Toniolo and Linder, Equation (2)

Percentage Accurate: 83.4% → 98.8%

Time: 16.5s

Alternatives: 12

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: 83.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{1 + \frac{t\_m}{l\_m} \cdot \frac{2}{\frac{l\_m}{t\_m}}}{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+143)
   (asin
    (pow
     (/
      (+ 1.0 (* (/ t_m l_m) (/ 2.0 (/ l_m t_m))))
      (- 1.0 (/ (/ Om Omc) (/ Omc Om))))
     -0.5))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+143) {
		tmp = asin(pow(((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))) / (1.0 - ((Om / Omc) / (Omc / Om)))), -0.5));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+143) then
        tmp = asin((((1.0d0 + ((t_m / l_m) * (2.0d0 / (l_m / t_m)))) / (1.0d0 - ((om / omc) / (omc / om)))) ** (-0.5d0)))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+143) {
		tmp = Math.asin(Math.pow(((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))) / (1.0 - ((Om / Omc) / (Omc / Om)))), -0.5));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+143:
		tmp = math.asin(math.pow(((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))) / (1.0 - ((Om / Omc) / (Omc / Om)))), -0.5))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+143)
		tmp = asin((Float64(Float64(1.0 + Float64(Float64(t_m / l_m) * Float64(2.0 / Float64(l_m / t_m)))) / Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))) ^ -0.5));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+143)
		tmp = asin((((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))) / (1.0 - ((Om / Omc) / (Omc / Om)))) ^ -0.5));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+143], N[ArcSin[N[Power[N[(N[(1.0 + N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000012e143

   1. Initial program 92.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 87.7%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   6. Applied egg-rr 92.0%

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

   if 5.00000000000000012e143 < (/.f64 t l)

   1. Initial program 39.3%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 36.5%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 27.2%

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

   7. Simplified 27.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 87.9%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

       7. sqrt-undiv N/A

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

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

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

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

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

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

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

       11. associate-/r/ N/A

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

       12. associate-*l/ N/A

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

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

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

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

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

       15. *-lowering-*.f64 81.9%

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

   11. Applied egg-rr 81.9%

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

   12. Taylor expanded in Om around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{1 + \frac{t}{\ell} \cdot \frac{2}{\frac{\ell}{t}}}{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+143)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_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+143) {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_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+143) then
        tmp = asin(sqrt(((1.0d0 - ((om * (om / omc)) / omc)) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_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+143) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_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+143:
		tmp = math.asin(math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_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+143)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+143)
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_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+143], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000012e143

   1. Initial program 92.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 87.7%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

      5. associate-/r/ N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

      16. /-lowering-/.f64 92.0%

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

   6. Applied egg-rr 92.0%

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

   if 5.00000000000000012e143 < (/.f64 t l)

   1. Initial program 39.3%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 36.5%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 27.2%

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

   7. Simplified 27.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 87.9%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

       7. sqrt-undiv N/A

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

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

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

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

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

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

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

       11. associate-/r/ N/A

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

       12. associate-*l/ N/A

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

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

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

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

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

       15. *-lowering-*.f64 81.9%

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

   11. Applied egg-rr 81.9%

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

   12. Taylor expanded in Om around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t\_m \cdot \frac{\frac{t\_m \cdot 2}{l\_m}}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{l\_m}{t\_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+26)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (+ 1.0 (* t_m (/ (/ (* t_m 2.0) l_m) l_m))))))
   (asin (/ (/ l_m t_m) (sqrt 2.0)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+26) {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_m))))));
	} else {
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	}
	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 ((t_m / l_m) <= 2d+26) then
        tmp = asin(sqrt(((1.0d0 - ((om * (om / omc)) / omc)) / (1.0d0 + (t_m * (((t_m * 2.0d0) / l_m) / l_m))))))
    else
        tmp = asin(((l_m / t_m) / sqrt(2.0d0)))
    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 ((t_m / l_m) <= 2e+26) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_m))))));
	} else {
		tmp = Math.asin(((l_m / t_m) / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 2e+26:
		tmp = math.asin(math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_m))))))
	else:
		tmp = math.asin(((l_m / t_m) / math.sqrt(2.0)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+26)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / Float64(1.0 + Float64(t_m * Float64(Float64(Float64(t_m * 2.0) / l_m) / l_m))))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) / sqrt(2.0)));
	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 ((t_m / l_m) <= 2e+26)
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_m))))));
	else
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	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[N[(t$95$m / l$95$m), $MachinePrecision], 2e+26], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(t$95$m * N[(N[(N[(t$95$m * 2.0), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e26

   1. Initial program 90.8%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 88.4%

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

   if 2.0000000000000001e26 < (/.f64 t l)

   1. Initial program 67.1%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 57.9%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 35.6%

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

   7. Simplified 35.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 91.5%

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

   10. Taylor expanded in Om around 0

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

       1. associate-/r* N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{t \cdot 2}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.2 \cdot 10^{-160}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;l\_m \leq 1.42 \cdot 10^{+109}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{2 \cdot \left(t\_m \cdot t\_m\right)}{l\_m \cdot l\_m}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 4.2e-160)
   (asin (/ l_m (* t_m (sqrt 2.0))))
   (if (<= l_m 1.42e+109)
     (asin (pow (+ 1.0 (/ (* 2.0 (* t_m t_m)) (* l_m l_m))) -0.5))
     (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 <= 4.2e-160) {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	} else if (l_m <= 1.42e+109) {
		tmp = asin(pow((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))), -0.5));
	} 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 <= 4.2d-160) then
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    else if (l_m <= 1.42d+109) then
        tmp = asin(((1.0d0 + ((2.0d0 * (t_m * t_m)) / (l_m * l_m))) ** (-0.5d0)))
    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 <= 4.2e-160) {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	} else if (l_m <= 1.42e+109) {
		tmp = Math.asin(Math.pow((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))), -0.5));
	} 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 <= 4.2e-160:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	elif l_m <= 1.42e+109:
		tmp = math.asin(math.pow((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))), -0.5))
	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 <= 4.2e-160)
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	elseif (l_m <= 1.42e+109)
		tmp = asin((Float64(1.0 + Float64(Float64(2.0 * Float64(t_m * t_m)) / Float64(l_m * l_m))) ^ -0.5));
	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 <= 4.2e-160)
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	elseif (l_m <= 1.42e+109)
		tmp = asin(((1.0 + ((2.0 * (t_m * t_m)) / (l_m * l_m))) ^ -0.5));
	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, 4.2e-160], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.42e+109], 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[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.2000000000000001e-160

   1. Initial program 82.5%

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

      1. 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 76.3%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 20.3%

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

   7. Simplified 20.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 33.3%

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

   10. Taylor expanded in Om around 0

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

   12. Simplified 37.4%

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

   if 4.2000000000000001e-160 < l < 1.4200000000000001e109

   1. Initial program 82.6%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 82.6%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   6. Applied egg-rr 82.7%

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

   7. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 78.0%

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

   9. Simplified 78.0%

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

   if 1.4200000000000001e109 < l

   1. Initial program 98.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 98.4%

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

   5. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in Om around 0

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

   10. Simplified 88.1%

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

Alternative 5: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{t\_m}{l\_m} \cdot \frac{2}{\frac{l\_m}{t\_m}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+143)
   (asin (pow (+ 1.0 (* (/ t_m l_m) (/ 2.0 (/ l_m t_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 / l_m) <= 5e+143) {
		tmp = asin(pow((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_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 / l_m) <= 5d+143) then
        tmp = asin(((1.0d0 + ((t_m / l_m) * (2.0d0 / (l_m / t_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 / l_m) <= 5e+143) {
		tmp = Math.asin(Math.pow((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_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 / l_m) <= 5e+143:
		tmp = math.asin(math.pow((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_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 (Float64(t_m / l_m) <= 5e+143)
		tmp = asin((Float64(1.0 + Float64(Float64(t_m / l_m) * Float64(2.0 / Float64(l_m / t_m)))) ^ -0.5));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+143)
		tmp = asin(((1.0 + ((t_m / l_m) * (2.0 / (l_m / t_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[N[(t$95$m / l$95$m), $MachinePrecision], 5e+143], N[ArcSin[N[Power[N[(1.0 + N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000012e143

   1. Initial program 92.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 87.7%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in Om around 0

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

   9. Simplified 91.6%

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

   if 5.00000000000000012e143 < (/.f64 t l)

   1. Initial program 39.3%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 36.5%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 27.2%

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

   7. Simplified 27.2%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 87.9%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

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

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

       7. sqrt-undiv N/A

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

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

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

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

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

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

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

       11. associate-/r/ N/A

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

       12. associate-*l/ N/A

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

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

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

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

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

       15. *-lowering-*.f64 81.9%

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

   11. Applied egg-rr 81.9%

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

   12. Taylor expanded in Om around 0

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

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

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

       2. sqrt-lowering-sqrt.f64 99.5%

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

   14. Simplified 99.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{t}{\ell} \cdot \frac{2}{\frac{\ell}{t}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 9 \cdot 10^{+232}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t\_m}{l\_m} \cdot \frac{2}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \frac{1}{t\_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 9e+232)
   (asin (sqrt (/ 1.0 (+ 1.0 (* (/ t_m l_m) (/ 2.0 (/ l_m t_m)))))))
   (asin (/ (* l_m (/ 1.0 t_m)) (sqrt 2.0)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 9e+232) {
		tmp = asin(sqrt((1.0 / (1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))))));
	} else {
		tmp = asin(((l_m * (1.0 / t_m)) / sqrt(2.0)));
	}
	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 (t_m <= 9d+232) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + ((t_m / l_m) * (2.0d0 / (l_m / t_m)))))))
    else
        tmp = asin(((l_m * (1.0d0 / t_m)) / sqrt(2.0d0)))
    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 (t_m <= 9e+232) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))))));
	} else {
		tmp = Math.asin(((l_m * (1.0 / t_m)) / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 9e+232:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))))))
	else:
		tmp = math.asin(((l_m * (1.0 / t_m)) / math.sqrt(2.0)))
	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 <= 9e+232)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(Float64(t_m / l_m) * Float64(2.0 / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * Float64(1.0 / t_m)) / sqrt(2.0)));
	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 (t_m <= 9e+232)
		tmp = asin(sqrt((1.0 / (1.0 + ((t_m / l_m) * (2.0 / (l_m / t_m)))))));
	else
		tmp = asin(((l_m * (1.0 / t_m)) / sqrt(2.0)));
	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[t$95$m, 9e+232], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 8.9999999999999995e232

   1. Initial program 86.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 82.3%

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

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

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

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

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

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

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

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

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

      5. associate-*l/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. div-inv N/A

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

      14. associate-*l* N/A

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

      15. associate-/r/ N/A

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

      16. clear-num N/A

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

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

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

   6. Applied egg-rr 86.7%

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

   7. Taylor expanded in Om around 0

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

   9. Simplified 86.3%

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

   if 8.9999999999999995e232 < t

   1. Initial program 59.2%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 59.2%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 22.1%

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

   7. Simplified 22.1%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 45.4%

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

   10. Taylor expanded in Om around 0

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

       1. associate-/r* N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 51.8%

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

   12. Simplified 51.8%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. /-lowering-/.f64 51.8%

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

   14. Applied egg-rr 51.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 9 \cdot 10^{+232}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{t}{\ell} \cdot \frac{2}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \frac{1}{t}}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.75 \cdot 10^{-157}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + t\_m \cdot \frac{\frac{t\_m \cdot 2}{l\_m}}{l\_m}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 2.75e-157)
   (asin (/ l_m (* t_m (sqrt 2.0))))
   (asin (sqrt (/ 1.0 (+ 1.0 (* t_m (/ (/ (* t_m 2.0) l_m) l_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 (l_m <= 2.75e-157) {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	} else {
		tmp = asin(sqrt((1.0 / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_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 (l_m <= 2.75d-157) then
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    else
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (t_m * (((t_m * 2.0d0) / l_m) / l_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 (l_m <= 2.75e-157) {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_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 l_m <= 2.75e-157:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	else:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_m))))))
	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 <= 2.75e-157)
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	else
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(t_m * Float64(Float64(Float64(t_m * 2.0) / l_m) / l_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 (l_m <= 2.75e-157)
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	else
		tmp = asin(sqrt((1.0 / (1.0 + (t_m * (((t_m * 2.0) / l_m) / l_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[l$95$m, 2.75e-157], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(t$95$m * N[(N[(N[(t$95$m * 2.0), $MachinePrecision] / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.7499999999999999e-157

   1. Initial program 82.5%

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

      1. 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 76.3%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 20.3%

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

   7. Simplified 20.3%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 33.3%

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

   10. Taylor expanded in Om around 0

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

   12. Simplified 37.4%

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

   if 2.7499999999999999e-157 < l

   1. Initial program 90.0%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 90.0%

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

   5. Taylor expanded in Om around 0

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

   7. Simplified 89.8%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.75 \cdot 10^{-157}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + t \cdot \frac{\frac{t \cdot 2}{\ell}}{\ell}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 3.8 \cdot 10^{+228}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{\frac{t\_m \cdot 2}{\frac{l\_m}{t\_m}}}{l\_m}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \frac{1}{t\_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 3.8e+228)
   (asin (pow (+ 1.0 (/ (/ (* t_m 2.0) (/ l_m t_m)) l_m)) -0.5))
   (asin (/ (* l_m (/ 1.0 t_m)) (sqrt 2.0)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 3.8e+228) {
		tmp = asin(pow((1.0 + (((t_m * 2.0) / (l_m / t_m)) / l_m)), -0.5));
	} else {
		tmp = asin(((l_m * (1.0 / t_m)) / sqrt(2.0)));
	}
	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 (t_m <= 3.8d+228) then
        tmp = asin(((1.0d0 + (((t_m * 2.0d0) / (l_m / t_m)) / l_m)) ** (-0.5d0)))
    else
        tmp = asin(((l_m * (1.0d0 / t_m)) / sqrt(2.0d0)))
    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 (t_m <= 3.8e+228) {
		tmp = Math.asin(Math.pow((1.0 + (((t_m * 2.0) / (l_m / t_m)) / l_m)), -0.5));
	} else {
		tmp = Math.asin(((l_m * (1.0 / t_m)) / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 3.8e+228:
		tmp = math.asin(math.pow((1.0 + (((t_m * 2.0) / (l_m / t_m)) / l_m)), -0.5))
	else:
		tmp = math.asin(((l_m * (1.0 / t_m)) / math.sqrt(2.0)))
	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 <= 3.8e+228)
		tmp = asin((Float64(1.0 + Float64(Float64(Float64(t_m * 2.0) / Float64(l_m / t_m)) / l_m)) ^ -0.5));
	else
		tmp = asin(Float64(Float64(l_m * Float64(1.0 / t_m)) / sqrt(2.0)));
	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 (t_m <= 3.8e+228)
		tmp = asin(((1.0 + (((t_m * 2.0) / (l_m / t_m)) / l_m)) ^ -0.5));
	else
		tmp = asin(((l_m * (1.0 / t_m)) / sqrt(2.0)));
	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[t$95$m, 3.8e+228], N[ArcSin[N[Power[N[(1.0 + N[(N[(N[(t$95$m * 2.0), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.8000000000000002e228

   1. Initial program 86.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 82.3%

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

      1. pow1/2 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-pow N/A

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

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

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

   6. Applied egg-rr 86.7%

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

   7. Taylor expanded in Om around 0

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

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

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 66.2%

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

   9. Simplified 66.2%

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

       1. times-frac N/A

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

       2. associate-*r/ N/A

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

       3. associate-*l* N/A

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

       4. associate-/r/ N/A

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

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

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

       6. associate-*l/ N/A

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

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

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

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

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

       9. /-lowering-/.f64 83.6%

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

   11. Applied egg-rr 83.6%

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

   if 3.8000000000000002e228 < t

   1. Initial program 59.2%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 59.2%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 22.1%

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

   7. Simplified 22.1%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 45.4%

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

   10. Taylor expanded in Om around 0

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

       1. associate-/r* N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 51.8%

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

   12. Simplified 51.8%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. /-lowering-/.f64 51.8%

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

   14. Applied egg-rr 51.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.8 \cdot 10^{+228}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{\frac{t \cdot 2}{\frac{\ell}{t}}}{\ell}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \frac{1}{t}}{\sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 7.4 \cdot 10^{+62}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{l\_m}{\sqrt{2}}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 7.4e+62) (asin (/ (/ l_m (sqrt 2.0)) t_m)) (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 <= 7.4e+62) {
		tmp = asin(((l_m / sqrt(2.0)) / t_m));
	} 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 <= 7.4d+62) then
        tmp = asin(((l_m / sqrt(2.0d0)) / t_m))
    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 <= 7.4e+62) {
		tmp = Math.asin(((l_m / Math.sqrt(2.0)) / t_m));
	} 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 <= 7.4e+62:
		tmp = math.asin(((l_m / math.sqrt(2.0)) / t_m))
	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 <= 7.4e+62)
		tmp = asin(Float64(Float64(l_m / sqrt(2.0)) / t_m));
	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 <= 7.4e+62)
		tmp = asin(((l_m / sqrt(2.0)) / t_m));
	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, 7.4e+62], N[ArcSin[N[(N[(l$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 7.40000000000000028e62

   1. Initial program 81.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 76.5%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. unpow2 N/A

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

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

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 23.9%

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

   7. Simplified 23.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. sqrt-div N/A

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

      4. pow1/2 N/A

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

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

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

   9. Applied egg-rr 34.0%

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

   10. Taylor expanded in Om around 0

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

       1. associate-/r* N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 37.5%

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

   12. Simplified 37.5%

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

       1. associate-/l/ N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 37.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{sqrt.f64}\left(2\right)\right), t\right)\right) \]

   14. Applied egg-rr 37.5%

       \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\frac{\ell}{\sqrt{2}}}{t}\right)} \]

   if 7.40000000000000028e62 < l

   1. Initial program 97.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 97.0%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 82.6%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

   7. Simplified 82.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)} \]

   8. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 86.0%

       \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 73.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.3 \cdot 10^{+62}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 2.3e+62) (asin (/ l_m (* t_m (sqrt 2.0)))) (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 <= 2.3e+62) {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	} 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 <= 2.3d+62) then
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    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 <= 2.3e+62) {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	} 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 <= 2.3e+62:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	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 <= 2.3e+62)
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	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 <= 2.3e+62)
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	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, 2.3e+62], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.29999999999999984e62

   1. Initial program 81.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 76.5%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{{\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}{{t}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)}{{t}^{2}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right) \cdot \frac{1}{2}}{{t}^{2}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right) \cdot \frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right) \cdot {\ell}^{2}\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \left(\ell \cdot \ell\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 23.9%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. Simplified 23.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{0.5}{t \cdot t}}}\right) \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)}{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{\sqrt{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)}}{\sqrt{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{{\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right)}^{\frac{1}{2}}}{\sqrt{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left({\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right)}^{\frac{1}{2}}\right), \left(\sqrt{\frac{t \cdot t}{\frac{1}{2}}}\right)\right)\right) \]

   9. Applied egg-rr 34.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{t \cdot \sqrt{2}}\right)} \]

   10. Taylor expanded in Om around 0

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\color{blue}{\ell}, \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 37.5%

       \[\leadsto \sin^{-1} \left(\frac{\color{blue}{\ell}}{t \cdot \sqrt{2}}\right) \]

   if 2.29999999999999984e62 < l

   1. Initial program 97.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 97.0%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 82.6%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

   7. Simplified 82.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)} \]

   8. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 86.0%

       \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 73.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

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+62) (asin (* l_m (/ (sqrt 0.5) t_m))) (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+62) {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	} 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+62) then
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    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+62) {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	} 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+62:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	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+62)
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	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+62)
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	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+62], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.19999999999999984e62

   1. Initial program 81.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 76.5%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \frac{{\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}{{t}^{2}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\frac{1}{2} \cdot \left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)}{{t}^{2}}\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right) \cdot \frac{1}{2}}{{t}^{2}}\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right) \cdot \frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right) \cdot {\ell}^{2}\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \left({\ell}^{2}\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \left(\ell \cdot \ell\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \left(\frac{\frac{1}{2}}{{t}^{2}}\right)\right)\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \left({t}^{2}\right)\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \left(t \cdot t\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 23.9%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right) \]

   7. Simplified 23.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{0.5}{t \cdot t}}}\right) \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{1}{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)}{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      3. sqrt-div N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{\sqrt{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)}}{\sqrt{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      4. pow1/2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{{\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right)}^{\frac{1}{2}}}{\sqrt{\frac{t \cdot t}{\frac{1}{2}}}}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left({\left(\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right) \cdot \left(\ell \cdot \ell\right)\right)}^{\frac{1}{2}}\right), \left(\sqrt{\frac{t \cdot t}{\frac{1}{2}}}\right)\right)\right) \]

   9. Applied egg-rr 34.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{t \cdot \sqrt{2}}\right)} \]
   10. Step-by-step derivation

       1. associate-/l* N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\ell \cdot \frac{\sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{t \cdot \sqrt{2}}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\frac{\sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{t \cdot \sqrt{2}} \cdot \ell\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{t \cdot \sqrt{2}}\right), \ell\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{\sqrt{2} \cdot t}\right), \ell\right)\right) \]

       5. associate-/r* N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{\sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{\sqrt{2}}}{t}\right), \ell\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{\sqrt{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}}{\sqrt{2}}\right), t\right), \ell\right)\right) \]

       7. sqrt-undiv N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\frac{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}{2}}\right), t\right), \ell\right)\right) \]

       8. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}}{2}\right)\right), t\right), \ell\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - \frac{Om}{\frac{Omc \cdot Omc}{Om}}\right), 2\right)\right), t\right), \ell\right)\right) \]

       10. --lowering--.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{Om}{\frac{Omc \cdot Omc}{Om}}\right)\right), 2\right)\right), t\right), \ell\right)\right) \]

       11. associate-/r/ N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{Om}{Omc \cdot Omc} \cdot Om\right)\right), 2\right)\right), t\right), \ell\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{Om \cdot Om}{Omc \cdot Omc}\right)\right), 2\right)\right), t\right), \ell\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left(Omc \cdot Omc\right)\right)\right), 2\right)\right), t\right), \ell\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right), 2\right)\right), t\right), \ell\right)\right) \]

       15. *-lowering-*.f64 32.7%

           \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right), 2\right)\right), t\right), \ell\right)\right) \]

   11. Applied egg-rr 32.7%

       \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{\frac{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}{2}}}{t} \cdot \ell\right)} \]

   12. Taylor expanded in Om around 0

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{\sqrt{\frac{1}{2}}}{t}\right)}, \ell\right)\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right), \ell\right)\right) \]

       2. sqrt-lowering-sqrt.f64 37.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right), \ell\right)\right) \]

   14. Simplified 37.5%

       \[\leadsto \sin^{-1} \left(\color{blue}{\frac{\sqrt{0.5}}{t}} \cdot \ell\right) \]

   if 3.19999999999999984e62 < l

   1. Initial program 97.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 97.0%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 82.6%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

   7. Simplified 82.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)} \]

   8. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 86.0%

       \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{+62}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} 1 \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc) :precision binary64 (asin 1.0))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin(1.0);
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(1.0d0)
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	return Math.asin(1.0);
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin(1.0)

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(1.0)
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(1.0);
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 85.0%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. Step-by-step derivation

   1. asin-lowering-asin.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   5. associate-+l- N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   6. sub0-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\right) \]

   7. distribute-frac-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)}\right)\right) \]

   8. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(\frac{{\left(\frac{Om}{Omc}\right)}^{2} - 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right)\right) \]

   9. distribute-frac-neg N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\mathsf{neg}\left(\left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   10. sub0-neg N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{0 - \left({\left(\frac{Om}{Omc}\right)}^{2} - 1\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   11. associate-+l- N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(0 - {\left(\frac{Om}{Omc}\right)}^{2}\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   12. neg-sub0 N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   14. sub-neg N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right)\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right), \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)\right)\right) \]

3. Simplified 80.9%

   \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 46.5%

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

7. Simplified 46.5%

   \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)} \]

8. Taylor expanded in Om around 0

   \[\leadsto \mathsf{asin.f64}\left(\color{blue}{1}\right) \]
9. Step-by-step derivation

10. Simplified 51.4%

    \[\leadsto \sin^{-1} \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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)))))))