Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.9%

Time: 15.2s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\left(l\_m \cdot 0.5\right) \cdot \sqrt{1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}}}{t\_m}}{\sqrt{0.5}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+148)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om (/ Omc Om)) Omc))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin
    (/
     (/ (* (* l_m 0.5) (sqrt (- 1.0 (/ Om (/ Omc (/ Om Omc)))))) t_m)
     (sqrt 0.5)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+148) {
		tmp = asin(sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin(((((l_m * 0.5) * sqrt((1.0 - (Om / (Omc / (Om / Omc)))))) / t_m) / sqrt(0.5)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+148) then
        tmp = asin(sqrt(((1.0d0 - ((om / (omc / om)) / omc)) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_m)))))))
    else
        tmp = asin(((((l_m * 0.5d0) * sqrt((1.0d0 - (om / (omc / (om / omc)))))) / t_m) / sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+148) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = Math.asin(((((l_m * 0.5) * Math.sqrt((1.0 - (Om / (Omc / (Om / Omc)))))) / t_m) / Math.sqrt(0.5)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+148:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))))
	else:
		tmp = math.asin(((((l_m * 0.5) * math.sqrt((1.0 - (Om / (Omc / (Om / Omc)))))) / t_m) / math.sqrt(0.5)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+148)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Float64(Omc / Om)) / Omc)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(Float64(Float64(l_m * 0.5) * sqrt(Float64(1.0 - Float64(Om / Float64(Omc / Float64(Om / Omc)))))) / t_m) / sqrt(0.5)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+148)
		tmp = asin(sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	else
		tmp = asin(((((l_m * 0.5) * sqrt((1.0 - (Om / (Omc / (Om / Omc)))))) / t_m) / sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+148], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[(l$95$m * 0.5), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(Om / N[(Omc / N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000024e148

   1. Initial program 90.4%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 90.4%

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

   4. Applied egg-rr 90.4%

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

      1. unpow2 N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r/ N/A

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

      4. associate-/r* N/A

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

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

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

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

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

      7. /-lowering-/.f64 90.4%

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

   6. Applied egg-rr 90.4%

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

   if 5.00000000000000024e148 < (/.f64 t l)

   1. Initial program 43.9%

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

   3. Taylor expanded in l around 0

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

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   5. Simplified 78.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \left(\frac{-0.125 \cdot \left(\ell \cdot \ell\right)}{\left(t \cdot t\right) \cdot \left(t \cdot \sqrt{0.5}\right)} + \frac{\sqrt{0.5}}{t}\right)\right)\right)} \]

   7. Applied egg-rr 18.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{t \cdot \left(-0.125 \cdot \left(\ell \cdot \ell\right)\right) + \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.5}{\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \sqrt{0.5}\right)} \cdot \left(\ell \cdot \sqrt{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}\right)\right)} \]

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 87.3%

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

   10. Simplified 87.3%

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

       1. associate-*l/ N/A

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

       2. associate-/r* N/A

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

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

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

   12. Applied egg-rr 99.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\left(\ell \cdot 0.5\right) \cdot \sqrt{1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}}}{t}}{\sqrt{0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{-177}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.3e-177)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om (/ Omc Om)) Omc))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m 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 (l_m <= 1.3e-177) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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 (l_m <= 1.3d-177) then
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    else
        tmp = asin(sqrt(((1.0d0 - ((om / (omc / om)) / omc)) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / 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 (l_m <= 1.3e-177) {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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 l_m <= 1.3e-177:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	else:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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 (l_m <= 1.3e-177)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Float64(Omc / Om)) / Omc)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / 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 (l_m <= 1.3e-177)
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt(((1.0 - ((Om / (Omc / Om)) / Omc)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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[l$95$m, 1.3e-177], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e-177

   1. Initial program 83.9%

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

   3. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 33.3%

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

   8. Simplified 33.3%

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

   if 1.3e-177 < l

   1. Initial program 85.2%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. unpow2 N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r/ N/A

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

      4. associate-/r* N/A

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

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

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

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

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

      7. /-lowering-/.f64 85.2%

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

   6. Applied egg-rr 85.2%

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

Alternative 3: 86.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{-177}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}}{1 + \frac{2}{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.3e-177)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin
    (sqrt
     (/
      (- 1.0 (/ Om (/ Omc (/ Om Omc))))
      (+ 1.0 (/ 2.0 (* (/ l_m t_m) (/ l_m 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 (l_m <= 1.3e-177) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin(sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / 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 (l_m <= 1.3d-177) then
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    else
        tmp = asin(sqrt(((1.0d0 - (om / (omc / (om / omc)))) / (1.0d0 + (2.0d0 / ((l_m / t_m) * (l_m / 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 (l_m <= 1.3e-177) {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / 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 l_m <= 1.3e-177:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	else:
		tmp = math.asin(math.sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / 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 (l_m <= 1.3e-177)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		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)))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 1.3e-177)
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt(((1.0 - (Om / (Omc / (Om / Omc)))) / (1.0 + (2.0 / ((l_m / t_m) * (l_m / 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[l$95$m, 1.3e-177], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e-177

   1. Initial program 83.9%

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

   3. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 33.3%

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

   8. Simplified 33.3%

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

   if 1.3e-177 < l

   1. Initial program 85.2%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 85.2%

      \[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{\frac{Omc}{\frac{Om}{Omc}}}}{1 + \frac{2}{\frac{\frac{\ell}{\frac{t}{\ell}}}{t}}}}\right)} \]
   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 85.3%

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

   6. Applied egg-rr 85.3%

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

Alternative 4: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.3 \cdot 10^{-177}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.3e-177)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m 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 (l_m <= 1.3e-177) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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 (l_m <= 1.3d-177) then
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    else
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / 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 (l_m <= 1.3e-177) {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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 l_m <= 1.3e-177:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	else:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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 (l_m <= 1.3e-177)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / 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 (l_m <= 1.3e-177)
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t_m / l_m) / (l_m / 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[l$95$m, 1.3e-177], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.3e-177

   1. Initial program 83.9%

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

   3. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.4%

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

   5. Simplified 67.4%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 33.3%

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

   8. Simplified 33.3%

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

   if 1.3e-177 < l

   1. Initial program 85.2%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 85.2%

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

   4. Applied egg-rr 85.2%

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

      1. unpow2 N/A

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

      2. associate-/r/ N/A

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

      3. associate-/r/ N/A

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

      4. associate-/r* N/A

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

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

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

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

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

      7. /-lowering-/.f64 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in Om around 0

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

   9. Simplified 84.0%

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

Alternative 5: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.5 \cdot 10^{+161}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{2}{\frac{l\_m}{\frac{t\_m \cdot t\_m}{l\_m}}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(0.5 \cdot \frac{l\_m}{t\_m \cdot \sqrt{0.5}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 2.5e+161)
   (asin (pow (+ 1.0 (/ 2.0 (/ l_m (/ (* t_m t_m) l_m)))) -0.5))
   (asin (* 0.5 (/ l_m (* t_m (sqrt 0.5)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 2.5e+161) {
		tmp = asin(pow((1.0 + (2.0 / (l_m / ((t_m * t_m) / l_m)))), -0.5));
	} else {
		tmp = asin((0.5 * (l_m / (t_m * sqrt(0.5)))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (t_m <= 2.5d+161) then
        tmp = asin(((1.0d0 + (2.0d0 / (l_m / ((t_m * t_m) / l_m)))) ** (-0.5d0)))
    else
        tmp = asin((0.5d0 * (l_m / (t_m * sqrt(0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 2.5e+161) {
		tmp = Math.asin(Math.pow((1.0 + (2.0 / (l_m / ((t_m * t_m) / l_m)))), -0.5));
	} else {
		tmp = Math.asin((0.5 * (l_m / (t_m * Math.sqrt(0.5)))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 2.5e+161:
		tmp = math.asin(math.pow((1.0 + (2.0 / (l_m / ((t_m * t_m) / l_m)))), -0.5))
	else:
		tmp = math.asin((0.5 * (l_m / (t_m * math.sqrt(0.5)))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 2.5e+161)
		tmp = asin((Float64(1.0 + Float64(2.0 / Float64(l_m / Float64(Float64(t_m * t_m) / l_m)))) ^ -0.5));
	else
		tmp = asin(Float64(0.5 * Float64(l_m / Float64(t_m * sqrt(0.5)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (t_m <= 2.5e+161)
		tmp = asin(((1.0 + (2.0 / (l_m / ((t_m * t_m) / l_m)))) ^ -0.5));
	else
		tmp = asin((0.5 * (l_m / (t_m * sqrt(0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 2.5e+161], N[ArcSin[N[Power[N[(1.0 + N[(2.0 / N[(l$95$m / N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(0.5 * N[(l$95$m / N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.4999999999999998e161

   1. Initial program 86.7%

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

   3. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 72.6%

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

   5. Simplified 72.6%

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

   7. Applied egg-rr 78.6%

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

   if 2.4999999999999998e161 < t

   1. Initial program 65.1%

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

   3. Taylor expanded in l around 0

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

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

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

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

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

   5. Simplified 56.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \left(\frac{-0.125 \cdot \left(\ell \cdot \ell\right)}{\left(t \cdot t\right) \cdot \left(t \cdot \sqrt{0.5}\right)} + \frac{\sqrt{0.5}}{t}\right)\right)\right)} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{t \cdot \left(-0.125 \cdot \left(\ell \cdot \ell\right)\right) + \left(t \cdot \left(t \cdot t\right)\right) \cdot 0.5}{\left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \sqrt{0.5}\right)} \cdot \left(\ell \cdot \sqrt{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}\right)\right)} \]

   8. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 67.2%

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

   10. Simplified 67.2%

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

   11. Taylor expanded in Om around 0

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

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

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 71.3%

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

   13. Simplified 71.3%

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

Alternative 6: 74.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{\frac{Omc}{Om}}}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.6e-5)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin (sqrt (- 1.0 (/ (/ Om (/ Omc Om)) Omc))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.6e-5) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 1.6d-5) then
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    else
        tmp = asin(sqrt((1.0d0 - ((om / (omc / om)) / omc))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.6e-5) {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.6e-5:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / (Omc / Om)) / Omc))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.6e-5)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Float64(Omc / Om)) / Omc))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 1.6e-5)
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt((1.0 - ((Om / (Omc / Om)) / Omc))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 1.6e-5], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.59999999999999993e-5

   1. Initial program 82.6%

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

   3. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 32.9%

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

   8. Simplified 32.9%

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

   if 1.59999999999999993e-5 < l

   1. Initial program 89.8%

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

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

   5. Simplified 67.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)} \]
   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 75.0%

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

   7. Applied egg-rr 75.0%

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

Alternative 7: 74.1% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 6e-6) (asin (/ (* l_m (sqrt 0.5)) t_m)) (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 6e-6) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 6d-6) then
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 6e-6) {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 6e-6:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 6e-6)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 6e-6)
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 6e-6], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.0000000000000002e-6

   1. Initial program 82.6%

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

   3. Taylor expanded in Om around 0

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

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

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

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

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

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

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 32.9%

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

   8. Simplified 32.9%

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

   if 6.0000000000000002e-6 < l

   1. Initial program 89.8%

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

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

   5. Simplified 67.2%

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

   6. Taylor expanded in Om around 0

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

   8. Simplified 73.9%

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

Alternative 8: 50.3% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

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

5. Simplified 47.3%

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

6. Taylor expanded in Om around 0

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

8. Simplified 53.2%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))