Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.9%

Time: 18.1s

Alternatives: 9

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: 84.0% 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: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\left(1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+141)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin (/ (* l_m (sqrt (* (- 1.0 (/ Om (/ Omc (/ Om Omc)))) 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+141) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin(((l_m * sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) * 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+141) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_m)))))))
    else
        tmp = asin(((l_m * sqrt(((1.0d0 - (om / (omc / (om / omc)))) * 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+141) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) * 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+141:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))))
	else:
		tmp = math.asin(((l_m * math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) * 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+141)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(Float64(1.0 - Float64(Om / Float64(Omc / Float64(Om / Omc)))) * 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+141)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	else
		tmp = asin(((l_m * sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) * 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+141], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000025e141

   1. Initial program 90.4%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.4%

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

   4. Applied egg-rr 90.4%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.4%

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

   6. Applied egg-rr 90.4%

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

   if 5.00000000000000025e141 < (/.f64 t l)

   1. Initial program 43.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 40.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 33.0%

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

   7. Simplified 33.0%

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

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

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

      2. associate-*r/ N/A

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

      3. sqrt-div N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

   9. Applied egg-rr 99.9%

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

Alternative 2: 98.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + 2 \cdot \frac{\frac{t\_m}{\frac{l\_m}{t\_m}}}{l\_m}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{\left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+14)
   (asin (pow (+ 1.0 (* 2.0 (/ (/ t_m (/ l_m t_m)) l_m))) -0.5))
   (asin (* (/ l_m t_m) (sqrt (* (- 1.0 (/ (/ Om Omc) (/ Omc Om))) 0.5))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+14) {
		tmp = asin(pow((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))), -0.5));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) * 0.5))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+14) then
        tmp = asin(((1.0d0 + (2.0d0 * ((t_m / (l_m / t_m)) / l_m))) ** (-0.5d0)))
    else
        tmp = asin(((l_m / t_m) * sqrt(((1.0d0 - ((om / omc) / (omc / om))) * 0.5d0))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+14) {
		tmp = Math.asin(Math.pow((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))), -0.5));
	} else {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) * 0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+14:
		tmp = math.asin(math.pow((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))), -0.5))
	else:
		tmp = math.asin(((l_m / t_m) * math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) * 0.5))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+14)
		tmp = asin((Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / Float64(l_m / t_m)) / l_m))) ^ -0.5));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) * 0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+14)
		tmp = asin(((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))) ^ -0.5));
	else
		tmp = asin(((l_m / t_m) * sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+14], N[ArcSin[N[Power[N[(1.0 + N[(2.0 * N[(N[(t$95$m / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5e14

   1. Initial program 89.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 77.7%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 76.3%

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

   9. Simplified 76.3%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-/r* N/A

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

       5. associate-*r/ N/A

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

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

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

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

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

       10. /-lowering-/.f64 85.7%

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

   11. Applied egg-rr 85.7%

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

   if 5e14 < (/.f64 t l)

   1. Initial program 66.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 43.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 34.7%

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

   7. Simplified 34.7%

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

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

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

      2. associate-*r/ N/A

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

      3. sqrt-div N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

   9. Applied egg-rr 99.6%

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

       1. sqrt-prod N/A

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

       2. associate-*r* N/A

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

       3. associate-*l/ N/A

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

       4. associate-/r/ N/A

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

       5. div-inv N/A

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

       6. times-frac N/A

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

       7. div-inv N/A

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

       8. clear-num N/A

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

       9. /-rgt-identity N/A

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

       10. sqrt-prod N/A

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

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

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

       12. /-lowering-/.f64 N/A

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

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

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

   11. Applied egg-rr 99.6%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + 2 \cdot \frac{\frac{t}{\frac{\ell}{t}}}{\ell}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\left(1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\right) \cdot 0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + 2 \cdot \frac{\frac{t\_m}{\frac{l\_m}{t\_m}}}{l\_m}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+14)
   (asin (pow (+ 1.0 (* 2.0 (/ (/ t_m (/ l_m t_m)) l_m))) -0.5))
   (asin (* (/ l_m t_m) (sqrt 0.5)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+14) {
		tmp = asin(pow((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))), -0.5));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+14) then
        tmp = asin(((1.0d0 + (2.0d0 * ((t_m / (l_m / t_m)) / l_m))) ** (-0.5d0)))
    else
        tmp = asin(((l_m / t_m) * sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+14) {
		tmp = Math.asin(Math.pow((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))), -0.5));
	} else {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+14:
		tmp = math.asin(math.pow((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))), -0.5))
	else:
		tmp = math.asin(((l_m / t_m) * math.sqrt(0.5)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+14)
		tmp = asin((Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / Float64(l_m / t_m)) / l_m))) ^ -0.5));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+14)
		tmp = asin(((1.0 + (2.0 * ((t_m / (l_m / t_m)) / l_m))) ^ -0.5));
	else
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+14], N[ArcSin[N[Power[N[(1.0 + N[(2.0 * N[(N[(t$95$m / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5e14

   1. Initial program 89.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 77.7%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 76.3%

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

   9. Simplified 76.3%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-/r* N/A

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

       5. associate-*r/ N/A

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

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

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

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

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

       10. /-lowering-/.f64 85.7%

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

   11. Applied egg-rr 85.7%

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

   if 5e14 < (/.f64 t l)

   1. Initial program 66.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 43.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 34.7%

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

   7. Simplified 34.7%

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

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

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

      2. associate-*r/ N/A

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

      3. sqrt-div N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in Om around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 98.0%

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

   12. Simplified 98.0%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 98.1%

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

   14. Applied egg-rr 98.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+14}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + 2 \cdot \frac{\frac{t}{\frac{\ell}{t}}}{\ell}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.05)
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (asin (* (/ l_m t_m) (sqrt 0.5)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.05) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.05d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin(((l_m / t_m) * sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.05) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.05:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	else:
		tmp = math.asin(((l_m / t_m) * math.sqrt(0.5)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.05)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.05)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.05], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.050000000000000003

   1. Initial program 89.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 77.7%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
   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 59.4%

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

   7. Simplified 59.4%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 67.0%

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

   9. Applied egg-rr 67.0%

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

   if 0.050000000000000003 < (/.f64 t l)

   1. Initial program 66.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 43.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 34.7%

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

   7. Simplified 34.7%

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

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

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

      2. associate-*r/ N/A

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

      3. sqrt-div N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in Om around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 98.0%

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

   12. Simplified 98.0%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 98.1%

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

   14. Applied egg-rr 98.1%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.05)
   (asin (+ 1.0 (* (/ Om Omc) (/ (* Om -0.5) Omc))))
   (asin (* (/ l_m t_m) (sqrt 0.5)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.05) {
		tmp = asin((1.0 + ((Om / Omc) * ((Om * -0.5) / Omc))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.05d0) then
        tmp = asin((1.0d0 + ((om / omc) * ((om * (-0.5d0)) / omc))))
    else
        tmp = asin(((l_m / t_m) * sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.05) {
		tmp = Math.asin((1.0 + ((Om / Omc) * ((Om * -0.5) / Omc))));
	} else {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.05:
		tmp = math.asin((1.0 + ((Om / Omc) * ((Om * -0.5) / Omc))))
	else:
		tmp = math.asin(((l_m / t_m) * math.sqrt(0.5)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.05)
		tmp = asin(Float64(1.0 + Float64(Float64(Om / Omc) * Float64(Float64(Om * -0.5) / Omc))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.05)
		tmp = asin((1.0 + ((Om / Omc) * ((Om * -0.5) / Omc))));
	else
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.05], N[ArcSin[N[(1.0 + N[(N[(Om / Omc), $MachinePrecision] * N[(N[(Om * -0.5), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.050000000000000003

   1. Initial program 89.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 77.7%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
   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 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 59.4%

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

   10. Simplified 59.4%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

       6. /-lowering-/.f64 66.7%

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

   12. Applied egg-rr 66.7%

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

   if 0.050000000000000003 < (/.f64 t l)

   1. Initial program 66.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 43.0%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 34.7%

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

   7. Simplified 34.7%

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

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

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

      2. associate-*r/ N/A

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

      3. sqrt-div N/A

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

      4. pow2 N/A

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

      5. sqrt-pow1 N/A

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

      6. metadata-eval N/A

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

      7. unpow1 N/A

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

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

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in Om around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 98.0%

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

   12. Simplified 98.0%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 98.1%

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

   14. Applied egg-rr 98.1%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 5.5e-6)
   (asin (* l_m (/ (sqrt 0.5) t_m)))
   (asin (+ 1.0 (* (/ Om Omc) (/ (* Om -0.5) 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 (l_m <= 5.5e-6) {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	} else {
		tmp = asin((1.0 + ((Om / Omc) * ((Om * -0.5) / 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 (l_m <= 5.5d-6) then
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    else
        tmp = asin((1.0d0 + ((om / omc) * ((om * (-0.5d0)) / 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 (l_m <= 5.5e-6) {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	} else {
		tmp = Math.asin((1.0 + ((Om / Omc) * ((Om * -0.5) / 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 l_m <= 5.5e-6:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	else:
		tmp = math.asin((1.0 + ((Om / Omc) * ((Om * -0.5) / Omc))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 5.5e-6)
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	else
		tmp = asin(Float64(1.0 + Float64(Float64(Om / Omc) * Float64(Float64(Om * -0.5) / 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 (l_m <= 5.5e-6)
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	else
		tmp = asin((1.0 + ((Om / Omc) * ((Om * -0.5) / 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[l$95$m, 5.5e-6], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 + N[(N[(Om / Omc), $MachinePrecision] * N[(N[(Om * -0.5), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.4999999999999999e-6

   1. Initial program 82.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 70.2%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 25.2%

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

   7. Simplified 25.2%

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

   8. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   9. 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 32.9%

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

   10. Simplified 32.9%

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

   if 5.4999999999999999e-6 < l

   1. Initial program 89.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 69.9%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
   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 67.2%

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

   7. Simplified 67.2%

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

   8. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 67.2%

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

   10. Simplified 67.2%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

       6. /-lowering-/.f64 74.1%

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

   12. Applied egg-rr 74.1%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.08 \cdot 10^{-134}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.08e-134)
   (asin (/ (* -0.5 (* Om Om)) (* Omc Omc)))
   (asin (+ 1.0 (* (/ Om Omc) (/ (* Om -0.5) 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 (l_m <= 1.08e-134) {
		tmp = asin(((-0.5 * (Om * Om)) / (Omc * Omc)));
	} else {
		tmp = asin((1.0 + ((Om / Omc) * ((Om * -0.5) / 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 (l_m <= 1.08d-134) then
        tmp = asin((((-0.5d0) * (om * om)) / (omc * omc)))
    else
        tmp = asin((1.0d0 + ((om / omc) * ((om * (-0.5d0)) / 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 (l_m <= 1.08e-134) {
		tmp = Math.asin(((-0.5 * (Om * Om)) / (Omc * Omc)));
	} else {
		tmp = Math.asin((1.0 + ((Om / Omc) * ((Om * -0.5) / 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 l_m <= 1.08e-134:
		tmp = math.asin(((-0.5 * (Om * Om)) / (Omc * Omc)))
	else:
		tmp = math.asin((1.0 + ((Om / Omc) * ((Om * -0.5) / Omc))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.08e-134)
		tmp = asin(Float64(Float64(-0.5 * Float64(Om * Om)) / Float64(Omc * Omc)));
	else
		tmp = asin(Float64(1.0 + Float64(Float64(Om / Omc) * Float64(Float64(Om * -0.5) / 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 (l_m <= 1.08e-134)
		tmp = asin(((-0.5 * (Om * Om)) / (Omc * Omc)));
	else
		tmp = asin((1.0 + ((Om / Omc) * ((Om * -0.5) / 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[l$95$m, 1.08e-134], N[ArcSin[N[(N[(-0.5 * N[(Om * Om), $MachinePrecision]), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 + N[(N[(Om / Omc), $MachinePrecision] * N[(N[(Om * -0.5), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.07999999999999999e-134

   1. Initial program 84.1%

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

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
   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 42.0%

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

   7. Simplified 42.0%

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

   8. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 42.0%

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

   10. Simplified 42.0%

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

   11. Taylor expanded in Om around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 15.3%

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

   13. Simplified 15.3%

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

   if 1.07999999999999999e-134 < l

   1. Initial program 85.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 68.3%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
   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 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 57.5%

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

   10. Simplified 57.5%

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

       1. associate-*r* N/A

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

       2. times-frac N/A

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

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

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

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

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

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

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

       6. /-lowering-/.f64 63.9%

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

   12. Applied egg-rr 63.9%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.08 \cdot 10^{-134}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.08 \cdot 10^{-134}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.08e-134) (asin (/ (* -0.5 (* Om Om)) (* Omc Omc))) (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.08e-134) {
		tmp = asin(((-0.5 * (Om * Om)) / (Omc * Omc)));
	} else {
		tmp = asin(1.0);
	}
	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 (l_m <= 1.08d-134) then
        tmp = asin((((-0.5d0) * (om * om)) / (omc * omc)))
    else
        tmp = asin(1.0d0)
    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 (l_m <= 1.08e-134) {
		tmp = Math.asin(((-0.5 * (Om * Om)) / (Omc * Omc)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.08e-134:
		tmp = math.asin(((-0.5 * (Om * Om)) / (Omc * Omc)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.08e-134)
		tmp = asin(Float64(Float64(-0.5 * Float64(Om * Om)) / Float64(Omc * Omc)));
	else
		tmp = asin(1.0);
	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 (l_m <= 1.08e-134)
		tmp = asin(((-0.5 * (Om * Om)) / (Omc * Omc)));
	else
		tmp = asin(1.0);
	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[l$95$m, 1.08e-134], N[ArcSin[N[(N[(-0.5 * N[(Om * Om), $MachinePrecision]), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.07999999999999999e-134

   1. Initial program 84.1%

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

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
   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 42.0%

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

   7. Simplified 42.0%

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

   8. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 42.0%

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

   10. Simplified 42.0%

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

   11. Taylor expanded in Om around inf

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

       1. associate-*r/ N/A

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

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

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

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

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

       4. unpow2 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 15.3%

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

   13. Simplified 15.3%

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

   if 1.07999999999999999e-134 < l

   1. Initial program 85.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 68.3%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
   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 57.5%

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

   7. Simplified 57.5%

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

   8. Taylor expanded in Om around 0

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

   10. Simplified 63.7%

       \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.3% 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 84.4%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
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 70.1%

   \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + \frac{\frac{2 \cdot \left(t \cdot t\right)}{\ell}}{\ell}}}\right)} \]
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 47.3%

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

7. Simplified 47.3%

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

8. Taylor expanded in Om around 0

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

10. Simplified 53.2%

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

Reproduce

herbie shell --seed 2024192 
(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)))))))