omc), 2.0)))) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin(((Math.sqrt(0.5) / (1.0 / l)) / t));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	return math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))

↓

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5e+125:
		tmp = -math.asin((math.sqrt(0.5) * (l / t)))
	elif (t / l) <= 2e+151:
		tmp = math.asin(math.sqrt(((1.0 + (1.0 + (-1.0 - math.pow((Om / Omc), 2.0)))) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin(((math.sqrt(0.5) / (1.0 / l)) / t))
	return tmp

Julia

function code(t, l, Om, Omc)
	return asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))))
end

↓

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5e+125)
		tmp = Float64(-asin(Float64(sqrt(0.5) * Float64(l / t))));
	elseif (Float64(t / l) <= 2e+151)
		tmp = asin(sqrt(Float64(Float64(1.0 + Float64(1.0 + Float64(-1.0 - (Float64(Om / Omc) ^ 2.0)))) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) / Float64(1.0 / l)) / t));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
end

↓

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5e+125)
		tmp = -asin((sqrt(0.5) * (l / t)));
	elseif ((t / l) <= 2e+151)
		tmp = asin(sqrt(((1.0 + (1.0 + (-1.0 - ((Om / Omc) ^ 2.0)))) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin(((sqrt(0.5) / (1.0 / l)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

↓

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+125], (-N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[N[(t / l), $MachinePrecision], 2e+151], N[ArcSin[N[Sqrt[N[(N[(1.0 + N[(1.0 + N[(-1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $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[(N[Sqrt[0.5], $MachinePrecision] / N[(1.0 / l), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.99999999999999962e125

   1. Initial program 29.6

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

   2. Taylor expanded in t around -inf 8.6

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

   3. Simplified 0.3

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

      [Start] 8.6	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 8.6	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 8.6	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 8.6	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 8.6	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 0.3	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [<=] 0.3	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]

   4. Taylor expanded in l around 0 8.6

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

   5. Simplified 0.3

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

      [Start] 8.6	\[ \sin^{-1} \left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      associate-*l/ [<=] 8.6	\[ \sin^{-1} \left(-\color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      associate-*l/ [=>] 8.6	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{0.5} \cdot \ell}{t}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      *-commutative [=>] 8.6	\[ \sin^{-1} \left(-\frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      unpow2 [=>] 8.6	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]
      unpow2 [=>] 8.6	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]
      times-frac [=>] 0.3	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]
      unpow2 [<=] 0.3	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]
      associate-*l/ [=>] 0.3	\[ \sin^{-1} \left(-\color{blue}{\frac{\left(\ell \cdot \sqrt{0.5}\right) \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t}}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(-\frac{\color{blue}{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\ell \cdot \sqrt{0.5}\right)}}{t}\right) \]
      associate-*l/ [<=] 0.3	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t} \cdot \left(\ell \cdot \sqrt{0.5}\right)}\right) \]
      *-commutative [=>] 0.3	\[ \sin^{-1} \left(-\color{blue}{\left(\ell \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t}}\right) \]
      *-commutative [<=] 0.3	\[ \sin^{-1} \left(-\color{blue}{\left(\sqrt{0.5} \cdot \ell\right)} \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t}\right) \]

   6. Taylor expanded in Om around 0 0.7

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

   7. Applied egg-rr 37.7

      \[\leadsto \color{blue}{\left(0 - e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\right)}\right) + 1} \]

   8. Simplified 0.7

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

      [Start] 37.7	\[ \left(0 - e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\right)}\right) + 1 \]
      associate-+l- [=>] 37.7	\[ \color{blue}{0 - \left(e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\right)} - 1\right)} \]
      expm1-def [=>] 0.7	\[ 0 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\right)\right)} \]
      expm1-log1p [=>] 0.7	\[ 0 - \color{blue}{\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)} \]
      sub0-neg [=>] 0.7	\[ \color{blue}{-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)} \]

   if -4.99999999999999962e125 < (/.f64 t l) < 2.00000000000000003e151

   1. Initial program 0.9

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

   2. Applied egg-rr 0.9

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

   if 2.00000000000000003e151 < (/.f64 t l)

   1. Initial program 33.2

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

   2. Taylor expanded in t around -inf 37.9

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

   3. Simplified 34.4

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

      [Start] 37.9	\[ \sin^{-1} \left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right) \]
      mul-1-neg [=>] 37.9	\[ \sin^{-1} \color{blue}{\left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
      *-commutative [=>] 37.9	\[ \sin^{-1} \left(-\color{blue}{\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}}\right) \]
      unpow2 [=>] 37.9	\[ \sin^{-1} \left(-\sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [=>] 37.9	\[ \sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      times-frac [=>] 34.4	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      unpow2 [<=] 34.4	\[ \sin^{-1} \left(-\sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right) \]
      *-commutative [=>] 34.4	\[ \sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t}\right) \]

   4. Taylor expanded in l around 0 37.9

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

   5. Simplified 34.4

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

      [Start] 37.9	\[ \sin^{-1} \left(-\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      associate-*l/ [<=] 37.9	\[ \sin^{-1} \left(-\color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      associate-*l/ [=>] 37.9	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{0.5} \cdot \ell}{t}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      *-commutative [=>] 37.9	\[ \sin^{-1} \left(-\frac{\color{blue}{\ell \cdot \sqrt{0.5}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \]
      unpow2 [=>] 37.9	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{\color{blue}{Om \cdot Om}}{{Omc}^{2}}}\right) \]
      unpow2 [=>] 37.9	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om \cdot Om}{\color{blue}{Omc \cdot Omc}}}\right) \]
      times-frac [=>] 34.4	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]
      unpow2 [<=] 34.4	\[ \sin^{-1} \left(-\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]
      associate-*l/ [=>] 34.4	\[ \sin^{-1} \left(-\color{blue}{\frac{\left(\ell \cdot \sqrt{0.5}\right) \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t}}\right) \]
      *-commutative [=>] 34.4	\[ \sin^{-1} \left(-\frac{\color{blue}{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\ell \cdot \sqrt{0.5}\right)}}{t}\right) \]
      associate-*l/ [<=] 34.4	\[ \sin^{-1} \left(-\color{blue}{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t} \cdot \left(\ell \cdot \sqrt{0.5}\right)}\right) \]
      *-commutative [=>] 34.4	\[ \sin^{-1} \left(-\color{blue}{\left(\ell \cdot \sqrt{0.5}\right) \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t}}\right) \]
      *-commutative [<=] 34.4	\[ \sin^{-1} \left(-\color{blue}{\left(\sqrt{0.5} \cdot \ell\right)} \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t}\right) \]

   6. Taylor expanded in Om around 0 34.4

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

   7. Applied egg-rr 0.6

      \[\leadsto \sin^{-1} \left(-\color{blue}{\frac{-\frac{\sqrt{0.5}}{\frac{1}{\ell}}}{t}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 + \left(1 + \left(-1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5}}{\frac{1}{\ell}}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	32832

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

Alternative 2
Error	0.8
Cost	27080

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5}}{\frac{1}{\ell}}}{t}\right)\\ \end{array} \]

Alternative 3
Error	0.8
Cost	14792

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5}}{\frac{1}{\ell}}}{t}\right)\\ \end{array} \]

Alternative 4
Error	1.2
Cost	14664

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+44}:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	1.3
Cost	14152

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+125}:\\ \;\;\;\;-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\sqrt{0.5}}{\frac{1}{\ell}}}{t}\right)\\ \end{array} \]

Alternative 6
Error	1.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 7
Error	1.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 8
Error	1.8
Cost	13640

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;-t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	1.8
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;-\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 10
Error	23.0
Cost	13385

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{-127} \lor \neg \left(\ell \leq 9 \cdot 10^{-59}\right):\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 11
Error	31.5
Cost	7104

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

Alternative 12
Error	31.7
Cost	6464

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

Reproduce

herbie shell --seed 2023039 
(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)))))))