Toniolo and Linder, Equation (2)

Percentage Accurate: 84.2% → 98.8%

Time: 18.3s

Alternatives: 8

Speedup: 2.0×

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: 84.2% 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: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + \frac{2}{\frac{\frac{l\_m}{t\_m}}{\frac{t\_m}{l\_m}}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 5e+152)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (+ 1.0 (/ 2.0 (/ (/ l_m t_m) (/ t_m l_m)))))))
   (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) {
	double tmp;
	if ((t_m / l_m) <= 5e+152) {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))));
	} else {
		tmp = asin((l_m * (sqrt(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) <= 5d+152) then
        tmp = asin(sqrt(((1.0d0 - ((om * (om / omc)) / omc)) / (1.0d0 + (2.0d0 / ((l_m / t_m) / (t_m / l_m)))))))
    else
        tmp = asin((l_m * (sqrt(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) <= 5e+152) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(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) <= 5e+152:
		tmp = math.asin(math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt(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) <= 5e+152)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / Float64(1.0 + Float64(2.0 / Float64(Float64(l_m / t_m) / Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(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) <= 5e+152)
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))));
	else
		tmp = asin((l_m * (sqrt(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], 5e+152], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 / N[(N[(l$95$m / t$95$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[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5e152

   1. Initial program 90.8%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 87.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{1}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{2}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\ell}{t \cdot \frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\frac{\ell}{t}}{\frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 90.9%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 90.9%

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

   if 5e152 < (/.f64 t l)

   1. Initial program 47.1%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 47.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot 2}{\ell}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{2}{\ell}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{1}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 47.1%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 47.1%

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

   7. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 47.1%

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

       1. frac-times N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot t}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{2 \cdot t}{\ell}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{2 \cdot t}}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{t}}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{t}{\ell}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       10. associate-/l/ N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\frac{\ell}{t}}{2}\right)\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), 2\right)\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 47.1%

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 47.1%

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

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\ell \cdot \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 99.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right) \]

   14. Simplified 99.5%

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

Alternative 2: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + \frac{t\_m}{\frac{l\_m}{t\_m}} \cdot \frac{2}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5 \cdot \left(1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+46)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (+ 1.0 (* (/ t_m (/ l_m t_m)) (/ 2.0 l_m))))))
   (asin (* (/ l_m t_m) (sqrt (* 0.5 (- 1.0 (/ Om (/ Omc (/ Om Omc))))))))))

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+46) {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))));
	}
	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+46) then
        tmp = asin(sqrt(((1.0d0 - ((om * (om / omc)) / omc)) / (1.0d0 + ((t_m / (l_m / t_m)) * (2.0d0 / l_m))))))
    else
        tmp = asin(((l_m / t_m) * sqrt((0.5d0 * (1.0d0 - (om / (omc / (om / omc))))))))
    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+46) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	} else {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))));
	}
	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+46:
		tmp = math.asin(math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))))
	else:
		tmp = math.asin(((l_m / t_m) * math.sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))))
	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+46)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / Float64(1.0 + Float64(Float64(t_m / Float64(l_m / t_m)) * Float64(2.0 / l_m))))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(Float64(0.5 * Float64(1.0 - Float64(Om / Float64(Omc / Float64(Om / Omc))))))));
	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+46)
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	else
		tmp = asin(((l_m / t_m) * sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))));
	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+46], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(t$95$m / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(0.5 * N[(1.0 - N[(Om / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2e46

   1. Initial program 89.9%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 87.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot 2}{\ell}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{2}{\ell}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{1}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 87.6%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 87.6%

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

   if 2e46 < (/.f64 t l)

   1. Initial program 69.1%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 65.5%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{1}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{2}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\ell}{t \cdot \frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\frac{\ell}{t}}{\frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 69.2%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 69.2%

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

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 82.0%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right)\right) \]

   9. Simplified 82.0%

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

       1. asin-lowering-asin.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \ell}{t}\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \frac{\ell}{t}\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \frac{\ell}{t}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right) \cdot \frac{\ell}{t}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right), \left(\frac{\ell}{t}\right)\right)\right) \]

   11. Applied egg-rr 99.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + \frac{t}{\frac{\ell}{t}} \cdot \frac{2}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5 \cdot \left(1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 100000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t\_m \cdot \frac{t\_m}{l\_m}\right)}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}\right)}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 100000000000.0)
   (asin (sqrt (/ 1.0 (+ 1.0 (/ (* 2.0 (* t_m (/ t_m l_m))) l_m)))))
   (asin (* l_m (/ (sqrt (* 0.5 (- 1.0 (/ Om (/ Omc (/ Om Omc)))))) 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) <= 100000000000.0) {
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))));
	} else {
		tmp = asin((l_m * (sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc)))))) / 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) <= 100000000000.0d0) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + ((2.0d0 * (t_m * (t_m / l_m))) / l_m)))))
    else
        tmp = asin((l_m * (sqrt((0.5d0 * (1.0d0 - (om / (omc / (om / omc)))))) / 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) <= 100000000000.0) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc)))))) / 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) <= 100000000000.0:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))))
	else:
		tmp = math.asin((l_m * (math.sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc)))))) / 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) <= 100000000000.0)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(Float64(2.0 * Float64(t_m * Float64(t_m / l_m))) / l_m)))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(0.5 * Float64(1.0 - Float64(Om / Float64(Omc / Float64(Om / Omc)))))) / 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) <= 100000000000.0)
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))));
	else
		tmp = asin((l_m * (sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc)))))) / 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], 100000000000.0], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(N[(2.0 * N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(0.5 * N[(1.0 - N[(Om / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1e11

   1. Initial program 89.5%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 87.1%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \ell\right)\right)\right), \ell\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 86.3%

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

   if 1e11 < (/.f64 t l)

   1. Initial program 73.6%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 70.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{1}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{2}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\ell}{t \cdot \frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\frac{\ell}{t}}{\frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 73.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 84.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right)\right) \]

   9. Simplified 84.5%

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

       1. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\left(\ell \cdot \frac{\sqrt{\frac{1}{2}}}{t}\right) \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\right) \]

       2. associate-*l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\ell \cdot \left(\frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\left(\frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right) \cdot \ell\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right), \ell\right)\right) \]

   11. Applied egg-rr 99.4%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 100000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot \frac{t}{\ell}\right)}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}\right)}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 100000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t\_m \cdot \frac{t\_m}{l\_m}\right)}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5 \cdot \left(1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}\right)}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 100000000000.0)
   (asin (sqrt (/ 1.0 (+ 1.0 (/ (* 2.0 (* t_m (/ t_m l_m))) l_m)))))
   (asin (* (/ l_m t_m) (sqrt (* 0.5 (- 1.0 (/ Om (/ Omc (/ Om Omc))))))))))

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) <= 100000000000.0) {
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))));
	}
	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) <= 100000000000.0d0) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + ((2.0d0 * (t_m * (t_m / l_m))) / l_m)))))
    else
        tmp = asin(((l_m / t_m) * sqrt((0.5d0 * (1.0d0 - (om / (omc / (om / omc))))))))
    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) <= 100000000000.0) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))));
	} else {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))));
	}
	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) <= 100000000000.0:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))))
	else:
		tmp = math.asin(((l_m / t_m) * math.sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))))
	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) <= 100000000000.0)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(Float64(2.0 * Float64(t_m * Float64(t_m / l_m))) / l_m)))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(Float64(0.5 * Float64(1.0 - Float64(Om / Float64(Omc / Float64(Om / Omc))))))));
	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) <= 100000000000.0)
		tmp = asin(sqrt((1.0 / (1.0 + ((2.0 * (t_m * (t_m / l_m))) / l_m)))));
	else
		tmp = asin(((l_m / t_m) * sqrt((0.5 * (1.0 - (Om / (Omc / (Om / Omc))))))));
	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], 100000000000.0], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(N[(2.0 * N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(0.5 * N[(1.0 - N[(Om / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1e11

   1. Initial program 89.5%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 87.1%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \ell\right)\right)\right), \ell\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 86.3%

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

   if 1e11 < (/.f64 t l)

   1. Initial program 73.6%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 70.6%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{1}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{2}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\ell}{t \cdot \frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\frac{\ell}{t}}{\frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 73.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in l around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 84.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right)\right) \]

   9. Simplified 84.5%

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

       1. asin-lowering-asin.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{\frac{1}{2}} \cdot \ell}{t}\right)\right) \]

       4. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \left(\sqrt{\frac{1}{2}} \cdot \frac{\ell}{t}\right)\right)\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \sqrt{\frac{1}{2}}\right) \cdot \frac{\ell}{t}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right) \cdot \frac{\ell}{t}\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}} \cdot \sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right), \left(\frac{\ell}{t}\right)\right)\right) \]

   11. Applied egg-rr 99.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 100000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2 \cdot \left(t \cdot \frac{t}{\ell}\right)}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5 \cdot \left(1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\frac{\frac{l\_m}{t\_m}}{\frac{t\_m}{l\_m}}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 5e+152)
   (asin (sqrt (/ 1.0 (+ 1.0 (/ 2.0 (/ (/ l_m t_m) (/ t_m l_m)))))))
   (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) {
	double tmp;
	if ((t_m / l_m) <= 5e+152) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))));
	} else {
		tmp = asin((l_m * (sqrt(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) <= 5d+152) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 / ((l_m / t_m) / (t_m / l_m)))))))
    else
        tmp = asin((l_m * (sqrt(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) <= 5e+152) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(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) <= 5e+152:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt(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) <= 5e+152)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 / Float64(Float64(l_m / t_m) / Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(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) <= 5e+152)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 / ((l_m / t_m) / (t_m / l_m)))))));
	else
		tmp = asin((l_m * (sqrt(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], 5e+152], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 / N[(N[(l$95$m / t$95$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[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5e152

   1. Initial program 90.8%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 87.9%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}\right)\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(2 \cdot \frac{1}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{2}{\frac{\ell}{t \cdot \frac{t}{\ell}}}\right)\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\ell}{t \cdot \frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \left(\frac{\frac{\ell}{t}}{\frac{t}{\ell}}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 90.9%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 90.0%

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

   if 5e152 < (/.f64 t l)

   1. Initial program 47.1%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 47.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot 2}{\ell}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{2}{\ell}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{1}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 47.1%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 47.1%

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

   7. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 47.1%

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

       1. frac-times N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot t}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{2 \cdot t}{\ell}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{2 \cdot t}}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{t}}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{t}{\ell}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       10. associate-/l/ N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\frac{\ell}{t}}{2}\right)\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), 2\right)\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 47.1%

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 47.1%

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

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\ell \cdot \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 99.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right) \]

   14. Simplified 99.5%

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

Alternative 6: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+54}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{t\_m}{\frac{l\_m}{\frac{t\_m}{\frac{l\_m}{2}}}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 1e+54)
   (asin (pow (+ 1.0 (/ t_m (/ l_m (/ t_m (/ l_m 2.0))))) -0.5))
   (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) {
	double tmp;
	if ((t_m / l_m) <= 1e+54) {
		tmp = asin(pow((1.0 + (t_m / (l_m / (t_m / (l_m / 2.0))))), -0.5));
	} else {
		tmp = asin((l_m * (sqrt(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) <= 1d+54) then
        tmp = asin(((1.0d0 + (t_m / (l_m / (t_m / (l_m / 2.0d0))))) ** (-0.5d0)))
    else
        tmp = asin((l_m * (sqrt(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) <= 1e+54) {
		tmp = Math.asin(Math.pow((1.0 + (t_m / (l_m / (t_m / (l_m / 2.0))))), -0.5));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(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) <= 1e+54:
		tmp = math.asin(math.pow((1.0 + (t_m / (l_m / (t_m / (l_m / 2.0))))), -0.5))
	else:
		tmp = math.asin((l_m * (math.sqrt(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) <= 1e+54)
		tmp = asin((Float64(1.0 + Float64(t_m / Float64(l_m / Float64(t_m / Float64(l_m / 2.0))))) ^ -0.5));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(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) <= 1e+54)
		tmp = asin(((1.0 + (t_m / (l_m / (t_m / (l_m / 2.0))))) ^ -0.5));
	else
		tmp = asin((l_m * (sqrt(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], 1e+54], N[ArcSin[N[Power[N[(1.0 + N[(t$95$m / N[(l$95$m / N[(t$95$m / N[(l$95$m / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.0000000000000001e54

   1. Initial program 90.0%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 87.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot 2}{\ell}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{2}{\ell}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{1}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 87.8%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 87.8%

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

   7. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 86.9%

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

       1. frac-times N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot t}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{2 \cdot t}{\ell}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{2 \cdot t}}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{t}}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{t}{\ell}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       10. associate-/l/ N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\frac{\ell}{t}}{2}\right)\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), 2\right)\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 89.2%

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 89.2%

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

       1. asin-lowering-asin.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1}{1 + \frac{\frac{t}{\ell}}{\frac{\frac{\ell}{t}}{2}}}}\right)\right) \]

       2. pow1/2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\left({\left(\frac{1}{1 + \frac{\frac{t}{\ell}}{\frac{\frac{\ell}{t}}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

       3. inv-pow N/A

          \[\leadsto \mathsf{asin.f64}\left(\left({\left({\left(1 + \frac{\frac{t}{\ell}}{\frac{\frac{\ell}{t}}{2}}\right)}^{-1}\right)}^{\frac{1}{2}}\right)\right) \]

       4. pow-pow N/A

          \[\leadsto \mathsf{asin.f64}\left(\left({\left(1 + \frac{\frac{t}{\ell}}{\frac{\frac{\ell}{t}}{2}}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right)\right) \]

       5. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{pow.f64}\left(\left(1 + \frac{\frac{t}{\ell}}{\frac{\frac{\ell}{t}}{2}}\right), \left(-1 \cdot \frac{1}{2}\right)\right)\right) \]

   13. Applied egg-rr 87.8%

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

   if 1.0000000000000001e54 < (/.f64 t l)

   1. Initial program 67.3%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 63.5%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot 2}{\ell}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{2}{\ell}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{1}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 63.5%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 63.5%

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

   7. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 63.5%

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

       1. frac-times N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot t}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{2 \cdot t}{\ell}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{2 \cdot t}}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{t}}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{t}{\ell}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       10. associate-/l/ N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\frac{\ell}{t}}{2}\right)\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), 2\right)\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 67.2%

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 67.2%

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

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\ell \cdot \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 99.2%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right) \]

   14. Simplified 99.2%

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

Alternative 7: 97.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{\frac{t\_m}{\frac{l\_m}{t\_m}}}{l\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.005)
   (asin (- 1.0 (/ (/ t_m (/ l_m t_m)) l_m)))
   (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) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin((1.0 - ((t_m / (l_m / t_m)) / l_m)));
	} else {
		tmp = asin((l_m * (sqrt(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) <= 0.005d0) then
        tmp = asin((1.0d0 - ((t_m / (l_m / t_m)) / l_m)))
    else
        tmp = asin((l_m * (sqrt(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) <= 0.005) {
		tmp = Math.asin((1.0 - ((t_m / (l_m / t_m)) / l_m)));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(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) <= 0.005:
		tmp = math.asin((1.0 - ((t_m / (l_m / t_m)) / l_m)))
	else:
		tmp = math.asin((l_m * (math.sqrt(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) <= 0.005)
		tmp = asin(Float64(1.0 - Float64(Float64(t_m / Float64(l_m / t_m)) / l_m)));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(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) <= 0.005)
		tmp = asin((1.0 - ((t_m / (l_m / t_m)) / l_m)));
	else
		tmp = asin((l_m * (sqrt(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], 0.005], N[ArcSin[N[(1.0 - N[(N[(t$95$m / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   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. Step-by-step derivation

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 87.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot 2}{\ell}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{2}{\ell}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{1}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 87.0%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 87.0%

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

   7. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 86.3%

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

   10. Taylor expanded in t around 0

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(1 + -1 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(1 + \left(\mathsf{neg}\left(\frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

       2. unsub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(1 - \frac{{t}^{2}}{{\ell}^{2}}\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({t}^{2}\right), \left({\ell}^{2}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(t \cdot t\right), \left({\ell}^{2}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left({\ell}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\ell \cdot \ell\right)\right)\right)\right) \]

       8. *-lowering-*.f64 61.4%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right) \]

   12. Simplified 61.4%

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

       1. associate-/r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{\frac{t \cdot t}{\ell}}{\ell}\right)\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t \cdot t}{\ell}\right), \ell\right)\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot t\right), \ell\right)\right)\right) \]

       4. associate-/r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \ell\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \ell\right)\right)\right) \]

       6. /-lowering-/.f64 68.0%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \ell\right)\right)\right) \]

   14. Applied egg-rr 68.0%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 74.1%

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

      1. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      5. associate-+l- N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

      7. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

      9. distribute-frac-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      10. sub0-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      11. associate-+l- N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      12. neg-sub0 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      14. sub-neg N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

   3. Simplified 71.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot 2}{\ell}\right)\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \left(\left(t \cdot \frac{t}{\ell}\right) \cdot \frac{2}{\ell}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(t \cdot \frac{1}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{t}{\frac{\ell}{t}}\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \left(\frac{\ell}{t}\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \left(\frac{2}{\ell}\right)\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 71.1%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, Omc\right)\right), Omc\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 71.1%

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

   7. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, t\right)\right), \mathsf{/.f64}\left(2, \ell\right)\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 70.4%

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

       1. frac-times N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{t \cdot 2}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot t}{\frac{\ell}{t} \cdot \ell}\right)\right)\right)\right)\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{2 \cdot t}{\ell}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       4. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{2 \cdot t}}}{\frac{\ell}{t}}\right)\right)\right)\right)\right) \]

       5. associate-/l/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       6. associate-/r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{\frac{\ell}{t}}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       7. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\frac{\frac{t}{\ell}}{\frac{\ell}{2 \cdot t}}\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\ell}{2 \cdot t}\right)\right)\right)\right)\right)\right) \]

       10. associate-/l/ N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{\frac{\ell}{t}}{2}\right)\right)\right)\right)\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\left(\frac{\ell}{t}\right), 2\right)\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 73.3%

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, t\right), 2\right)\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 73.3%

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

   12. Taylor expanded in t around inf

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   13. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\ell \cdot \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right)\right)\right) \]

       4. sqrt-lowering-sqrt.f64 98.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right) \]

   14. Simplified 98.5%

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

Alternative 8: 51.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} 1 \end{array} \]

FPCore

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

C

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

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(1.0d0)
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(1.0);
}

Python

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

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(1.0)
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(1.0);
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[1.0], $MachinePrecision]

Derivation

1. Initial program 85.7%

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

   1. asin-lowering-asin.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   10. sub0-neg N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   11. associate-+l- N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

3. Simplified 83.1%

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

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 49.0%

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

7. Simplified 49.0%

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

8. Taylor expanded in Om around 0

   \[\leadsto \mathsf{asin.f64}\left(\color{blue}{1}\right) \]
9. Step-by-step derivation

10. Simplified 53.6%

    \[\leadsto \sin^{-1} \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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)))))))