Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.4%

Time: 18.3s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\frac{\sqrt{0.5}}{t\_m} + \frac{\left(l\_m \cdot l\_m\right) \cdot -0.125}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot \sqrt{0.5}\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+152)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m)))))))
   (asin
    (*
     l_m
     (+
      (/ (sqrt 0.5) t_m)
      (/ (* (* l_m l_m) -0.125) (* (* t_m t_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+152) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = asin((l_m * ((sqrt(0.5) / t_m) + (((l_m * l_m) * -0.125) / ((t_m * t_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+152) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_m)))))))
    else
        tmp = asin((l_m * ((sqrt(0.5d0) / t_m) + (((l_m * l_m) * (-0.125d0)) / ((t_m * t_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+152) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = Math.asin((l_m * ((Math.sqrt(0.5) / t_m) + (((l_m * l_m) * -0.125) / ((t_m * t_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+152:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))))
	else:
		tmp = math.asin((l_m * ((math.sqrt(0.5) / t_m) + (((l_m * l_m) * -0.125) / ((t_m * t_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+152)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(Float64(sqrt(0.5) / t_m) + Float64(Float64(Float64(l_m * l_m) * -0.125) / Float64(Float64(t_m * t_m) * Float64(t_m * sqrt(0.5)))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+152)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	else
		tmp = asin((l_m * ((sqrt(0.5) / t_m) + (((l_m * l_m) * -0.125) / ((t_m * t_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+152], 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[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision] + N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * -0.125), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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. *-lowering-*.f64 N/A

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

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

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

      4. /-lowering-/.f64 94.7%

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

   4. Applied egg-rr 94.7%

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

   if 5e152 < (/.f64 t l)

   1. Initial program 59.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 56.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 59.9%

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

   8. Applied egg-rr 59.9%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 59.9%

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

   12. Taylor expanded in l around 0

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} + \frac{\sqrt{\frac{1}{2}}}{t}\right)\right)}\right) \]
   13. Step-by-step derivation

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

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{-1}{8} \cdot \frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} + \frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{\sqrt{\frac{1}{2}}}{t} + \frac{-1}{8} \cdot \frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(\frac{\sqrt{\frac{1}{2}}}{t}\right), \left(\frac{-1}{8} \cdot \frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right), \left(\frac{-1}{8} \cdot \frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \left(\frac{-1}{8} \cdot \frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}}\right)\right)\right)\right) \]

       6. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \left(\frac{\frac{-1}{8} \cdot {\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot {\ell}^{2}\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\left({\ell}^{2} \cdot \frac{-1}{8}\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{-1}{8}\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{-1}{8}\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       12. unpow3 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \left(\left(\left(t \cdot t\right) \cdot t\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \left(\left({t}^{2} \cdot t\right) \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right) \]

       14. associate-*l* N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \left({t}^{2} \cdot \left(t \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left({t}^{2}\right), \left(t \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\left(t \cdot t\right), \left(t \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(t \cdot \sqrt{\frac{1}{2}}\right)\right)\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \left(\sqrt{\frac{1}{2}}\right)\right)\right)\right)\right)\right)\right) \]

       19. sqrt-lowering-sqrt.f64 99.7%

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{-1}{8}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right) \]

   14. Simplified 99.7%

       \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\frac{\sqrt{0.5}}{t} + \frac{\left(\ell \cdot \ell\right) \cdot -0.125}{\left(t \cdot t\right) \cdot \left(t \cdot \sqrt{0.5}\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+147)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m)))))))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+147) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+147) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_m)))))))
    else
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+147) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+147:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))))
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+147)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+147)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+147], 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[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.0000000000000002e147

   1. Initial program 94.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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. *-lowering-*.f64 N/A

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

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

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

      4. /-lowering-/.f64 94.7%

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

   4. Applied egg-rr 94.7%

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

   if 5.0000000000000002e147 < (/.f64 t l)

   1. Initial program 62.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 56.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 62.0%

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

   6. Applied egg-rr 62.0%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 62.0%

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

   8. Applied egg-rr 62.0%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 62.0%

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

   12. Taylor expanded in t around inf

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 99.6%

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

   14. Simplified 99.6%

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

Alternative 3: 85.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.35 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}{1 + \frac{t\_m}{\frac{l\_m}{t\_m}} \cdot \frac{2}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 2.35e+208)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om (/ Omc Om)) Omc))
      (+ 1.0 (* (/ t_m (/ l_m t_m)) (/ 2.0 l_m))))))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 2.35e+208) {
		tmp = asin(sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (t_m <= 2.35d+208) then
        tmp = asin(sqrt(((1.0d0 - ((om / (omc / om)) / omc)) / (1.0d0 + ((t_m / (l_m / t_m)) * (2.0d0 / l_m))))))
    else
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 2.35e+208) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 2.35e+208:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))))
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 2.35e+208)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Float64(Omc / Om)) / Omc)) / Float64(1.0 + Float64(Float64(t_m / Float64(l_m / t_m)) * Float64(2.0 / l_m))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (t_m <= 2.35e+208)
		tmp = asin(sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 2.35e+208], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(t$95$m / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.34999999999999993e208

   1. Initial program 91.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 77.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

      16. /-lowering-/.f64 90.3%

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

   8. Applied egg-rr 90.3%

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

   if 2.34999999999999993e208 < t

   1. Initial program 61.7%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 45.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 61.7%

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

   8. Applied egg-rr 61.7%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 61.7%

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

   12. Taylor expanded in t around inf

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 77.1%

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

   14. Simplified 77.1%

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

Alternative 4: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.4 \cdot 10^{-160}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{elif}\;l\_m \leq 2.3 \cdot 10^{+115}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{l\_m \cdot l\_m}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.4e-160)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (if (<= l_m 2.3e+115)
     (asin (pow (+ 1.0 (/ (* 2.0 (* t_m t_m)) (* l_m l_m))) -0.5))
     (asin (sqrt (- 1.0 (/ (/ Om (/ Omc Om)) Omc)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.4e-160) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else if (l_m <= 2.3e+115) {
		tmp = asin(pow((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))), -0.5));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 1.4d-160) then
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    else if (l_m <= 2.3d+115) then
        tmp = asin(((1.0d0 + ((2.0d0 * (t_m * t_m)) / (l_m * l_m))) ** (-0.5d0)))
    else
        tmp = asin(sqrt((1.0d0 - ((om / (omc / om)) / omc))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.4e-160) {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	} else if (l_m <= 2.3e+115) {
		tmp = Math.asin(Math.pow((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))), -0.5));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.4e-160:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	elif l_m <= 2.3e+115:
		tmp = math.asin(math.pow((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))), -0.5))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.4e-160)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	elseif (l_m <= 2.3e+115)
		tmp = asin((Float64(1.0 + Float64(Float64(2.0 * Float64(t_m * t_m)) / Float64(l_m * l_m))) ^ -0.5));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Float64(Omc / Om)) / Omc))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 1.4e-160)
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	elseif (l_m <= 2.3e+115)
		tmp = asin(((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))) ^ -0.5));
	else
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 1.4e-160], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 2.3e+115], N[ArcSin[N[Power[N[(1.0 + N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.40000000000000008e-160

   1. Initial program 91.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 74.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 87.4%

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

   8. Applied egg-rr 87.4%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 86.6%

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

   12. Taylor expanded in t around inf

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 39.7%

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

   14. Simplified 39.7%

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

   if 1.40000000000000008e-160 < l < 2.30000000000000004e115

   1. Initial program 80.5%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 73.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 80.6%

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

   8. Applied egg-rr 80.6%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. sqrt-pow1 N/A

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

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

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

   10. Applied egg-rr 80.6%

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

   11. Taylor expanded in Om around 0

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)}, \frac{-1}{2}\right)\right) \]
   12. 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 78.8%

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

   13. Simplified 78.8%

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

   if 2.30000000000000004e115 < l

   1. Initial program 98.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 78.8%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
   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 74.1%

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

   7. Simplified 74.1%

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

      1. associate-/r* N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 87.0%

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

   9. Applied egg-rr 87.0%

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

Alternative 5: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 1.8 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t\_m}{\frac{l\_m}{t\_m}} \cdot \frac{2}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 1.8e+208)
   (asin (sqrt (/ 1.0 (+ 1.0 (* (/ t_m (/ l_m t_m)) (/ 2.0 l_m))))))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 1.8e+208) {
		tmp = asin(sqrt((1.0 / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (t_m <= 1.8d+208) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + ((t_m / (l_m / t_m)) * (2.0d0 / l_m))))))
    else
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 1.8e+208) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 1.8e+208:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))))
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 1.8e+208)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(Float64(t_m / Float64(l_m / t_m)) * Float64(2.0 / l_m))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (t_m <= 1.8e+208)
		tmp = asin(sqrt((1.0 / (1.0 + ((t_m / (l_m / t_m)) * (2.0 / l_m))))));
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 1.8e+208], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(N[(t$95$m / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.80000000000000001e208

   1. Initial program 91.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 77.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. un-div-inv N/A

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

      8. clear-num N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

      16. /-lowering-/.f64 90.3%

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

   8. Applied egg-rr 90.3%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 89.8%

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

   if 1.80000000000000001e208 < t

   1. Initial program 61.7%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 45.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 61.7%

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

   8. Applied egg-rr 61.7%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 61.7%

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

   12. Taylor expanded in t around inf

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 77.1%

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

   14. Simplified 77.1%

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

Alternative 6: 85.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 7.1 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{t\_m \cdot \left(\frac{t\_m}{l\_m} \cdot 2\right)}{l\_m}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 7.1e+208)
   (asin (pow (+ 1.0 (/ (* t_m (* (/ t_m l_m) 2.0)) l_m)) -0.5))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 7.1e+208) {
		tmp = asin(pow((1.0 + ((t_m * ((t_m / l_m) * 2.0)) / l_m)), -0.5));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (t_m <= 7.1d+208) then
        tmp = asin(((1.0d0 + ((t_m * ((t_m / l_m) * 2.0d0)) / l_m)) ** (-0.5d0)))
    else
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 7.1e+208) {
		tmp = Math.asin(Math.pow((1.0 + ((t_m * ((t_m / l_m) * 2.0)) / l_m)), -0.5));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 7.1e+208:
		tmp = math.asin(math.pow((1.0 + ((t_m * ((t_m / l_m) * 2.0)) / l_m)), -0.5))
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 7.1e+208)
		tmp = asin((Float64(1.0 + Float64(Float64(t_m * Float64(Float64(t_m / l_m) * 2.0)) / l_m)) ^ -0.5));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (t_m <= 7.1e+208)
		tmp = asin(((1.0 + ((t_m * ((t_m / l_m) * 2.0)) / l_m)) ^ -0.5));
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 7.1e+208], N[ArcSin[N[Power[N[(1.0 + N[(N[(t$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.10000000000000019e208

   1. Initial program 91.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 77.3%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 84.1%

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

   6. Applied egg-rr 84.1%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 89.5%

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

   8. Applied egg-rr 89.5%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 89.0%

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

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

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

       2. inv-pow N/A

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

       3. sqrt-pow1 N/A

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

       4. metadata-eval N/A

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

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

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

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

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

       7. associate-/r/ N/A

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

       8. associate-*r/ N/A

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

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

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

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

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

       11. associate-/l* N/A

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

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

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

       13. /-lowering-/.f64 89.8%

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

   13. Applied egg-rr 89.8%

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

   if 7.10000000000000019e208 < t

   1. Initial program 61.7%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 45.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 50.9%

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

   6. Applied egg-rr 50.9%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 61.7%

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

   8. Applied egg-rr 61.7%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 61.7%

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

   12. Taylor expanded in t around inf

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 77.1%

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

   14. Simplified 77.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.1 \cdot 10^{+208}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{t \cdot \left(\frac{t}{\ell} \cdot 2\right)}{\ell}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 7.6e-25)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin (sqrt (- 1.0 (/ (/ Om (/ Omc Om)) Omc))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 7.6e-25) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 7.6d-25) then
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    else
        tmp = asin(sqrt((1.0d0 - ((om / (omc / om)) / omc))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 7.6e-25) {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 7.6e-25:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 7.6e-25)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Float64(Omc / Om)) / Omc))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 7.6e-25)
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 7.6e-25], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.5999999999999996e-25

   1. Initial program 88.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 74.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 85.9%

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

   8. Applied egg-rr 85.9%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 85.3%

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

   12. Taylor expanded in t around inf

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 36.8%

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

   14. Simplified 36.8%

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

   if 7.5999999999999996e-25 < l

   1. Initial program 92.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 76.2%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
   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 66.5%

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

   7. Simplified 66.5%

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

      1. associate-/r* N/A

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

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

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

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

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

      6. /-lowering-/.f64 77.2%

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

   9. Applied egg-rr 77.2%

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

Alternative 8: 74.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 7.6e-25)
   (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 <= 7.6e-25) {
		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 <= 7.6d-25) 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 <= 7.6e-25) {
		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 <= 7.6e-25:
		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 <= 7.6e-25)
		tmp = asin(Float64(Float64(l_m * 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 <= 7.6e-25)
		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, 7.6e-25], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $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 < 7.5999999999999996e-25

   1. Initial program 88.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 74.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 79.9%

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

   6. Applied egg-rr 79.9%

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

      1. associate-*r* N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. clear-num N/A

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

      9. div-inv N/A

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

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

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

      11. /-lowering-/.f64 85.9%

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

   8. Applied egg-rr 85.9%

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

   9. Taylor expanded in Om around 0

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

   11. Simplified 85.3%

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

   12. Taylor expanded in t around inf

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

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

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

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

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

       3. sqrt-lowering-sqrt.f64 36.8%

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

   14. Simplified 36.8%

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

   if 7.5999999999999996e-25 < l

   1. Initial program 92.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 76.2%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
   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 66.5%

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

   7. Simplified 66.5%

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

   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 66.5%

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

   10. Simplified 66.5%

       \[\leadsto \sin^{-1} \color{blue}{\left(1 + \frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)} \]
   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. *-commutative N/A

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

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

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

       7. /-lowering-/.f64 77.2%

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

   12. Applied egg-rr 77.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 7.6 \cdot 10^{-25}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 9e-163)
   (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 <= 9e-163) {
		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 <= 9d-163) 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 <= 9e-163) {
		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 <= 9e-163:
		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 <= 9e-163)
		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 <= 9e-163)
		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, 9e-163], 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 < 8.9999999999999995e-163

   1. Initial program 90.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 75.1%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
   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 46.4%

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

   7. Simplified 46.4%

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

   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 45.9%

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

   10. Simplified 45.9%

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

   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. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 15.7%

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

   13. Simplified 15.7%

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

   if 8.9999999999999995e-163 < l

   1. Initial program 88.2%

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

      1. 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 75.3%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
   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 55.9%

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

   7. Simplified 55.9%

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

   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 55.9%

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

   10. Simplified 55.9%

       \[\leadsto \sin^{-1} \color{blue}{\left(1 + \frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)} \]
   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. *-commutative N/A

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

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

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

       7. /-lowering-/.f64 64.4%

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

   12. Applied egg-rr 64.4%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 9 \cdot 10^{-163}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + \frac{Om}{Omc} \cdot \frac{Om \cdot -0.5}{Omc}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.2 \cdot 10^{-162}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 3.2e-162) (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 <= 3.2e-162) {
		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 <= 3.2d-162) 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 <= 3.2e-162) {
		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 <= 3.2e-162:
		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 <= 3.2e-162)
		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 <= 3.2e-162)
		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, 3.2e-162], 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 < 3.19999999999999975e-162

   1. Initial program 90.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 75.1%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
   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 46.4%

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

   7. Simplified 46.4%

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

   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 45.9%

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

   10. Simplified 45.9%

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

   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. *-commutative N/A

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

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

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 15.7%

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

   13. Simplified 15.7%

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

   if 3.19999999999999975e-162 < l

   1. Initial program 88.2%

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

      1. 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 75.3%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
   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 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in Om around 0

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

   10. Simplified 64.4%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-162}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5 \cdot \left(Om \cdot Om\right)}{Omc \cdot Omc}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.5% 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 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 75.2%

   \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 + t \cdot \left(2 \cdot \frac{t}{\ell \cdot \ell}\right)}}\right)} \]
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 50.1%

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

7. Simplified 50.1%

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

8. Taylor expanded in Om around 0

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

10. Simplified 55.6%

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

Reproduce

herbie shell --seed 2024288 
(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)))))))