Toniolo and Linder, Equation (2)

Percentage Accurate: 83.3% → 98.8%

Time: 20.4s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t\_1} \cdot \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t_m l_m) 1e+146)
     (asin (sqrt (/ t_1 (+ (* 2.0 (/ (/ t_m l_m) (/ l_m t_m))) 1.0))))
     (asin (* (sqrt t_1) (* l_m (/ (sqrt 0.5) t_m)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t_m / l_m) <= 1e+146) {
		tmp = asin(sqrt((t_1 / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))));
	} else {
		tmp = asin((sqrt(t_1) * (l_m * (sqrt(0.5) / t_m))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t_m / l_m) <= 1d+146) then
        tmp = asin(sqrt((t_1 / ((2.0d0 * ((t_m / l_m) / (l_m / t_m))) + 1.0d0))))
    else
        tmp = asin((sqrt(t_1) * (l_m * (sqrt(0.5d0) / t_m))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t_m / l_m) <= 1e+146) {
		tmp = Math.asin(Math.sqrt((t_1 / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))));
	} else {
		tmp = Math.asin((Math.sqrt(t_1) * (l_m * (Math.sqrt(0.5) / t_m))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t_m / l_m) <= 1e+146:
		tmp = math.asin(math.sqrt((t_1 / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))))
	else:
		tmp = math.asin((math.sqrt(t_1) * (l_m * (math.sqrt(0.5) / t_m))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+146)
		tmp = asin(sqrt(Float64(t_1 / Float64(Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m))) + 1.0))));
	else
		tmp = asin(Float64(sqrt(t_1) * Float64(l_m * Float64(sqrt(0.5) / t_m))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t_m / l_m) <= 1e+146)
		tmp = asin(sqrt((t_1 / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))));
	else
		tmp = asin((sqrt(t_1) * (l_m * (sqrt(0.5) / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+146], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.99999999999999934e145

   1. Initial program 91.4%

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

      1. unpow2 91.4%

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

      2. clear-num 91.4%

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

      3. un-div-inv 91.4%

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

   4. Applied egg-rr 91.4%

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

      1. unpow2 91.4%

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

      2. clear-num 91.4%

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

      3. un-div-inv 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 9.99999999999999934e145 < (/.f64 t l)

   1. Initial program 47.5%

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

   3. Taylor expanded in t around inf 93.6%

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

      1. *-commutative 93.6%

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

      2. unpow2 93.6%

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

      3. unpow2 93.6%

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

      4. times-frac 99.6%

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

      5. unpow2 99.6%

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

      6. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

      1. unpow2 47.5%

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

      2. clear-num 47.5%

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

      3. un-div-inv 47.5%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\frac{1}{\frac{\mathsf{hypot}\left(1, \frac{\sqrt{\frac{2 \cdot t\_m}{l\_m}}}{\sqrt{\frac{l\_m}{t\_m}}}\right)}{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin
  (/
   1.0
   (/
    (hypot 1.0 (/ (sqrt (/ (* 2.0 t_m) l_m)) (sqrt (/ l_m t_m))))
    (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))))))

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 / (hypot(1.0, (sqrt(((2.0 * t_m) / l_m)) / sqrt((l_m / t_m)))) / sqrt((1.0 - pow((Om / Omc), 2.0))))));
}

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 / (Math.hypot(1.0, (Math.sqrt(((2.0 * t_m) / l_m)) / Math.sqrt((l_m / t_m)))) / Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin((1.0 / (math.hypot(1.0, (math.sqrt(((2.0 * t_m) / l_m)) / math.sqrt((l_m / t_m)))) / math.sqrt((1.0 - math.pow((Om / Omc), 2.0))))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(1.0 / Float64(hypot(1.0, Float64(sqrt(Float64(Float64(2.0 * t_m) / l_m)) / sqrt(Float64(l_m / t_m)))) / sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((1.0 / (hypot(1.0, (sqrt(((2.0 * t_m) / l_m)) / sqrt((l_m / t_m)))) / sqrt((1.0 - ((Om / Omc) ^ 2.0))))));
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[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[N[(N[(2.0 * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(l$95$m / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 85.7%

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

   1. sqrt-div 85.7%

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

   2. clear-num 85.7%

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

   3. add-sqr-sqrt 85.7%

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

   4. hypot-1-def 85.7%

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

   5. *-commutative 85.7%

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

   6. sqrt-prod 86.0%

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

   7. sqrt-pow1 97.5%

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

   8. metadata-eval 97.5%

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

   9. pow1 97.5%

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

4. Applied egg-rr 97.5%

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

   1. *-commutative 97.5%

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

   2. pow1 97.5%

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

   3. metadata-eval 97.5%

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

   4. sqrt-pow1 86.0%

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

   5. sqrt-prod 85.7%

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

   6. unpow2 85.7%

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

   7. clear-num 85.7%

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

   8. div-inv 85.7%

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

   9. associate-*r/ 85.7%

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

   10. sqrt-div 48.0%

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

6. Applied egg-rr 48.0%

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

   1. associate-*r/ 48.0%

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

8. Simplified 48.0%

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

Alternative 3: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\frac{1}{\frac{\mathsf{hypot}\left(1, \sqrt{2} \cdot \frac{t\_m}{l\_m}\right)}{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin
  (/
   1.0
   (/
    (hypot 1.0 (* (sqrt 2.0) (/ t_m l_m)))
    (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))))))

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 / (hypot(1.0, (sqrt(2.0) * (t_m / l_m))) / sqrt((1.0 - pow((Om / Omc), 2.0))))));
}

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 / (Math.hypot(1.0, (Math.sqrt(2.0) * (t_m / l_m))) / Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin((1.0 / (math.hypot(1.0, (math.sqrt(2.0) * (t_m / l_m))) / math.sqrt((1.0 - math.pow((Om / Omc), 2.0))))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(1.0 / Float64(hypot(1.0, Float64(sqrt(2.0) * Float64(t_m / l_m))) / sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((1.0 / (hypot(1.0, (sqrt(2.0) * (t_m / l_m))) / sqrt((1.0 - ((Om / Omc) ^ 2.0))))));
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[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 85.7%

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

   1. sqrt-div 85.7%

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

   2. clear-num 85.7%

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

   3. add-sqr-sqrt 85.7%

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

   4. hypot-1-def 85.7%

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

   5. *-commutative 85.7%

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

   6. sqrt-prod 86.0%

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

   7. sqrt-pow1 97.5%

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

   8. metadata-eval 97.5%

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

   9. pow1 97.5%

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

4. Applied egg-rr 97.5%

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

5. Final simplification 97.5%

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

Alternative 4: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \frac{t\_m \cdot \sqrt{2}}{l\_m}\right)}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
   (hypot 1.0 (/ (* t_m (sqrt 2.0)) l_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin((sqrt((1.0 - pow((Om / Omc), 2.0))) / hypot(1.0, ((t_m * sqrt(2.0)) / l_m))));
}

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((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) / Math.hypot(1.0, ((t_m * Math.sqrt(2.0)) / l_m))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) / math.hypot(1.0, ((t_m * math.sqrt(2.0)) / l_m))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) / hypot(1.0, Float64(Float64(t_m * sqrt(2.0)) / l_m))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) / hypot(1.0, ((t_m * sqrt(2.0)) / l_m))));
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[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 85.7%

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

   1. sqrt-div 85.7%

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

   2. frac-2neg 85.7%

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

   3. add-sqr-sqrt 85.7%

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

   4. hypot-1-def 85.7%

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

   5. *-commutative 85.7%

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

   6. sqrt-prod 86.0%

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

   7. sqrt-pow1 97.5%

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

   8. metadata-eval 97.5%

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

   9. pow1 97.5%

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

4. Applied egg-rr 97.5%

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

   1. associate-*l/ 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 5: 98.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{hypot}\left(1, \sqrt{2} \cdot \frac{t\_m}{l\_m}\right)}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
   (hypot 1.0 (* (sqrt 2.0) (/ t_m l_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin((sqrt((1.0 - pow((Om / Omc), 2.0))) / hypot(1.0, (sqrt(2.0) * (t_m / l_m)))));
}

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((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) / Math.hypot(1.0, (Math.sqrt(2.0) * (t_m / l_m)))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) / math.hypot(1.0, (math.sqrt(2.0) * (t_m / l_m)))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) / hypot(1.0, Float64(sqrt(2.0) * Float64(t_m / l_m)))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) / hypot(1.0, (sqrt(2.0) * (t_m / l_m)))));
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[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 85.7%

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

   1. sqrt-div 85.7%

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

   2. div-inv 85.7%

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

   3. add-sqr-sqrt 85.7%

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

   4. hypot-1-def 85.7%

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

   5. *-commutative 85.7%

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

   6. sqrt-prod 86.0%

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

   7. sqrt-pow1 97.5%

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

   8. metadata-eval 97.5%

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

   9. pow1 97.5%

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

4. Applied egg-rr 97.5%

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

   1. associate-*r/ 97.5%

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

   2. *-rgt-identity 97.5%

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

6. Simplified 97.5%

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

7. Final simplification 97.5%

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

Alternative 6: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 10^{+292}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{2 \cdot \frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}} + 1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (pow (/ t_m l_m) 2.0) 1e+292)
   (asin (pow (+ (/ (* 2.0 (/ t_m l_m)) (/ l_m t_m)) 1.0) -0.5))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (pow((t_m / l_m), 2.0) <= 1e+292) {
		tmp = asin(pow((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0), -0.5));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (((t_m / l_m) ** 2.0d0) <= 1d+292) then
        tmp = asin(((((2.0d0 * (t_m / l_m)) / (l_m / t_m)) + 1.0d0) ** (-0.5d0)))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.pow((t_m / l_m), 2.0) <= 1e+292) {
		tmp = Math.asin(Math.pow((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0), -0.5));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.pow((t_m / l_m), 2.0) <= 1e+292:
		tmp = math.asin(math.pow((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0), -0.5))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if ((Float64(t_m / l_m) ^ 2.0) <= 1e+292)
		tmp = asin((Float64(Float64(Float64(2.0 * Float64(t_m / l_m)) / Float64(l_m / t_m)) + 1.0) ^ -0.5));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (((t_m / l_m) ^ 2.0) <= 1e+292)
		tmp = asin(((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0) ^ -0.5));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision], 1e+292], N[ArcSin[N[Power[N[(N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) #s(literal 2 binary64)) < 1e292

   1. Initial program 98.4%

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

      1. sqrt-div 98.4%

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

      2. clear-num 98.4%

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

      3. add-sqr-sqrt 98.4%

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

      4. hypot-1-def 98.4%

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

      5. *-commutative 98.4%

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

      6. sqrt-prod 98.3%

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

      7. sqrt-pow1 98.3%

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

      8. metadata-eval 98.3%

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

      9. pow1 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. pow1 98.3%

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

      3. metadata-eval 98.3%

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

      4. sqrt-pow1 98.3%

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

      5. sqrt-prod 98.4%

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

      6. unpow2 98.4%

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

      7. clear-num 98.4%

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

      8. div-inv 98.4%

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

      9. associate-*r/ 98.4%

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

      10. sqrt-div 48.4%

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

   6. Applied egg-rr 48.4%

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

      1. associate-*r/ 48.4%

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

   8. Simplified 48.4%

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

   9. Taylor expanded in Om around 0 47.4%

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

       1. inv-pow 47.4%

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

       2. /-rgt-identity 47.4%

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

       3. hypot-undefine 47.4%

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

       4. sqrt-pow2 47.4%

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

       5. metadata-eval 47.4%

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

       6. sqrt-undiv 47.4%

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

       7. sqrt-undiv 97.3%

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

       8. add-sqr-sqrt 97.4%

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

       9. associate-/l* 97.4%

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

       10. metadata-eval 97.4%

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

   11. Applied egg-rr 97.4%

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

   if 1e292 < (pow.f64 (/.f64 t l) #s(literal 2 binary64))

   1. Initial program 48.4%

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

   3. Taylor expanded in t around inf 69.7%

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

      1. *-commutative 69.7%

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

      2. unpow2 69.7%

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

      3. unpow2 69.7%

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

      4. times-frac 74.4%

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

      5. unpow2 74.4%

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

      6. associate-/l* 74.4%

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

   5. Simplified 74.4%

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

   6. Taylor expanded in Om around 0 74.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 10^{+292}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{2 \cdot \frac{t}{\ell}}{\frac{\ell}{t}} + 1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) #s(literal 2 binary64)) < 5.00000000000000024e-5

   1. Initial program 97.9%

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

   3. Taylor expanded in t around 0 87.5%

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

      1. unpow2 87.5%

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

      2. unpow2 87.5%

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

      3. times-frac 96.7%

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

      4. unpow2 96.7%

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

   5. Simplified 96.7%

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

      1. unpow2 97.9%

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

      2. clear-num 97.9%

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

      3. un-div-inv 97.9%

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

   7. Applied egg-rr 96.7%

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

   if 5.00000000000000024e-5 < (pow.f64 (/.f64 t l) #s(literal 2 binary64))

   1. Initial program 73.8%

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

   3. Taylor expanded in t around inf 56.2%

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

      1. *-commutative 56.2%

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

      2. unpow2 56.2%

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

      3. unpow2 56.2%

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

      4. times-frac 62.1%

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

      5. unpow2 62.1%

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

      6. associate-/l* 62.1%

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

   5. Simplified 62.1%

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

   6. Taylor expanded in Om around 0 61.7%

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

4. Final simplification 79.1%

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

Alternative 8: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - t\_1}{2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot \left(t\_1 \cdot -0.5 + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (/ (/ Om Omc) (/ Omc Om))))
   (if (<= (/ t_m l_m) 1e+146)
     (asin (sqrt (/ (- 1.0 t_1) (+ (* 2.0 (/ (/ t_m l_m) (/ l_m t_m))) 1.0))))
     (asin (* (* l_m (/ (sqrt 0.5) t_m)) (+ (* t_1 -0.5) 1.0))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = (Om / Omc) / (Omc / Om);
	double tmp;
	if ((t_m / l_m) <= 1e+146) {
		tmp = asin(sqrt(((1.0 - t_1) / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))));
	} else {
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * ((t_1 * -0.5) + 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) :: t_1
    real(8) :: tmp
    t_1 = (om / omc) / (omc / om)
    if ((t_m / l_m) <= 1d+146) then
        tmp = asin(sqrt(((1.0d0 - t_1) / ((2.0d0 * ((t_m / l_m) / (l_m / t_m))) + 1.0d0))))
    else
        tmp = asin(((l_m * (sqrt(0.5d0) / t_m)) * ((t_1 * (-0.5d0)) + 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 t_1 = (Om / Omc) / (Omc / Om);
	double tmp;
	if ((t_m / l_m) <= 1e+146) {
		tmp = Math.asin(Math.sqrt(((1.0 - t_1) / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))));
	} else {
		tmp = Math.asin(((l_m * (Math.sqrt(0.5) / t_m)) * ((t_1 * -0.5) + 1.0)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = (Om / Omc) / (Omc / Om)
	tmp = 0
	if (t_m / l_m) <= 1e+146:
		tmp = math.asin(math.sqrt(((1.0 - t_1) / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))))
	else:
		tmp = math.asin(((l_m * (math.sqrt(0.5) / t_m)) * ((t_1 * -0.5) + 1.0)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(Float64(Om / Omc) / Float64(Omc / Om))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+146)
		tmp = asin(sqrt(Float64(Float64(1.0 - t_1) / Float64(Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m))) + 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * Float64(sqrt(0.5) / t_m)) * Float64(Float64(t_1 * -0.5) + 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)
	t_1 = (Om / Omc) / (Omc / Om);
	tmp = 0.0;
	if ((t_m / l_m) <= 1e+146)
		tmp = asin(sqrt(((1.0 - t_1) / ((2.0 * ((t_m / l_m) / (l_m / t_m))) + 1.0))));
	else
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * ((t_1 * -0.5) + 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_] := Block[{t$95$1 = N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+146], N[ArcSin[N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.99999999999999934e145

   1. Initial program 91.4%

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

      1. unpow2 91.4%

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

      2. clear-num 91.4%

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

      3. un-div-inv 91.4%

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

   4. Applied egg-rr 91.4%

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

      1. unpow2 91.4%

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

      2. clear-num 91.4%

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

      3. un-div-inv 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 9.99999999999999934e145 < (/.f64 t l)

   1. Initial program 47.5%

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

   3. Taylor expanded in t around inf 93.6%

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

      1. *-commutative 93.6%

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

      2. unpow2 93.6%

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

      3. unpow2 93.6%

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

      4. times-frac 99.6%

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

      5. unpow2 99.6%

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

      6. associate-/l* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in Om around 0 93.6%

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

      1. unpow2 93.6%

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

      2. unpow2 93.6%

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

      3. times-frac 99.6%

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

      4. unpow2 99.6%

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

   8. Simplified 99.6%

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

      1. unpow2 47.5%

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

      2. clear-num 47.5%

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

      3. un-div-inv 47.5%

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

   10. Applied egg-rr 99.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \left(\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}} \cdot -0.5 + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{2 \cdot \frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}} + 1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot \left(\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}} \cdot -0.5 + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 4e+30)
   (asin (pow (+ (/ (* 2.0 (/ t_m l_m)) (/ l_m t_m)) 1.0) -0.5))
   (asin
    (*
     (* l_m (/ (sqrt 0.5) t_m))
     (+ (* (/ (/ Om Omc) (/ Omc Om)) -0.5) 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 ((t_m / l_m) <= 4e+30) {
		tmp = asin(pow((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0), -0.5));
	} else {
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * ((((Om / Omc) / (Omc / Om)) * -0.5) + 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 ((t_m / l_m) <= 4d+30) then
        tmp = asin(((((2.0d0 * (t_m / l_m)) / (l_m / t_m)) + 1.0d0) ** (-0.5d0)))
    else
        tmp = asin(((l_m * (sqrt(0.5d0) / t_m)) * ((((om / omc) / (omc / om)) * (-0.5d0)) + 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 ((t_m / l_m) <= 4e+30) {
		tmp = Math.asin(Math.pow((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0), -0.5));
	} else {
		tmp = Math.asin(((l_m * (Math.sqrt(0.5) / t_m)) * ((((Om / Omc) / (Omc / Om)) * -0.5) + 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 (t_m / l_m) <= 4e+30:
		tmp = math.asin(math.pow((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0), -0.5))
	else:
		tmp = math.asin(((l_m * (math.sqrt(0.5) / t_m)) * ((((Om / Omc) / (Omc / Om)) * -0.5) + 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 (Float64(t_m / l_m) <= 4e+30)
		tmp = asin((Float64(Float64(Float64(2.0 * Float64(t_m / l_m)) / Float64(l_m / t_m)) + 1.0) ^ -0.5));
	else
		tmp = asin(Float64(Float64(l_m * Float64(sqrt(0.5) / t_m)) * Float64(Float64(Float64(Float64(Om / Omc) / Float64(Omc / Om)) * -0.5) + 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 ((t_m / l_m) <= 4e+30)
		tmp = asin(((((2.0 * (t_m / l_m)) / (l_m / t_m)) + 1.0) ^ -0.5));
	else
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * ((((Om / Omc) / (Omc / Om)) * -0.5) + 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[N[(t$95$m / l$95$m), $MachinePrecision], 4e+30], N[ArcSin[N[Power[N[(N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.0000000000000001e30

   1. Initial program 90.6%

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

      1. sqrt-div 90.6%

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

      2. clear-num 90.6%

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

      3. add-sqr-sqrt 90.6%

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

      4. hypot-1-def 90.6%

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

      5. *-commutative 90.6%

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

      6. sqrt-prod 91.0%

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

      7. sqrt-pow1 98.3%

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

      8. metadata-eval 98.3%

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

      9. pow1 98.3%

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

   4. Applied egg-rr 98.3%

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

      1. *-commutative 98.3%

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

      2. pow1 98.3%

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

      3. metadata-eval 98.3%

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

      4. sqrt-pow1 91.0%

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

      5. sqrt-prod 90.6%

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

      6. unpow2 90.6%

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

      7. clear-num 90.6%

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

      8. div-inv 90.6%

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

      9. associate-*r/ 90.6%

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

      10. sqrt-div 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. associate-*r/ 35.8%

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

   8. Simplified 35.8%

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

   9. Taylor expanded in Om around 0 35.0%

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

       1. inv-pow 35.0%

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

       2. /-rgt-identity 35.0%

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

       3. hypot-undefine 35.0%

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

       4. sqrt-pow2 35.0%

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

       5. metadata-eval 35.0%

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

       6. sqrt-undiv 35.0%

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

       7. sqrt-undiv 89.8%

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

       8. add-sqr-sqrt 89.9%

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

       9. associate-/l* 89.9%

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

       10. metadata-eval 89.9%

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

   11. Applied egg-rr 89.9%

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

   if 4.0000000000000001e30 < (/.f64 t l)

   1. Initial program 67.1%

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

   3. Taylor expanded in t around inf 90.1%

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

      1. *-commutative 90.1%

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

      2. unpow2 90.1%

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

      3. unpow2 90.1%

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

      4. times-frac 99.4%

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

      5. unpow2 99.4%

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

      6. associate-/l* 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in Om around 0 90.0%

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

      1. unpow2 90.0%

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

      2. unpow2 90.0%

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

      3. times-frac 99.3%

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

      4. unpow2 99.3%

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

   8. Simplified 99.3%

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

      1. unpow2 67.1%

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

      2. clear-num 67.1%

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

      3. un-div-inv 67.1%

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

   10. Applied egg-rr 99.3%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left({\left(\frac{2 \cdot \frac{t}{\ell}}{\frac{\ell}{t}} + 1\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \left(\frac{\frac{Om}{Omc}}{\frac{Omc}{Om}} \cdot -0.5 + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t\_m}{l\_m}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.01)
   (asin (- 1.0 (pow (/ t_m l_m) 2.0)))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.01) {
		tmp = asin((1.0 - pow((t_m / l_m), 2.0)));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.01d0) then
        tmp = asin((1.0d0 - ((t_m / l_m) ** 2.0d0)))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.01) {
		tmp = Math.asin((1.0 - Math.pow((t_m / l_m), 2.0)));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.01:
		tmp = math.asin((1.0 - math.pow((t_m / l_m), 2.0)))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.01)
		tmp = asin(Float64(1.0 - (Float64(t_m / l_m) ^ 2.0)));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.01)
		tmp = asin((1.0 - ((t_m / l_m) ^ 2.0)));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.01], N[ArcSin[N[(1.0 - N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0100000000000000002

   1. Initial program 90.0%

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

      1. sqrt-div 90.0%

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

      2. clear-num 90.0%

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

      3. add-sqr-sqrt 90.0%

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

      4. hypot-1-def 90.0%

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

      5. *-commutative 90.0%

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

      6. sqrt-prod 90.4%

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

      7. sqrt-pow1 98.2%

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

      8. metadata-eval 98.2%

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

      9. pow1 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. pow1 98.2%

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

      3. metadata-eval 98.2%

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

      4. sqrt-pow1 90.4%

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

      5. sqrt-prod 90.0%

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

      6. unpow2 90.0%

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

      7. clear-num 90.0%

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

      8. div-inv 90.0%

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

      9. associate-*r/ 90.0%

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

      10. sqrt-div 31.4%

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

   6. Applied egg-rr 31.4%

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

      1. associate-*r/ 31.4%

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

   8. Simplified 31.4%

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

   9. Taylor expanded in Om around 0 30.6%

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

   10. Taylor expanded in t around 0 53.8%

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

       1. associate-/l* 53.8%

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

       2. *-commutative 53.8%

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

       3. associate-*l* 53.8%

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

       4. associate-*r/ 53.8%

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

       5. associate-*l/ 53.8%

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

       6. unpow2 53.8%

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

       7. rem-square-sqrt 53.8%

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

       8. metadata-eval 53.8%

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

       9. associate-*r/ 53.8%

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

       10. mul-1-neg 53.8%

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

       11. unsub-neg 53.8%

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

       12. unpow2 53.8%

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

       13. unpow2 53.8%

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

       14. times-frac 64.4%

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

       15. unpow2 64.4%

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

   12. Simplified 64.4%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 73.5%

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

   3. Taylor expanded in t around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. unpow2 87.1%

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

      3. unpow2 87.1%

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

      4. times-frac 96.8%

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

      5. unpow2 96.8%

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

      6. associate-/l* 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in Om around 0 96.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t\_m}{l\_m}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.01)
   (asin (- 1.0 (pow (/ t_m l_m) 2.0)))
   (asin (/ l_m (* t_m (sqrt 2.0))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.01) {
		tmp = asin((1.0 - pow((t_m / l_m), 2.0)));
	} else {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.01d0) then
        tmp = asin((1.0d0 - ((t_m / l_m) ** 2.0d0)))
    else
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.01) {
		tmp = Math.asin((1.0 - Math.pow((t_m / l_m), 2.0)));
	} else {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.01:
		tmp = math.asin((1.0 - math.pow((t_m / l_m), 2.0)))
	else:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.01)
		tmp = asin(Float64(1.0 - (Float64(t_m / l_m) ^ 2.0)));
	else
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.01)
		tmp = asin((1.0 - ((t_m / l_m) ^ 2.0)));
	else
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.01], N[ArcSin[N[(1.0 - N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0100000000000000002

   1. Initial program 90.0%

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

      1. sqrt-div 90.0%

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

      2. clear-num 90.0%

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

      3. add-sqr-sqrt 90.0%

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

      4. hypot-1-def 90.0%

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

      5. *-commutative 90.0%

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

      6. sqrt-prod 90.4%

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

      7. sqrt-pow1 98.2%

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

      8. metadata-eval 98.2%

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

      9. pow1 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. *-commutative 98.2%

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

      2. pow1 98.2%

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

      3. metadata-eval 98.2%

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

      4. sqrt-pow1 90.4%

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

      5. sqrt-prod 90.0%

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

      6. unpow2 90.0%

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

      7. clear-num 90.0%

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

      8. div-inv 90.0%

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

      9. associate-*r/ 90.0%

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

      10. sqrt-div 31.4%

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

   6. Applied egg-rr 31.4%

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

      1. associate-*r/ 31.4%

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

   8. Simplified 31.4%

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

   9. Taylor expanded in Om around 0 30.6%

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

   10. Taylor expanded in t around 0 53.8%

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

       1. associate-/l* 53.8%

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

       2. *-commutative 53.8%

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

       3. associate-*l* 53.8%

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

       4. associate-*r/ 53.8%

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

       5. associate-*l/ 53.8%

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

       6. unpow2 53.8%

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

       7. rem-square-sqrt 53.8%

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

       8. metadata-eval 53.8%

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

       9. associate-*r/ 53.8%

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

       10. mul-1-neg 53.8%

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

       11. unsub-neg 53.8%

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

       12. unpow2 53.8%

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

       13. unpow2 53.8%

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

       14. times-frac 64.4%

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

       15. unpow2 64.4%

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

   12. Simplified 64.4%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 73.5%

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

      1. sqrt-div 73.5%

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

      2. frac-2neg 73.5%

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

      3. add-sqr-sqrt 73.5%

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

      4. hypot-1-def 73.5%

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

      5. *-commutative 73.5%

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

      6. sqrt-prod 73.3%

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

      7. sqrt-pow1 95.5%

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

      8. metadata-eval 95.5%

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

      9. pow1 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. associate-*l/ 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in Om around 0 94.8%

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

   8. Taylor expanded in t around inf 96.0%

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

Alternative 12: 71.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.6 \cdot 10^{-86}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{l\_m}{t\_m}}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 2.6e-86) (asin (/ (/ l_m t_m) (sqrt 2.0))) (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 2.6e-86) {
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 2.6d-86) then
        tmp = asin(((l_m / t_m) / sqrt(2.0d0)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 2.6e-86) {
		tmp = Math.asin(((l_m / t_m) / Math.sqrt(2.0)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 2.6e-86:
		tmp = math.asin(((l_m / t_m) / math.sqrt(2.0)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 2.6e-86)
		tmp = asin(Float64(Float64(l_m / t_m) / sqrt(2.0)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 2.6e-86)
		tmp = asin(((l_m / t_m) / sqrt(2.0)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 2.6e-86], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.6000000000000001e-86

   1. Initial program 84.2%

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

      1. sqrt-div 84.2%

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

      2. clear-num 84.2%

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

      3. add-sqr-sqrt 84.2%

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

      4. hypot-1-def 84.2%

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

      5. *-commutative 84.2%

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

      6. sqrt-prod 84.0%

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

      7. sqrt-pow1 96.9%

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

      8. metadata-eval 96.9%

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

      9. pow1 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. *-commutative 96.9%

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

      2. pow1 96.9%

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

      3. metadata-eval 96.9%

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

      4. sqrt-pow1 84.0%

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

      5. sqrt-prod 84.2%

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

      6. unpow2 84.2%

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

      7. clear-num 84.1%

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

      8. div-inv 84.1%

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

      9. associate-*r/ 84.1%

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

      10. sqrt-div 45.5%

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

   6. Applied egg-rr 45.5%

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

      1. associate-*r/ 45.5%

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

   8. Simplified 45.5%

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

   9. Taylor expanded in Om around 0 45.5%

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

   10. Taylor expanded in t around inf 38.0%

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

       1. associate-/r* 38.0%

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

   12. Simplified 38.0%

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

   if 2.6000000000000001e-86 < l

   1. Initial program 89.3%

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

   3. Taylor expanded in t around 0 64.7%

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

      1. unpow2 64.7%

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

      2. unpow2 64.7%

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

      3. times-frac 72.6%

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

      4. unpow2 72.6%

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

   5. Simplified 72.6%

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

   6. Taylor expanded in Om around 0 70.6%

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

Alternative 13: 71.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3 \cdot 10^{-86}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 3e-86) (asin (/ l_m (* t_m (sqrt 2.0)))) (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 3e-86) {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l_m <= 3d-86) then
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 3e-86) {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 3e-86:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 3e-86)
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (l_m <= 3e-86)
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 3e-86], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.0000000000000001e-86

   1. Initial program 84.2%

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

      1. sqrt-div 84.2%

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

      2. frac-2neg 84.2%

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

      3. add-sqr-sqrt 84.2%

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

      4. hypot-1-def 84.2%

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

      5. *-commutative 84.2%

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

      6. sqrt-prod 84.0%

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

      7. sqrt-pow1 96.8%

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

      8. metadata-eval 96.8%

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

      9. pow1 96.8%

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

   4. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in Om around 0 96.7%

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

   8. Taylor expanded in t around inf 38.0%

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

   if 3.0000000000000001e-86 < l

   1. Initial program 89.3%

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

   3. Taylor expanded in t around 0 64.7%

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

      1. unpow2 64.7%

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

      2. unpow2 64.7%

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

      3. times-frac 72.6%

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

      4. unpow2 72.6%

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

   5. Simplified 72.6%

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

   6. Taylor expanded in Om around 0 70.6%

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

Alternative 14: 50.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in t around 0 46.0%

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

   1. unpow2 46.0%

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

   2. unpow2 46.0%

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

   3. times-frac 50.8%

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

   4. unpow2 50.8%

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

5. Simplified 50.8%

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

6. Taylor expanded in Om around 0 50.2%

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

Reproduce

herbie shell --seed 2024117 
(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)))))))