Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.4%

Time: 14.2s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

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)))))))

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))))));
}

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

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))))));
}

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))))))

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

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

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]

Initial Program: 83.6% accurate, 1.0× speedup

Math

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

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)))))))

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))))));
}

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

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))))));
}

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))))))

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

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

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]

Alternative 1: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 40000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -1e+161)
     (asin (* (sqrt t_1) (* (/ l t) (- (sqrt 0.5)))))
     (if (<= (/ t l) 40000000.0)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
       (asin
        (*
         (/ (* l (sqrt 0.5)) t)
         (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -1e+161) {
		tmp = asin((sqrt(t_1) * ((l / t) * -sqrt(0.5))));
	} else if ((t / l) <= 40000000.0) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((((l * sqrt(0.5)) / t) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this 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) / (omc / om))
    if ((t / l) <= (-1d+161)) then
        tmp = asin((sqrt(t_1) * ((l / t) * -sqrt(0.5d0))))
    else if ((t / l) <= 40000000.0d0) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((((l * sqrt(0.5d0)) / t) * sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -1e+161) {
		tmp = Math.asin((Math.sqrt(t_1) * ((l / t) * -Math.sqrt(0.5))));
	} else if ((t / l) <= 40000000.0) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((((l * Math.sqrt(0.5)) / t) * Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -1e+161:
		tmp = math.asin((math.sqrt(t_1) * ((l / t) * -math.sqrt(0.5))))
	elif (t / l) <= 40000000.0:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((((l * math.sqrt(0.5)) / t) * math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -1e+161)
		tmp = asin(Float64(sqrt(t_1) * Float64(Float64(l / t) * Float64(-sqrt(0.5)))));
	elseif (Float64(t / l) <= 40000000.0)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(Float64(Float64(l * sqrt(0.5)) / t) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -1e+161)
		tmp = asin((sqrt(t_1) * ((l / t) * -sqrt(0.5))));
	elseif ((t / l) <= 40000000.0)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((((l * sqrt(0.5)) / t) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+161], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 40000000.0], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e161

   1. Initial program 50.6%

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

      1. unpow2 50.6%

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

      2. clear-num 50.6%

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

      3. clear-num 50.6%

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

      4. frac-times 50.6%

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

      5. metadata-eval 50.6%

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

   3. Applied egg-rr 50.6%

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

   4. Taylor expanded in t around -inf 91.3%

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

      1. mul-1-neg 91.3%

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

      2. associate-/l* 91.2%

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

      3. *-commutative 91.2%

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

      4. distribute-rgt-neg-in 91.2%

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

      5. unpow2 91.2%

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

      6. unpow2 91.2%

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

      7. times-frac 99.5%

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

      8. unpow2 99.5%

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

      9. mul-1-neg 99.5%

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

      10. associate-/r/ 99.6%

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

      11. associate-*r* 99.6%

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

      12. neg-mul-1 99.6%

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

   6. Simplified 99.6%

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

      1. unpow2 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

   8. Applied egg-rr 99.6%

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

   if -1e161 < (/.f64 t l) < 4e7

   1. Initial program 98.6%

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

      1. unpow2 98.6%

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

      2. clear-num 98.6%

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

      3. frac-times 96.1%

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

      4. *-un-lft-identity 96.1%

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

   3. Applied egg-rr 96.1%

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

      1. unpow2 14.8%

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

      2. clear-num 14.8%

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

      3. un-div-inv 14.8%

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

   5. Applied egg-rr 96.1%

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

      1. *-un-lft-identity 96.1%

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

      2. frac-times 98.6%

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

      3. clear-num 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 4e7 < (/.f64 t l)

   1. Initial program 65.2%

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

   2. Taylor expanded in t around inf 85.7%

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

      1. unpow2 85.7%

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

      2. unpow2 85.7%

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

      3. frac-times 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 40000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (pow (/ t l) 2.0) 5e+300)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
   (asin (/ (hypot 1.0 (/ Om Omc)) (hypot 1.0 (* (/ t l) (sqrt 2.0)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (pow((t / l), 2.0) <= 5e+300) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((hypot(1.0, (Om / Omc)) / hypot(1.0, ((t / l) * sqrt(2.0)))));
	}
	return tmp;
}

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (Math.pow((t / l), 2.0) <= 5e+300) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((Math.hypot(1.0, (Om / Omc)) / Math.hypot(1.0, ((t / l) * Math.sqrt(2.0)))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if math.pow((t / l), 2.0) <= 5e+300:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((math.hypot(1.0, (Om / Omc)) / math.hypot(1.0, ((t / l) * math.sqrt(2.0)))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if ((Float64(t / l) ^ 2.0) <= 5e+300)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(hypot(1.0, Float64(Om / Omc)) / hypot(1.0, Float64(Float64(t / l) * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (((t / l) ^ 2.0) <= 5e+300)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((hypot(1.0, (Om / Omc)) / hypot(1.0, ((t / l) * sqrt(2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision], 5e+300], 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[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[1.0 ^ 2 + N[(Om / Omc), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) 2) < 5.00000000000000026e300

   1. Initial program 98.8%

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

      1. unpow2 98.8%

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

      2. clear-num 98.8%

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

      3. frac-times 93.9%

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

      4. *-un-lft-identity 93.9%

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

   3. Applied egg-rr 93.9%

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

      1. unpow2 12.9%

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

      2. clear-num 12.9%

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

      3. un-div-inv 12.9%

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

   5. Applied egg-rr 93.9%

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

      1. *-un-lft-identity 93.9%

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

      2. frac-times 98.8%

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

      3. clear-num 98.8%

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

   7. Applied egg-rr 98.8%

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

   if 5.00000000000000026e300 < (pow.f64 (/.f64 t l) 2)

   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. Step-by-step derivation

      1. unpow2 47.3%

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

      2. clear-num 47.3%

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

      3. clear-num 47.3%

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

      4. frac-times 47.3%

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

      5. metadata-eval 47.3%

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

   3. Applied egg-rr 47.3%

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

      1. unpow2 67.3%

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

      2. clear-num 67.3%

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

      3. un-div-inv 67.3%

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

   5. Applied egg-rr 47.3%

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

   7. Applied egg-rr 46.1%

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

      1. expm1-def 98.1%

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

      2. expm1-log1p 98.1%

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

   9. Simplified 98.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\mathsf{hypot}\left(1, \frac{Om}{Omc}\right)}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\\ \end{array} \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om))))
   (hypot 1.0 (* (/ t l) (sqrt 2.0))))))

C

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

Java

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

Python

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

Julia

t = abs(t)
function code(t, l, Om, Omc)
	return asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))) / hypot(1.0, Float64(Float64(t / l) * sqrt(2.0)))))
end

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.5%

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

   1. sqrt-div 82.5%

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

   2. div-inv 82.5%

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

   3. add-sqr-sqrt 82.5%

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

   4. hypot-1-def 82.5%

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

   5. *-commutative 82.5%

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

   6. sqrt-prod 82.4%

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

   7. unpow2 82.4%

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

   8. sqrt-prod 54.1%

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

   9. add-sqr-sqrt 98.7%

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

3. Applied egg-rr 98.7%

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

   1. associate-*r/ 98.7%

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

   2. *-rgt-identity 98.7%

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

5. Simplified 98.7%

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

   1. unpow2 30.1%

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

   2. clear-num 30.1%

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

   3. un-div-inv 30.1%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 4: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 40000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -1e+161)
     (asin (* (sqrt t_1) (/ (- l) (* t (sqrt 2.0)))))
     (if (<= (/ t l) 40000000.0)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
       (asin
        (*
         (/ (* l (sqrt 0.5)) t)
         (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -1e+161) {
		tmp = asin((sqrt(t_1) * (-l / (t * sqrt(2.0)))));
	} else if ((t / l) <= 40000000.0) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((((l * sqrt(0.5)) / t) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this 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) / (omc / om))
    if ((t / l) <= (-1d+161)) then
        tmp = asin((sqrt(t_1) * (-l / (t * sqrt(2.0d0)))))
    else if ((t / l) <= 40000000.0d0) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((((l * sqrt(0.5d0)) / t) * sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -1e+161) {
		tmp = Math.asin((Math.sqrt(t_1) * (-l / (t * Math.sqrt(2.0)))));
	} else if ((t / l) <= 40000000.0) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((((l * Math.sqrt(0.5)) / t) * Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -1e+161:
		tmp = math.asin((math.sqrt(t_1) * (-l / (t * math.sqrt(2.0)))))
	elif (t / l) <= 40000000.0:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((((l * math.sqrt(0.5)) / t) * math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -1e+161)
		tmp = asin(Float64(sqrt(t_1) * Float64(Float64(-l) / Float64(t * sqrt(2.0)))));
	elseif (Float64(t / l) <= 40000000.0)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(Float64(Float64(l * sqrt(0.5)) / t) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -1e+161)
		tmp = asin((sqrt(t_1) * (-l / (t * sqrt(2.0)))));
	elseif ((t / l) <= 40000000.0)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((((l * sqrt(0.5)) / t) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+161], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 40000000.0], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e161

   1. Initial program 50.6%

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

      1. sqrt-div 50.6%

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

      2. div-inv 50.6%

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

      3. add-sqr-sqrt 50.6%

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

      4. hypot-1-def 50.6%

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

      5. *-commutative 50.6%

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

      6. sqrt-prod 50.6%

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

      7. unpow2 50.6%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in t around -inf 91.3%

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

      1. mul-1-neg 91.3%

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

      2. *-commutative 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

      4. unpow2 91.3%

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

      5. unpow2 91.3%

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

      6. times-frac 99.6%

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

      7. unpow2 99.6%

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

   8. Simplified 99.6%

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

      1. unpow2 99.6%

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

      2. clear-num 99.6%

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

      3. un-div-inv 99.6%

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

   10. Applied egg-rr 99.6%

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

   if -1e161 < (/.f64 t l) < 4e7

   1. Initial program 98.6%

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

      1. unpow2 98.6%

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

      2. clear-num 98.6%

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

      3. frac-times 96.1%

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

      4. *-un-lft-identity 96.1%

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

   3. Applied egg-rr 96.1%

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

      1. unpow2 14.8%

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

      2. clear-num 14.8%

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

      3. un-div-inv 14.8%

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

   5. Applied egg-rr 96.1%

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

      1. *-un-lft-identity 96.1%

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

      2. frac-times 98.6%

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

      3. clear-num 98.6%

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

   7. Applied egg-rr 98.6%

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

   if 4e7 < (/.f64 t l)

   1. Initial program 65.2%

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

   2. Taylor expanded in t around inf 85.7%

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

      1. unpow2 85.7%

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

      2. unpow2 85.7%

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

      3. frac-times 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 40000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 5: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 40000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) 40000000.0)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
   (asin (* (/ (* l (sqrt 0.5)) t) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= 40000000.0) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((((l * sqrt(0.5)) / t) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this 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) <= 40000000.0d0) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((((l * sqrt(0.5d0)) / t) * sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= 40000000.0) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((((l * Math.sqrt(0.5)) / t) * Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= 40000000.0:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((((l * math.sqrt(0.5)) / t) * math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= 40000000.0)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(Float64(Float64(l * sqrt(0.5)) / t) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= 40000000.0)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((((l * sqrt(0.5)) / t) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], 40000000.0], 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[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4e7

   1. Initial program 89.2%

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

      1. unpow2 89.2%

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

      2. clear-num 89.2%

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

      3. frac-times 87.2%

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

      4. *-un-lft-identity 87.2%

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

   3. Applied egg-rr 87.2%

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

      1. unpow2 31.4%

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

      2. clear-num 31.4%

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

      3. un-div-inv 31.4%

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

   5. Applied egg-rr 87.2%

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

      1. *-un-lft-identity 87.2%

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

      2. frac-times 89.2%

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

      3. clear-num 89.2%

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

   7. Applied egg-rr 89.2%

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

   if 4e7 < (/.f64 t l)

   1. Initial program 65.2%

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

   2. Taylor expanded in t around inf 85.7%

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

      1. unpow2 85.7%

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

      2. unpow2 85.7%

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

      3. frac-times 99.5%

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

   4. Applied egg-rr 99.5%

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 40000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 6: 83.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right) \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (sqrt
   (/ (- 1.0 (/ (/ Om Omc) (/ Omc Om))) (+ 1.0 (* 2.0 (* (/ t l) (/ t l))))))))

C

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

Fortran

NOTE: t should be positive before calling this 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
    code = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := 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[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.5%

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

   1. unpow2 82.5%

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

   2. clear-num 82.5%

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

   3. frac-times 79.1%

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

   4. *-un-lft-identity 79.1%

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

3. Applied egg-rr 79.1%

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

   1. unpow2 30.1%

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

   2. clear-num 30.1%

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

   3. un-div-inv 30.1%

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

5. Applied egg-rr 79.1%

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

   1. *-un-lft-identity 79.1%

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

   2. frac-times 82.5%

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

   3. clear-num 82.5%

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

7. Applied egg-rr 82.5%

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

8. Final simplification 82.5%

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

Alternative 7: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right) \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	return asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
}

Fortran

NOTE: t should be positive before calling this 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
    code = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	return Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	return math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))

Julia

t = abs(t)
function code(t, l, Om, Omc)
	return asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))))
end

MATLAB

t = abs(t)
function tmp = code(t, l, Om, Omc)
	tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.5%

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

2. Taylor expanded in t around 0 43.6%

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

   1. unpow2 43.6%

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

   2. unpow2 43.6%

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

   3. times-frac 48.9%

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

   4. unpow2 48.9%

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

4. Simplified 48.9%

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

   1. unpow2 30.1%

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

   2. clear-num 30.1%

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

   3. un-div-inv 30.1%

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

6. Applied egg-rr 48.9%

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

7. Final simplification 48.9%

   \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right) \]

Reproduce

herbie shell --seed 2023306 
(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)))))))