Toniolo and Linder, Equation (2)

Percentage Accurate: 84.1% → 98.1%

Time: 19.5s

Alternatives: 9

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: 84.1% 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: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot \frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{t\_1} \cdot \frac{1}{\frac{t\_m}{\sqrt{0.5}}}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_m l_m) 5e+77)
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (* t_m (/ t_m l_m)) l_m))))))
     (asin (* l_m (* (sqrt t_1) (/ 1.0 (/ t_m (sqrt 0.5)))))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l_m) <= 5e+77) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	} else {
		tmp = asin((l_m * (sqrt(t_1) * (1.0 / (t_m / sqrt(0.5))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t_m / l_m) <= 5d+77) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m * (t_m / l_m)) / l_m))))))
    else
        tmp = asin((l_m * (sqrt(t_1) * (1.0d0 / (t_m / sqrt(0.5d0))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l_m) <= 5e+77) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(t_1) * (1.0 / (t_m / Math.sqrt(0.5))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t_m / l_m) <= 5e+77:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))))
	else:
		tmp = math.asin((l_m * (math.sqrt(t_1) * (1.0 / (t_m / math.sqrt(0.5))))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+77)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l_m)) / l_m))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(t_1) * Float64(1.0 / Float64(t_m / sqrt(0.5))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+77)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	else
		tmp = asin((l_m * (sqrt(t_1) * (1.0 / (t_m / sqrt(0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+77], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(1.0 / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000004e77

   1. Initial program 89.6%

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

      1. unpow2 89.6%

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

      2. associate-*r/ 89.6%

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

   4. Applied egg-rr 89.6%

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

   if 5.00000000000000004e77 < (/.f64 t l)

   1. Initial program 48.2%

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

   3. Taylor expanded in t around inf 85.2%

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

      1. associate-/l* 85.3%

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

      2. associate-*l* 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

      5. times-frac 99.5%

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

      6. unpow2 99.5%

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

   5. Simplified 99.5%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. unpow-1 99.6%

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

   9. Simplified 99.6%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\frac{t}{\sqrt{0.5}}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.8× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
   (hypot 1.0 (* (/ t_m l_m) (sqrt 2.0))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.8%

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

   1. sqrt-div 82.8%

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

   2. add-sqr-sqrt 82.8%

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

   3. hypot-1-def 82.8%

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

   4. *-commutative 82.8%

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

   5. sqrt-prod 82.8%

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

   6. unpow2 82.8%

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

   7. sqrt-prod 57.7%

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

   8. add-sqr-sqrt 99.1%

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

4. Applied egg-rr 99.1%

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

5. Final simplification 99.1%

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

Alternative 3: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{t\_1 \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_m l_m) 2e+59)
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m)))))))
     (asin (* l_m (/ (sqrt (* t_1 0.5)) t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l_m) <= 2e+59) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = asin((l_m * (sqrt((t_1 * 0.5)) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t_m / l_m) <= 2d+59) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_m)))))))
    else
        tmp = asin((l_m * (sqrt((t_1 * 0.5d0)) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l_m) <= 2e+59) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt((t_1 * 0.5)) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t_m / l_m) <= 2e+59:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt((t_1 * 0.5)) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+59)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(t_1 * 0.5)) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+59)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	else
		tmp = asin((l_m * (sqrt((t_1 * 0.5)) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+59], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.99999999999999994e59

   1. Initial program 89.4%

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

      1. unpow2 89.4%

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

   4. Applied egg-rr 89.4%

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

   if 1.99999999999999994e59 < (/.f64 t l)

   1. Initial program 52.7%

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

   3. Taylor expanded in t around inf 86.4%

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

      1. associate-/l* 86.5%

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

      2. associate-*l* 86.5%

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

      3. unpow2 86.5%

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

      4. unpow2 86.5%

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

      5. times-frac 99.5%

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

      6. unpow2 99.5%

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

   5. Simplified 99.5%

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

      1. pow1 99.5%

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

      2. associate-*l/ 99.5%

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

      3. sqrt-unprod 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. unpow1 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot \frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{t\_1 \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_m l_m) 2e+59)
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (* t_m (/ t_m l_m)) l_m))))))
     (asin (* l_m (/ (sqrt (* t_1 0.5)) t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l_m) <= 2e+59) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	} else {
		tmp = asin((l_m * (sqrt((t_1 * 0.5)) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t_m / l_m) <= 2d+59) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m * (t_m / l_m)) / l_m))))))
    else
        tmp = asin((l_m * (sqrt((t_1 * 0.5d0)) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t_m / l_m) <= 2e+59) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt((t_1 * 0.5)) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t_m / l_m) <= 2e+59:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))))
	else:
		tmp = math.asin((l_m * (math.sqrt((t_1 * 0.5)) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+59)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l_m)) / l_m))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(t_1 * 0.5)) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+59)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	else
		tmp = asin((l_m * (sqrt((t_1 * 0.5)) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+59], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(t$95$1 * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.99999999999999994e59

   1. Initial program 89.4%

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

      1. unpow2 89.4%

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

      2. associate-*r/ 89.4%

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

   4. Applied egg-rr 89.4%

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

   if 1.99999999999999994e59 < (/.f64 t l)

   1. Initial program 52.7%

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

   3. Taylor expanded in t around inf 86.4%

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

      1. associate-/l* 86.5%

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

      2. associate-*l* 86.5%

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

      3. unpow2 86.5%

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

      4. unpow2 86.5%

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

      5. times-frac 99.5%

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

      6. unpow2 99.5%

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

   5. Simplified 99.5%

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

      1. pow1 99.5%

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

      2. associate-*l/ 99.5%

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

      3. sqrt-unprod 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. unpow1 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+59)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m)))))))
   (asin (* l_m (/ (sqrt (* (- 1.0 (pow (/ Om Omc) 2.0)) 0.5)) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+59) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = asin((l_m * (sqrt(((1.0 - pow((Om / Omc), 2.0)) * 0.5)) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 2d+59) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_m)))))))
    else
        tmp = asin((l_m * (sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) * 0.5d0)) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+59) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) * 0.5)) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 2e+59:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) * 0.5)) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+59)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) * 0.5)) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+59)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	else
		tmp = asin((l_m * (sqrt(((1.0 - ((Om / Omc) ^ 2.0)) * 0.5)) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+59], 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$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.99999999999999994e59

   1. Initial program 89.4%

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

      1. unpow2 89.4%

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

   4. Applied egg-rr 89.4%

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

      1. unpow2 89.4%

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

      2. clear-num 89.4%

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

      3. un-div-inv 89.4%

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

   6. Applied egg-rr 89.4%

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

   if 1.99999999999999994e59 < (/.f64 t l)

   1. Initial program 52.7%

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

   3. Taylor expanded in t around inf 86.4%

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

      1. associate-/l* 86.5%

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

      2. associate-*l* 86.5%

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

      3. unpow2 86.5%

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

      4. unpow2 86.5%

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

      5. times-frac 99.5%

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

      6. unpow2 99.5%

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

   5. Simplified 99.5%

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

      1. pow1 99.5%

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

      2. associate-*l/ 99.5%

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

      3. sqrt-unprod 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. unpow1 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{1}{\frac{t\_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+77)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m)))))))
   (asin (* l_m (/ 1.0 (/ t_m (sqrt 0.5)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+77) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = asin((l_m * (1.0 / (t_m / sqrt(0.5)))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+77) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_m)))))))
    else
        tmp = asin((l_m * (1.0d0 / (t_m / sqrt(0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+77) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = Math.asin((l_m * (1.0 / (t_m / Math.sqrt(0.5)))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+77:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))))
	else:
		tmp = math.asin((l_m * (1.0 / (t_m / math.sqrt(0.5)))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+77)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(1.0 / Float64(t_m / sqrt(0.5)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+77)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	else
		tmp = asin((l_m * (1.0 / (t_m / sqrt(0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+77], 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$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(1.0 / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000004e77

   1. Initial program 89.6%

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

      1. unpow2 89.6%

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

   4. Applied egg-rr 89.6%

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

      1. unpow2 89.6%

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

      2. clear-num 89.6%

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

      3. un-div-inv 89.6%

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

   6. Applied egg-rr 89.6%

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

   if 5.00000000000000004e77 < (/.f64 t l)

   1. Initial program 48.2%

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

   3. Taylor expanded in t around inf 85.2%

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

      1. associate-/l* 85.3%

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

      2. associate-*l* 85.3%

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

      3. unpow2 85.3%

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

      4. unpow2 85.3%

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

      5. times-frac 99.5%

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

      6. unpow2 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in Om around 0 97.8%

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

      1. associate-/l* 98.0%

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

   8. Simplified 98.0%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

   10. Applied egg-rr 98.1%

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

       1. unpow-1 99.6%

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

   12. Simplified 98.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{1}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.9× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.2)
   (asin (sqrt (- 1.0 (/ (* Om (/ Om Omc)) Omc))))
   (asin (* l_m (/ 1.0 (/ t_m (sqrt 0.5)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.2) {
		tmp = asin(sqrt((1.0 - ((Om * (Om / Omc)) / Omc))));
	} else {
		tmp = asin((l_m * (1.0 / (t_m / sqrt(0.5)))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.2d0) then
        tmp = asin(sqrt((1.0d0 - ((om * (om / omc)) / omc))))
    else
        tmp = asin((l_m * (1.0d0 / (t_m / sqrt(0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.2) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om * (Om / Omc)) / Omc))));
	} else {
		tmp = Math.asin((l_m * (1.0 / (t_m / Math.sqrt(0.5)))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.2:
		tmp = math.asin(math.sqrt((1.0 - ((Om * (Om / Omc)) / Omc))))
	else:
		tmp = math.asin((l_m * (1.0 / (t_m / math.sqrt(0.5)))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.2)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc))));
	else
		tmp = asin(Float64(l_m * Float64(1.0 / Float64(t_m / sqrt(0.5)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.2)
		tmp = asin(sqrt((1.0 - ((Om * (Om / Omc)) / Omc))));
	else
		tmp = asin((l_m * (1.0 / (t_m / sqrt(0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.2], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(1.0 / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.20000000000000001

   1. Initial program 88.8%

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

   3. Taylor expanded in t around 0 59.2%

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

      1. unpow2 59.2%

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

      2. unpow2 59.2%

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

      3. times-frac 70.5%

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

      4. unpow2 70.5%

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

   5. Simplified 70.5%

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

      1. unpow2 70.5%

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

      2. associate-*r/ 70.5%

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

   7. Applied egg-rr 70.5%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 63.0%

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

   3. Taylor expanded in t around inf 87.2%

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

      1. associate-/l* 87.4%

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

      2. associate-*l* 87.4%

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

      3. unpow2 87.4%

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

      4. unpow2 87.4%

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

      5. times-frac 97.5%

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

      6. unpow2 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in Om around 0 96.2%

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

      1. associate-/l* 96.4%

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

   8. Simplified 96.4%

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

      1. clear-num 97.5%

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

      2. inv-pow 97.5%

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

   10. Applied egg-rr 96.4%

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

       1. unpow-1 97.5%

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

   12. Simplified 96.4%

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

4. Final simplification 76.4%

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

Alternative 8: 48.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(l\_m \cdot \frac{1}{\frac{t\_m}{\sqrt{0.5}}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (* l_m (/ 1.0 (/ t_m (sqrt 0.5))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin((l_m * (1.0 / (t_m / sqrt(0.5)))));
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin((l_m * (1.0d0 / (t_m / sqrt(0.5d0)))))
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	return Math.asin((l_m * (1.0 / (t_m / Math.sqrt(0.5)))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin((l_m * (1.0 / (t_m / math.sqrt(0.5)))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(l_m * Float64(1.0 / Float64(t_m / sqrt(0.5)))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((l_m * (1.0 / (t_m / sqrt(0.5)))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[N[(l$95$m * N[(1.0 / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in t around inf 25.1%

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

   1. associate-/l* 25.1%

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

   2. associate-*l* 25.1%

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

   3. unpow2 25.1%

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

   4. unpow2 25.1%

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

   5. times-frac 28.0%

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

   6. unpow2 28.0%

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

5. Simplified 28.0%

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

6. Taylor expanded in Om around 0 27.7%

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

   1. associate-/l* 27.7%

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

8. Simplified 27.7%

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

   1. clear-num 28.0%

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

   2. inv-pow 28.0%

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

10. Applied egg-rr 27.7%

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

    1. unpow-1 28.0%

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

12. Simplified 27.7%

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

13. Final simplification 27.7%

    \[\leadsto \sin^{-1} \left(\ell \cdot \frac{1}{\frac{t}{\sqrt{0.5}}}\right) \]
14. Add Preprocessing

Alternative 9: 48.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc) :precision binary64 (asin (* l_m (/ (sqrt 0.5) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin((l_m * (sqrt(0.5) / t_m)));
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin((l_m * (sqrt(0.5d0) / t_m)))
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	return Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin((l_m * (math.sqrt(0.5) / t_m)))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(l_m * Float64(sqrt(0.5) / t_m)))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((l_m * (sqrt(0.5) / t_m)));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.8%

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

3. Taylor expanded in t around inf 25.1%

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

   1. associate-/l* 25.1%

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

   2. associate-*l* 25.1%

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

   3. unpow2 25.1%

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

   4. unpow2 25.1%

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

   5. times-frac 28.0%

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

   6. unpow2 28.0%

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

5. Simplified 28.0%

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

6. Taylor expanded in Om around 0 27.7%

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

   1. associate-/l* 27.7%

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

8. Simplified 27.7%

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

9. Final simplification 27.7%

   \[\leadsto \sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024052 
(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)))))))