Toniolo and Linder, Equation (2)

Percentage Accurate: 84.2% → 98.6%

Time: 15.2s

Alternatives: 8

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\frac{1 - \frac{Om \cdot t\_1}{\frac{Omc}{\frac{Om}{Omc}}}}{1 + t\_1}}{1 + \frac{2}{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{l\_m}{t\_m}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (/ (/ Om (/ Omc Om)) Omc)))
   (if (<= (/ t_m l_m) 2e+143)
     (asin
      (sqrt
       (/
        (/ (- 1.0 (/ (* Om t_1) (/ Omc (/ Om Omc)))) (+ 1.0 t_1))
        (+ 1.0 (/ 2.0 (* (/ l_m t_m) (/ l_m t_m)))))))
     (asin (/ (/ l_m t_m) (sqrt 2.0))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = (Om / (Omc / Om)) / Omc;
	double tmp;
	if ((t_m / l_m) <= 2e+143) {
		tmp = asin(sqrt((((1.0 - ((Om * t_1) / (Omc / (Om / Omc)))) / (1.0 + t_1)) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))));
	} else {
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (om / (omc / om)) / omc
    if ((t_m / l_m) <= 2d+143) then
        tmp = asin(sqrt((((1.0d0 - ((om * t_1) / (omc / (om / omc)))) / (1.0d0 + t_1)) / (1.0d0 + (2.0d0 / ((l_m / t_m) * (l_m / t_m)))))))
    else
        tmp = asin(((l_m / t_m) / sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = (Om / (Omc / Om)) / Omc;
	double tmp;
	if ((t_m / l_m) <= 2e+143) {
		tmp = Math.asin(Math.sqrt((((1.0 - ((Om * t_1) / (Omc / (Om / Omc)))) / (1.0 + t_1)) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))));
	} else {
		tmp = Math.asin(((l_m / t_m) / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = (Om / (Omc / Om)) / Omc
	tmp = 0
	if (t_m / l_m) <= 2e+143:
		tmp = math.asin(math.sqrt((((1.0 - ((Om * t_1) / (Omc / (Om / Omc)))) / (1.0 + t_1)) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))))
	else:
		tmp = math.asin(((l_m / t_m) / math.sqrt(2.0)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(Float64(Om / Float64(Omc / Om)) / Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+143)
		tmp = asin(sqrt(Float64(Float64(Float64(1.0 - Float64(Float64(Om * t_1) / Float64(Omc / Float64(Om / Omc)))) / Float64(1.0 + t_1)) / Float64(1.0 + Float64(2.0 / Float64(Float64(l_m / t_m) * Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = (Om / (Omc / Om)) / Omc;
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+143)
		tmp = asin(sqrt((((1.0 - ((Om * t_1) / (Omc / (Om / Omc)))) / (1.0 + t_1)) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))));
	else
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+143], N[ArcSin[N[Sqrt[N[(N[(N[(1.0 - N[(N[(Om * t$95$1), $MachinePrecision] / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 / N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

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

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. unpow2 N/A

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

      17. clear-num N/A

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

      18. clear-num N/A

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

      19. frac-times N/A

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

      20. metadata-eval N/A

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

   4. Applied egg-rr 87.9%

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

      1. associate-/r/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 91.9%

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

   6. Applied egg-rr 91.9%

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

   8. Applied egg-rr 91.9%

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

   if 2e143 < (/.f64 t l)

   1. Initial program 53.7%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 97.5%

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

   10. Simplified 97.5%

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

       1. clear-num N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 99.8%

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

   12. Applied egg-rr 99.8%

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

4. Final simplification 92.9%

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

Alternative 2: 98.6% accurate, 1.8× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+143)
   (asin
    (sqrt
     (/
      (- 1.0 (/ Om (/ Omc (/ Om Omc))))
      (+ 1.0 (/ 2.0 (* (/ l_m t_m) (/ l_m t_m)))))))
   (asin (/ (/ l_m t_m) (sqrt 2.0)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+143) {
		tmp = asin(sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))));
	} else {
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 2d+143) then
        tmp = asin(sqrt(((1.0d0 - (om / (omc / (om / omc)))) / (1.0d0 + (2.0d0 / ((l_m / t_m) * (l_m / t_m)))))))
    else
        tmp = asin(((l_m / t_m) / sqrt(2.0d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+143) {
		tmp = Math.asin(Math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))));
	} else {
		tmp = Math.asin(((l_m / t_m) / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 2e+143:
		tmp = math.asin(math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))))
	else:
		tmp = math.asin(((l_m / t_m) / math.sqrt(2.0)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+143)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om / Float64(Omc / Float64(Om / Omc)))) / Float64(1.0 + Float64(2.0 / Float64(Float64(l_m / t_m) * Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) / sqrt(2.0)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+143)
		tmp = asin(sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / t_m)))))));
	else
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+143], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 / N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

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

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. unpow2 N/A

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

      17. clear-num N/A

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

      18. clear-num N/A

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

      19. frac-times N/A

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

      20. metadata-eval N/A

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

   4. Applied egg-rr 87.9%

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

      1. associate-/r/ N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 91.9%

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

   6. Applied egg-rr 91.9%

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

   if 2e143 < (/.f64 t l)

   1. Initial program 53.7%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 97.5%

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

   10. Simplified 97.5%

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

       1. clear-num N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 99.8%

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

   12. Applied egg-rr 99.8%

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

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 20000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}}{1 + \frac{2}{l\_m \cdot \frac{\frac{l\_m}{t\_m}}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 20000000.0)
   (asin
    (sqrt
     (/
      (- 1.0 (/ Om (/ Omc (/ Om Omc))))
      (+ 1.0 (/ 2.0 (* l_m (/ (/ l_m t_m) t_m)))))))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 20000000.0) {
		tmp = asin(sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / (l_m * ((l_m / t_m) / t_m)))))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 20000000.0d0) then
        tmp = asin(sqrt(((1.0d0 - (om / (omc / (om / omc)))) / (1.0d0 + (2.0d0 / (l_m * ((l_m / t_m) / t_m)))))))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 20000000.0) {
		tmp = Math.asin(Math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / (l_m * ((l_m / t_m) / t_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 20000000.0:
		tmp = math.asin(math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / (l_m * ((l_m / t_m) / t_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 20000000.0)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om / Float64(Omc / Float64(Om / Omc)))) / Float64(1.0 + Float64(2.0 / Float64(l_m * Float64(Float64(l_m / t_m) / t_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 20000000.0)
		tmp = asin(sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / (l_m * ((l_m / t_m) / t_m)))))));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 20000000.0], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 / N[(l$95$m * N[(N[(l$95$m / t$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2e7

   1. Initial program 90.6%

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. *-lft-identity N/A

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

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

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

      10. *-commutative N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. unpow2 N/A

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

      17. clear-num N/A

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

      18. clear-num N/A

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

      19. frac-times N/A

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

      20. metadata-eval N/A

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

   4. Applied egg-rr 89.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 2e7 < (/.f64 t l)

   1. Initial program 75.5%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 66.5%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 42.9%

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

   7. Simplified 42.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 74.4%

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

   9. Applied egg-rr 74.4%

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

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   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. *-lowering-*.f64 N/A

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

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

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

       4. sqrt-lowering-sqrt.f64 98.3%

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

   12. Simplified 98.3%

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

Alternative 4: 97.7% accurate, 1.9× speedup

Math

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

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) 5e+143)
   (asin (pow (+ 1.0 (/ (/ 2.0 (/ l_m t_m)) (/ l_m t_m))) -0.5))
   (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) <= 5e+143) {
		tmp = asin(pow((1.0 + ((2.0 / (l_m / t_m)) / (l_m / t_m))), -0.5));
	} 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) <= 5d+143) then
        tmp = asin(((1.0d0 + ((2.0d0 / (l_m / t_m)) / (l_m / t_m))) ** (-0.5d0)))
    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) <= 5e+143) {
		tmp = Math.asin(Math.pow((1.0 + ((2.0 / (l_m / t_m)) / (l_m / t_m))), -0.5));
	} 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) <= 5e+143:
		tmp = math.asin(math.pow((1.0 + ((2.0 / (l_m / t_m)) / (l_m / t_m))), -0.5))
	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) <= 5e+143)
		tmp = asin((Float64(1.0 + Float64(Float64(2.0 / Float64(l_m / t_m)) / Float64(l_m / t_m))) ^ -0.5));
	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) <= 5e+143)
		tmp = asin(((1.0 + ((2.0 / (l_m / t_m)) / (l_m / t_m))) ^ -0.5));
	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], 5e+143], N[ArcSin[N[Power[N[(1.0 + N[(N[(2.0 / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $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) < 5.00000000000000012e143

   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. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.1%

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

   7. Simplified 67.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 90.4%

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

   9. Applied egg-rr 90.4%

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

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

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

       2. pow1/2 N/A

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

       3. associate-/r* N/A

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

       4. pow-flip N/A

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

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

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

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

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

       7. associate-/r* N/A

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

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

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

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

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

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

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

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

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

       12. metadata-eval 90.5%

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

   11. Applied egg-rr 90.5%

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

   if 5.00000000000000012e143 < (/.f64 t l)

   1. Initial program 53.7%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 53.7%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 97.5%

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

   10. Simplified 97.5%

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

       1. clear-num N/A

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

       2. associate-/r* N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 99.8%

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

   12. Applied egg-rr 99.8%

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

Alternative 5: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.0002)
   (asin (sqrt (- 1.0 (/ (/ Om (/ Omc Om)) Omc))))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.0002d0) then
        tmp = asin(sqrt((1.0d0 - ((om / (omc / om)) / omc))))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.0002:
		tmp = math.asin(math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.0002)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Float64(Omc / Om)) / Omc))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.0002)
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0002], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 90.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   4. 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 60.9%

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

   5. Simplified 60.9%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)} \]
   6. 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. associate-/r/ N/A

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

      3. associate-/r/ N/A

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

      4. associate-/r* N/A

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

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

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

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

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

      7. /-lowering-/.f64 66.6%

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

   7. Applied egg-rr 66.6%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 77.5%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 44.9%

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

   7. Simplified 44.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 76.5%

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

   9. Applied egg-rr 76.5%

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

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   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. *-lowering-*.f64 N/A

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

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

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

       4. sqrt-lowering-sqrt.f64 94.8%

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

   12. Simplified 94.8%

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

Alternative 6: 96.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.0002)
   (asin (- 1.0 (/ (/ t_m l_m) (/ l_m t_m))))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = asin((1.0 - ((t_m / l_m) / (l_m / t_m))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.0002d0) then
        tmp = asin((1.0d0 - ((t_m / l_m) / (l_m / t_m))))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = Math.asin((1.0 - ((t_m / l_m) / (l_m / t_m))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.0002:
		tmp = math.asin((1.0 - ((t_m / l_m) / (l_m / t_m))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.0002)
		tmp = asin(Float64(1.0 - Float64(Float64(t_m / l_m) / Float64(l_m / t_m))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.0002)
		tmp = asin((1.0 - ((t_m / l_m) / (l_m / t_m))));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0002], N[ArcSin[N[(1.0 - N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 90.3%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 88.7%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 73.1%

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

   7. Simplified 73.1%

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

   8. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 56.2%

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

   10. Simplified 56.2%

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

       1. clear-num N/A

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

       2. frac-times N/A

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

       3. associate-/r* N/A

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

       4. clear-num N/A

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

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

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

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

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

       7. /-lowering-/.f64 64.0%

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

   12. Applied egg-rr 64.0%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 77.5%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 44.9%

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

   7. Simplified 44.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. div-inv N/A

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

      5. associate-/r* N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 76.5%

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

   9. Applied egg-rr 76.5%

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

   10. Taylor expanded in l around 0

       \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   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. *-lowering-*.f64 N/A

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

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

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

       4. sqrt-lowering-sqrt.f64 94.8%

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

   12. Simplified 94.8%

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

Alternative 7: 59.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 8.6 \cdot 10^{+110}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t\_m \cdot t\_m}{l\_m \cdot l\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 8.6e+110)
   (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 <= 8.6e+110) {
		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 <= 8.6d+110) 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 <= 8.6e+110) {
		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 <= 8.6e+110:
		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 <= 8.6e+110)
		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 <= 8.6e+110)
		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, 8.6e+110], 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 < 8.60000000000000014e110

   1. Initial program 84.9%

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

      1. clear-num N/A

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

      2. sqrt-div N/A

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

      3. metadata-eval N/A

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

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

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

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

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

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

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

   4. Applied egg-rr 80.9%

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

   5. Taylor expanded in Om around 0

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

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

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

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

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 67.2%

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

   7. Simplified 67.2%

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

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + \frac{{t}^{2}}{{\ell}^{2}}\right)}\right)\right) \]
   9. 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 56.2%

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

   10. Simplified 56.2%

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

   if 8.60000000000000014e110 < l

   1. Initial program 97.7%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
   4. 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 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in Om around 0

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

   8. Simplified 68.5%

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

Alternative 8: 50.0% 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 86.8%

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

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)}\right) \]
4. 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 45.4%

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

5. Simplified 45.4%

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

6. Taylor expanded in Om around 0

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

8. Simplified 48.8%

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

Reproduce

herbie shell --seed 2024191 
(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)))))))