Toniolo and Linder, Equation (2)

Percentage Accurate: 84.1% → 98.8%

Time: 16.4s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}} \cdot 2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{l\_m}{t\_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

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) 1e+154)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (+ 1.0 (* (/ (/ t_m l_m) (/ l_m t_m)) 2.0)))))
   (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) <= 1e+154) {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (((t_m / l_m) / (l_m / t_m)) * 2.0)))));
	} 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) <= 1d+154) then
        tmp = asin(sqrt(((1.0d0 - ((om * (om / omc)) / omc)) / (1.0d0 + (((t_m / l_m) / (l_m / t_m)) * 2.0d0)))))
    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) <= 1e+154) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (((t_m / l_m) / (l_m / t_m)) * 2.0)))));
	} 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) <= 1e+154:
		tmp = math.asin(math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (((t_m / l_m) / (l_m / t_m)) * 2.0)))))
	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) <= 1e+154)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / Float64(1.0 + Float64(Float64(Float64(t_m / l_m) / Float64(l_m / t_m)) * 2.0)))));
	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) <= 1e+154)
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (((t_m / l_m) / (l_m / t_m)) * 2.0)))));
	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], 1e+154], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $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) < 1.00000000000000004e154

   1. Initial program 91.9%

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

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

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. sub0-neg N/A

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

      7. distribute-frac-neg N/A

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

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

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

      9. distribute-frac-neg N/A

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

      10. sub0-neg N/A

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

      11. associate-+l- N/A

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

      12. neg-sub0 N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

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

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

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   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 91.9%

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

   6. Applied egg-rr 91.9%

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

   if 1.00000000000000004e154 < (/.f64 t l)

   1. Initial program 37.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 37.3%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 37.3%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 85.9%

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

   9. Simplified 85.9%

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

   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 99.6%

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

   12. Simplified 99.6%

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

Alternative 2: 87.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{l\_m}{\sqrt{\frac{-2}{-1 + \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t\_m \cdot \frac{\frac{t\_m \cdot 2}{l\_m}}{l\_m}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 5e-133)
   (asin (/ (/ l_m (sqrt (/ -2.0 (+ -1.0 (/ (/ Om Omc) (/ Omc Om)))))) t_m))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (+ 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 <= 5e-133) {
		tmp = asin(((l_m / sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (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 <= 5d-133) then
        tmp = asin(((l_m / sqrt(((-2.0d0) / ((-1.0d0) + ((om / omc) / (omc / om)))))) / t_m))
    else
        tmp = asin(sqrt(((1.0d0 - ((om * (om / omc)) / omc)) / (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 <= 5e-133) {
		tmp = Math.asin(((l_m / Math.sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (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 <= 5e-133:
		tmp = math.asin(((l_m / math.sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m))
	else:
		tmp = math.asin(math.sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (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 <= 5e-133)
		tmp = asin(Float64(Float64(l_m / sqrt(Float64(-2.0 / Float64(-1.0 + Float64(Float64(Om / Omc) / Float64(Omc / Om)))))) / t_m));
	else
		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))))));
	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 <= 5e-133)
		tmp = asin(((l_m / sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m));
	else
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (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, 5e-133], N[ArcSin[N[(N[(l$95$m / N[Sqrt[N[(-2.0 / N[(-1.0 + N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999999e-133

   1. Initial program 82.7%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 82.6%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 31.2%

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

   9. Simplified 31.2%

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

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

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

       2. associate-*r/ N/A

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

       3. clear-num N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

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

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

   11. Applied egg-rr 37.7%

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

       1. associate-/r* N/A

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

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

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

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

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

       4. unpow1/2 N/A

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

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

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

       6. associate-*l/ N/A

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

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

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-/r* N/A

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

       12. associate-/r/ N/A

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

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

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

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

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

       15. /-lowering-/.f64 39.0%

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

   13. Applied egg-rr 39.0%

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

   if 4.9999999999999999e-133 < l

   1. Initial program 86.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 86.5%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{\sqrt{\frac{-2}{-1 + \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{t \cdot 2}{\ell}}{\ell}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.15 \cdot 10^{-131}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{l\_m}{\sqrt{\frac{-2}{-1 + \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}}}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m \cdot 2}}\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.15e-131)
   (asin (/ (/ l_m (sqrt (/ -2.0 (+ -1.0 (/ (/ Om Omc) (/ Omc Om)))))) t_m))
   (asin (pow (+ 1.0 (/ (/ t_m l_m) (/ l_m (* t_m 2.0)))) -0.5))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.15e-131) {
		tmp = asin(((l_m / sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m));
	} else {
		tmp = asin(pow((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))), -0.5));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 1.15d-131) then
        tmp = asin(((l_m / sqrt(((-2.0d0) / ((-1.0d0) + ((om / omc) / (omc / om)))))) / t_m))
    else
        tmp = asin(((1.0d0 + ((t_m / l_m) / (l_m / (t_m * 2.0d0)))) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.15e-131) {
		tmp = Math.asin(((l_m / Math.sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m));
	} else {
		tmp = Math.asin(Math.pow((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))), -0.5));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.15e-131:
		tmp = math.asin(((l_m / math.sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m))
	else:
		tmp = math.asin(math.pow((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))), -0.5))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.15e-131)
		tmp = asin(Float64(Float64(l_m / sqrt(Float64(-2.0 / Float64(-1.0 + Float64(Float64(Om / Omc) / Float64(Omc / Om)))))) / t_m));
	else
		tmp = asin((Float64(1.0 + Float64(Float64(t_m / l_m) / Float64(l_m / Float64(t_m * 2.0)))) ^ -0.5));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 1.15e-131)
		tmp = asin(((l_m / sqrt((-2.0 / (-1.0 + ((Om / Omc) / (Omc / Om)))))) / t_m));
	else
		tmp = asin(((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 1.15e-131], N[ArcSin[N[(N[(l$95$m / N[Sqrt[N[(-2.0 / N[(-1.0 + N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Power[N[(1.0 + N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / N[(t$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.15000000000000011e-131

   1. Initial program 82.7%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   6. Applied egg-rr 82.6%

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

   7. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

      13. sqrt-lowering-sqrt.f64 31.2%

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

   9. Simplified 31.2%

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

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

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

       2. associate-*r/ N/A

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

       3. clear-num N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

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

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

   11. Applied egg-rr 37.7%

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

       1. associate-/r* N/A

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

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

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

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

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

       4. unpow1/2 N/A

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

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

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

       6. associate-*l/ N/A

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

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

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

       8. metadata-eval N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-/r* N/A

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

       12. associate-/r/ N/A

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

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

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

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

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

       15. /-lowering-/.f64 39.0%

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

   13. Applied egg-rr 39.0%

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

   if 1.15000000000000011e-131 < l

   1. Initial program 86.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 86.5%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 61.1%

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

   7. Simplified 61.1%

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

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

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

      2. inv-pow N/A

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

      3. sqrt-pow1 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/r/ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

   9. Applied egg-rr 85.6%

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

Alternative 4: 86.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t\_m} \cdot \frac{l\_m}{{0.5}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m \cdot 2}}\right)}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 8.2e-133)
   (asin (* (/ 1.0 t_m) (/ l_m (pow 0.5 -0.5))))
   (asin (pow (+ 1.0 (/ (/ t_m l_m) (/ l_m (* t_m 2.0)))) -0.5))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 8.2e-133) {
		tmp = asin(((1.0 / t_m) * (l_m / pow(0.5, -0.5))));
	} else {
		tmp = asin(pow((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))), -0.5));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 8.2d-133) then
        tmp = asin(((1.0d0 / t_m) * (l_m / (0.5d0 ** (-0.5d0)))))
    else
        tmp = asin(((1.0d0 + ((t_m / l_m) / (l_m / (t_m * 2.0d0)))) ** (-0.5d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 8.2e-133) {
		tmp = Math.asin(((1.0 / t_m) * (l_m / Math.pow(0.5, -0.5))));
	} else {
		tmp = Math.asin(Math.pow((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))), -0.5));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 8.2e-133:
		tmp = math.asin(((1.0 / t_m) * (l_m / math.pow(0.5, -0.5))))
	else:
		tmp = math.asin(math.pow((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))), -0.5))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 8.2e-133)
		tmp = asin(Float64(Float64(1.0 / t_m) * Float64(l_m / (0.5 ^ -0.5))));
	else
		tmp = asin((Float64(1.0 + Float64(Float64(t_m / l_m) / Float64(l_m / Float64(t_m * 2.0)))) ^ -0.5));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 8.2e-133)
		tmp = asin(((1.0 / t_m) * (l_m / (0.5 ^ -0.5))));
	else
		tmp = asin(((1.0 + ((t_m / l_m) / (l_m / (t_m * 2.0)))) ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 8.2e-133], N[ArcSin[N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(l$95$m / N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Power[N[(1.0 + N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / N[(t$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.20000000000000045e-133

   1. Initial program 82.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 62.5%

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

   7. Simplified 62.5%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 39.0%

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

   10. Simplified 39.0%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 39.0%

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

   12. Applied egg-rr 39.0%

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

       1. clear-num N/A

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

       2. associate-*l/ N/A

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

       3. div-inv N/A

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

       4. times-frac N/A

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

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

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

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

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

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

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

       8. pow1/2 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval 39.0%

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

   14. Applied egg-rr 39.0%

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

   if 8.20000000000000045e-133 < l

   1. Initial program 86.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 86.5%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 61.1%

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

   7. Simplified 61.1%

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

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

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

      2. inv-pow N/A

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

      3. sqrt-pow1 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/r/ N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

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

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

   9. Applied egg-rr 85.6%

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

Alternative 5: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t\_m} \cdot \frac{l\_m}{{0.5}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 3e-38)
   (asin (* (/ 1.0 t_m) (/ l_m (pow 0.5 -0.5))))
   (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))))

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 <= 3e-38) {
		tmp = asin(((1.0 / t_m) * (l_m / pow(0.5, -0.5))));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	}
	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 <= 3d-38) then
        tmp = asin(((1.0d0 / t_m) * (l_m / (0.5d0 ** (-0.5d0)))))
    else
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    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 <= 3e-38) {
		tmp = Math.asin(((1.0 / t_m) * (l_m / Math.pow(0.5, -0.5))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	}
	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 <= 3e-38:
		tmp = math.asin(((1.0 / t_m) * (l_m / math.pow(0.5, -0.5))))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))))
	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 <= 3e-38)
		tmp = asin(Float64(Float64(1.0 / t_m) * Float64(l_m / (0.5 ^ -0.5))));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))));
	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 <= 3e-38)
		tmp = asin(((1.0 / t_m) * (l_m / (0.5 ^ -0.5))));
	else
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	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, 3e-38], N[ArcSin[N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(l$95$m / N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.99999999999999989e-38

   1. Initial program 79.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.9%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 40.4%

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

   10. Simplified 40.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 40.4%

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

   12. Applied egg-rr 40.4%

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

       1. clear-num N/A

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

       2. associate-*l/ N/A

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

       3. div-inv N/A

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

       4. times-frac N/A

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

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

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

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

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

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

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

       8. pow1/2 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval 40.3%

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

   14. Applied egg-rr 40.3%

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

   if 2.99999999999999989e-38 < l

   1. Initial program 94.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 94.4%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   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 65.4%

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

   7. Simplified 65.4%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 73.3%

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

   9. Applied egg-rr 73.3%

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

Alternative 6: 74.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.5 \cdot 10^{-37}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t\_m} \cdot \frac{l\_m}{{0.5}^{-0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 2.5e-37)
   (asin (* (/ 1.0 t_m) (/ l_m (pow 0.5 -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 <= 2.5e-37) {
		tmp = asin(((1.0 / t_m) * (l_m / pow(0.5, -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 <= 2.5d-37) then
        tmp = asin(((1.0d0 / t_m) * (l_m / (0.5d0 ** (-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 <= 2.5e-37) {
		tmp = Math.asin(((1.0 / t_m) * (l_m / Math.pow(0.5, -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 <= 2.5e-37:
		tmp = math.asin(((1.0 / t_m) * (l_m / math.pow(0.5, -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 <= 2.5e-37)
		tmp = asin(Float64(Float64(1.0 / t_m) * Float64(l_m / (0.5 ^ -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 <= 2.5e-37)
		tmp = asin(((1.0 / t_m) * (l_m / (0.5 ^ -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, 2.5e-37], N[ArcSin[N[(N[(1.0 / t$95$m), $MachinePrecision] * N[(l$95$m / N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.4999999999999999e-37

   1. Initial program 79.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.9%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 40.4%

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

   10. Simplified 40.4%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 40.4%

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

   12. Applied egg-rr 40.4%

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

       1. clear-num N/A

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

       2. associate-*l/ N/A

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

       3. div-inv N/A

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

       4. times-frac N/A

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

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

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

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

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

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

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

       8. pow1/2 N/A

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

       9. pow-flip N/A

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

       10. metadata-eval N/A

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval 40.3%

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

   14. Applied egg-rr 40.3%

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

   if 2.4999999999999999e-37 < l

   1. Initial program 94.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 94.4%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   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 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in Om around 0

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

   10. Simplified 72.5%

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

Alternative 7: 74.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.12 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.12e-38) (asin (* (/ l_m t_m) (sqrt 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 <= 1.12e-38) {
		tmp = asin(((l_m / t_m) * sqrt(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 <= 1.12d-38) then
        tmp = asin(((l_m / t_m) * sqrt(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 <= 1.12e-38) {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(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 <= 1.12e-38:
		tmp = math.asin(((l_m / t_m) * math.sqrt(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 <= 1.12e-38)
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(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 <= 1.12e-38)
		tmp = asin(((l_m / t_m) * sqrt(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, 1.12e-38], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.1200000000000001e-38

   1. Initial program 79.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.9%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 40.4%

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

   10. Simplified 40.4%

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

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

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

       5. /-lowering-/.f64 40.4%

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

   12. Applied egg-rr 40.4%

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

   if 1.1200000000000001e-38 < l

   1. Initial program 94.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 94.4%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
   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 65.4%

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

   7. Simplified 65.4%

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

   8. Taylor expanded in Om around 0

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

   10. Simplified 72.5%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.12 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 6 \cdot 10^{+91}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t\_m \cdot t\_m}{l\_m \cdot l\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 6e+91)
   (asin (/ 1.0 (+ 1.0 (/ (* t_m t_m) (* l_m l_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 <= 6e+91) {
		tmp = asin((1.0 / (1.0 + ((t_m * t_m) / (l_m * l_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 <= 6d+91) then
        tmp = asin((1.0d0 / (1.0d0 + ((t_m * t_m) / (l_m * l_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 <= 6e+91) {
		tmp = Math.asin((1.0 / (1.0 + ((t_m * t_m) / (l_m * l_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 <= 6e+91:
		tmp = math.asin((1.0 / (1.0 + ((t_m * t_m) / (l_m * l_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 <= 6e+91)
		tmp = asin(Float64(1.0 / Float64(1.0 + Float64(Float64(t_m * t_m) / Float64(l_m * l_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 <= 6e+91)
		tmp = asin((1.0 / (1.0 + ((t_m * t_m) / (l_m * l_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, 6e+91], N[ArcSin[N[(1.0 / N[(1.0 + N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.00000000000000012e91

   1. Initial program 80.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 75.3%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 62.1%

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

   7. Simplified 62.1%

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

   9. Applied egg-rr 79.4%

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

   10. Taylor expanded in t around 0

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

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 51.0%

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

   12. Simplified 51.0%

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

   if 6.00000000000000012e91 < l

   1. Initial program 99.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 99.3%

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

   5. Taylor expanded in t around 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 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in Om around 0

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

   10. Simplified 82.9%

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

Alternative 9: 50.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 84.0%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
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 79.9%

   \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + t \cdot \frac{\frac{2 \cdot t}{\ell}}{\ell}}}\right)} \]
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 45.0%

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

7. Simplified 45.0%

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

8. Taylor expanded in Om around 0

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

10. Simplified 50.7%

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

Reproduce

herbie shell --seed 2024138 
(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)))))))