Toniolo and Linder, Equation (2)

Percentage Accurate: 84.2% → 98.4%

Time: 14.6s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 + t_1 \cdot -0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (pow (/ Om Omc) 2.0)))
   (if (<= (/ t l) -2e+156)
     (asin (/ (- l) (/ t (sqrt 0.5))))
     (if (<= (/ t l) 5e+115)
       (asin (sqrt (/ (- 1.0 t_1) (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (/ (sqrt (+ 0.5 (* t_1 -0.5))) (/ t l)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -2e+156) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 5e+115) {
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (om / omc) ** 2.0d0
    if ((t / l) <= (-2d+156)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 5d+115) then
        tmp = asin(sqrt(((1.0d0 - t_1) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt((0.5d0 + (t_1 * (-0.5d0)))) / (t / l)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -2e+156) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 5e+115) {
		tmp = Math.asin(Math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -2e+156:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 5e+115:
		tmp = math.asin(math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt((0.5 + (t_1 * -0.5))) / (t / l)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(Om / Omc) ^ 2.0
	tmp = 0.0
	if (Float64(t / l) <= -2e+156)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 5e+115)
		tmp = asin(sqrt(Float64(Float64(1.0 - t_1) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(sqrt(Float64(0.5 + Float64(t_1 * -0.5))) / Float64(t / l)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = (Om / Omc) ^ 2.0;
	tmp = 0.0;
	if ((t / l) <= -2e+156)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 5e+115)
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+156], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+115], N[ArcSin[N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(0.5 + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e156

   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. Taylor expanded in t around -inf 85.4%

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

      1. mul-1-neg 85.4%

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

      2. *-commutative 85.4%

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

      3. distribute-rgt-neg-in 85.4%

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

      4. unpow2 85.4%

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

      5. unpow2 85.4%

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

      6. times-frac 99.6%

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

      7. unpow2 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/l* 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in Om around 0 99.7%

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

   if -2e156 < (/.f64 t l) < 5.00000000000000008e115

   1. Initial program 98.9%

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

      1. unpow2 98.9%

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

      2. clear-num 98.9%

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

      3. un-div-inv 98.9%

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

   3. Applied egg-rr 98.9%

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

   if 5.00000000000000008e115 < (/.f64 t l)

   1. Initial program 58.3%

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

   2. Taylor expanded in t around inf 86.9%

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

      1. *-commutative 86.9%

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

      2. unpow2 86.9%

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

      3. unpow2 86.9%

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

      4. associate-/l* 86.9%

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

   4. Simplified 86.9%

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

      1. expm1-log1p-u 86.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-udef 35.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]

      3. associate-*r/ 35.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)}\right)} - 1 \]

      4. pow1/2 35.4%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right)}^{0.5}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      5. times-frac 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}^{0.5} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      6. unpow2 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{0.5} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      7. pow1/2 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5} \cdot \color{blue}{{0.5}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      8. pow-prod-down 42.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

   6. Applied egg-rr 42.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.8× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/ (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (hypot 1.0 (/ (* t (sqrt 2.0)) l)))))

C

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

Java

t = Math.abs(t);
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))) / Math.hypot(1.0, ((t * Math.sqrt(2.0)) / l))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, 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 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.4%

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

   1. sqrt-div 84.3%

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

   2. div-inv 84.3%

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

   3. add-sqr-sqrt 84.3%

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

   4. hypot-1-def 84.3%

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

   5. *-commutative 84.3%

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

   6. sqrt-prod 84.3%

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

   7. unpow2 84.3%

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

   8. sqrt-prod 59.1%

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

   9. add-sqr-sqrt 98.7%

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

3. Applied egg-rr 98.7%

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

   1. associate-*r/ 98.7%

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

   2. *-rgt-identity 98.7%

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

   3. associate-*l/ 98.7%

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

5. Simplified 98.7%

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

6. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+39}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - t_1}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 + t_1 \cdot -0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (pow (/ Om Omc) 2.0)))
   (if (<= (/ t l) -5e+24)
     (asin (/ (- l) (/ t (sqrt 0.5))))
     (if (<= (/ t l) 1e+39)
       (asin (sqrt (/ (- 1.0 t_1) (+ 1.0 (* 2.0 (/ (* t (/ t l)) l))))))
       (asin (/ (sqrt (+ 0.5 (* t_1 -0.5))) (/ t l)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -5e+24) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 1e+39) {
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = asin((sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (om / omc) ** 2.0d0
    if ((t / l) <= (-5d+24)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 1d+39) then
        tmp = asin(sqrt(((1.0d0 - t_1) / (1.0d0 + (2.0d0 * ((t * (t / l)) / l))))))
    else
        tmp = asin((sqrt((0.5d0 + (t_1 * (-0.5d0)))) / (t / l)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -5e+24) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 1e+39) {
		tmp = Math.asin(Math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = Math.asin((Math.sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -5e+24:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 1e+39:
		tmp = math.asin(math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t * (t / l)) / l))))))
	else:
		tmp = math.asin((math.sqrt((0.5 + (t_1 * -0.5))) / (t / l)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(Om / Omc) ^ 2.0
	tmp = 0.0
	if (Float64(t / l) <= -5e+24)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 1e+39)
		tmp = asin(sqrt(Float64(Float64(1.0 - t_1) / Float64(1.0 + Float64(2.0 * Float64(Float64(t * Float64(t / l)) / l))))));
	else
		tmp = asin(Float64(sqrt(Float64(0.5 + Float64(t_1 * -0.5))) / Float64(t / l)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = (Om / Omc) ^ 2.0;
	tmp = 0.0;
	if ((t / l) <= -5e+24)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 1e+39)
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	else
		tmp = asin((sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5e+24], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+39], N[ArcSin[N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(0.5 + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.00000000000000045e24

   1. Initial program 69.2%

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

   2. Taylor expanded in t around -inf 86.2%

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

      1. mul-1-neg 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt-neg-in 86.2%

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

      4. unpow2 86.2%

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

      5. unpow2 86.2%

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

      6. times-frac 99.4%

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

      7. unpow2 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in Om around 0 99.5%

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

   if -5.00000000000000045e24 < (/.f64 t l) < 9.9999999999999994e38

   1. Initial program 98.7%

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

      1. unpow2 98.7%

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

      2. associate-*r/ 98.0%

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

   3. Applied egg-rr 98.0%

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

   if 9.9999999999999994e38 < (/.f64 t l)

   1. Initial program 67.2%

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

   2. Taylor expanded in t around inf 87.8%

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

      1. *-commutative 87.8%

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

      2. unpow2 87.8%

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

      3. unpow2 87.8%

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

      4. associate-/l* 87.9%

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

   4. Simplified 87.9%

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

      1. expm1-log1p-u 87.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-udef 28.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]

      3. associate-*r/ 28.5%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)}\right)} - 1 \]

      4. pow1/2 28.5%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right)}^{0.5}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      5. times-frac 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}^{0.5} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      6. unpow2 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{0.5} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      7. pow1/2 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5} \cdot \color{blue}{{0.5}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      8. pow-prod-down 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

   6. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p 99.5%

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

   8. Simplified 99.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+39}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 4: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 + t_1 \cdot -0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (pow (/ Om Omc) 2.0)))
   (if (<= (/ t l) -50.0)
     (asin (/ (- l) (/ t (sqrt 0.5))))
     (if (<= (/ t l) 2e-8)
       (asin (sqrt (- 1.0 t_1)))
       (asin (/ (sqrt (+ 0.5 (* t_1 -0.5))) (/ t l)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = asin(sqrt((1.0 - t_1)));
	} else {
		tmp = asin((sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (om / omc) ** 2.0d0
    if ((t / l) <= (-50.0d0)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 2d-8) then
        tmp = asin(sqrt((1.0d0 - t_1)))
    else
        tmp = asin((sqrt((0.5d0 + (t_1 * (-0.5d0)))) / (t / l)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = Math.asin(Math.sqrt((1.0 - t_1)));
	} else {
		tmp = Math.asin((Math.sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -50.0:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 2e-8:
		tmp = math.asin(math.sqrt((1.0 - t_1)))
	else:
		tmp = math.asin((math.sqrt((0.5 + (t_1 * -0.5))) / (t / l)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(Om / Omc) ^ 2.0
	tmp = 0.0
	if (Float64(t / l) <= -50.0)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 2e-8)
		tmp = asin(sqrt(Float64(1.0 - t_1)));
	else
		tmp = asin(Float64(sqrt(Float64(0.5 + Float64(t_1 * -0.5))) / Float64(t / l)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = (Om / Omc) ^ 2.0;
	tmp = 0.0;
	if ((t / l) <= -50.0)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 2e-8)
		tmp = asin(sqrt((1.0 - t_1)));
	else
		tmp = asin((sqrt((0.5 + (t_1 * -0.5))) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -50.0], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e-8], N[ArcSin[N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(0.5 + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -50

   1. Initial program 71.5%

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

   2. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. *-commutative 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. unpow2 87.2%

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

      5. unpow2 87.2%

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

      6. times-frac 99.4%

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

      7. unpow2 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in Om around 0 99.5%

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

   if -50 < (/.f64 t l) < 2e-8

   1. Initial program 98.6%

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

   2. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. unpow2 84.9%

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

      3. times-frac 98.5%

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

      4. unpow2 98.5%

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

   4. Simplified 98.5%

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

   if 2e-8 < (/.f64 t l)

   1. Initial program 72.9%

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

   2. Taylor expanded in t around inf 86.7%

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

      1. *-commutative 86.7%

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

      2. unpow2 86.7%

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

      3. unpow2 86.7%

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

      4. associate-/l* 86.8%

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

   4. Simplified 86.8%

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

      1. expm1-log1p-u 86.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-udef 27.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]

      3. associate-*r/ 27.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \color{blue}{\left(\frac{\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)}\right)} - 1 \]

      4. pow1/2 27.3%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right)}^{0.5}} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      5. times-frac 31.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - \color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}^{0.5} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      6. unpow2 31.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - \color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{0.5} \cdot \sqrt{0.5}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      7. pow1/2 31.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}^{0.5} \cdot \color{blue}{{0.5}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

      8. pow-prod-down 31.8%

         \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{\color{blue}{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}}{\frac{t}{\ell}}\right)\right)} - 1 \]

   6. Applied egg-rr 31.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 97.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{{\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right) \cdot 0.5\right)}^{0.5}}{\frac{t}{\ell}}\right)\right)\right)} \]

      2. expm1-log1p 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 5: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50.0)
   (asin (/ (- l) (/ t (sqrt 0.5))))
   (if (<= (/ t l) 2e-8)
     (asin (sqrt (- 1.0 (pow (/ Om Omc) 2.0))))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = asin(sqrt((1.0 - pow((Om / Omc), 2.0))));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: tmp
    if ((t / l) <= (-50.0d0)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 2d-8) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) ** 2.0d0))))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = Math.asin(Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50.0:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 2e-8:
		tmp = math.asin(math.sqrt((1.0 - math.pow((Om / Omc), 2.0))))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50.0)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 2e-8)
		tmp = asin(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50.0)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 2e-8)
		tmp = asin(sqrt((1.0 - ((Om / Omc) ^ 2.0))));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50.0], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e-8], N[ArcSin[N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -50

   1. Initial program 71.5%

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

   2. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. *-commutative 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. unpow2 87.2%

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

      5. unpow2 87.2%

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

      6. times-frac 99.4%

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

      7. unpow2 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in Om around 0 99.5%

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

   if -50 < (/.f64 t l) < 2e-8

   1. Initial program 98.6%

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

   2. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. unpow2 84.9%

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

      3. times-frac 98.5%

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

      4. unpow2 98.5%

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

   4. Simplified 98.5%

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

   if 2e-8 < (/.f64 t l)

   1. Initial program 72.9%

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

   2. Taylor expanded in t around inf 86.7%

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

      1. *-commutative 86.7%

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

      2. unpow2 86.7%

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

      3. unpow2 86.7%

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

      4. associate-/l* 86.8%

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

   4. Simplified 86.8%

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

   5. Taylor expanded in Om around 0 96.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 6: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+221} \lor \neg \left(\frac{t}{\ell} \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (or (<= (/ t l) -1e+221) (not (<= (/ t l) 2e-8)))
   (asin (/ (sqrt 0.5) (/ t l)))
   (asin (+ 1.0 (* -0.5 (* (/ Om Omc) (/ Om Omc)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (((t / l) <= -1e+221) || !((t / l) <= 2e-8)) {
		tmp = asin((sqrt(0.5) / (t / l)));
	} else {
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: tmp
    if (((t / l) <= (-1d+221)) .or. (.not. ((t / l) <= 2d-8))) then
        tmp = asin((sqrt(0.5d0) / (t / l)))
    else
        tmp = asin((1.0d0 + ((-0.5d0) * ((om / omc) * (om / omc)))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (((t / l) <= -1e+221) || !((t / l) <= 2e-8)) {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	} else {
		tmp = Math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if ((t / l) <= -1e+221) or not ((t / l) <= 2e-8):
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	else:
		tmp = math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if ((Float64(t / l) <= -1e+221) || !(Float64(t / l) <= 2e-8))
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	else
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (((t / l) <= -1e+221) || ~(((t / l) <= 2e-8)))
		tmp = asin((sqrt(0.5) / (t / l)));
	else
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[Or[LessEqual[N[(t / l), $MachinePrecision], -1e+221], N[Not[LessEqual[N[(t / l), $MachinePrecision], 2e-8]], $MachinePrecision]], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 + N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < -1e221 or 2e-8 < (/.f64 t l)

   1. Initial program 69.4%

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

   2. Taylor expanded in t around inf 76.2%

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

      1. *-commutative 76.2%

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

      2. unpow2 76.2%

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

      3. unpow2 76.2%

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

      4. associate-/l* 76.2%

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

   4. Simplified 76.2%

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

   5. Taylor expanded in Om around 0 86.8%

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

   if -1e221 < (/.f64 t l) < 2e-8

   1. Initial program 94.0%

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

   2. Taylor expanded in t around 0 65.6%

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

      1. unpow2 65.6%

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

      2. unpow2 65.6%

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

   4. Simplified 65.6%

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

   5. Taylor expanded in Om around 0 65.1%

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

      1. unpow2 65.1%

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

      2. unpow2 65.1%

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

      3. times-frac 75.4%

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

      4. unpow2 75.4%

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

   7. Simplified 75.4%

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

      1. unpow2 75.4%

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

   9. Applied egg-rr 75.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+221} \lor \neg \left(\frac{t}{\ell} \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \end{array} \]

Alternative 7: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50.0)
   (asin (* (sqrt 0.5) (/ (- l) t)))
   (if (<= (/ t l) 2e-8)
     (asin (+ 1.0 (* -0.5 (* (/ Om Omc) (/ Om Omc)))))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = asin((sqrt(0.5) * (-l / t)));
	} else if ((t / l) <= 2e-8) {
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: tmp
    if ((t / l) <= (-50.0d0)) then
        tmp = asin((sqrt(0.5d0) * (-l / t)))
    else if ((t / l) <= 2d-8) then
        tmp = asin((1.0d0 + ((-0.5d0) * ((om / omc) * (om / omc)))))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = Math.asin((Math.sqrt(0.5) * (-l / t)));
	} else if ((t / l) <= 2e-8) {
		tmp = Math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50.0:
		tmp = math.asin((math.sqrt(0.5) * (-l / t)))
	elif (t / l) <= 2e-8:
		tmp = math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50.0)
		tmp = asin(Float64(sqrt(0.5) * Float64(Float64(-l) / t)));
	elseif (Float64(t / l) <= 2e-8)
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50.0)
		tmp = asin((sqrt(0.5) * (-l / t)));
	elseif ((t / l) <= 2e-8)
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[((-l) / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e-8], N[ArcSin[N[(1.0 + N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -50

   1. Initial program 71.5%

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

   2. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. *-commutative 87.2%

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

      3. unpow2 87.2%

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

      4. unpow2 87.2%

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

      5. associate-/l* 86.2%

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

   4. Simplified 86.2%

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

   5. Taylor expanded in t around 0 87.2%

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

      1. associate-*r/ 87.1%

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

   7. Simplified 87.1%

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

   8. Taylor expanded in Om around 0 99.4%

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

   if -50 < (/.f64 t l) < 2e-8

   1. Initial program 98.6%

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

   2. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. unpow2 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in Om around 0 84.2%

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

      1. unpow2 84.2%

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

      2. unpow2 84.2%

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

      3. times-frac 97.7%

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

      4. unpow2 97.7%

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

   7. Simplified 97.7%

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

      1. unpow2 97.7%

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

   9. Applied egg-rr 97.7%

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

   if 2e-8 < (/.f64 t l)

   1. Initial program 72.9%

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

   2. Taylor expanded in t around inf 86.7%

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

      1. *-commutative 86.7%

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

      2. unpow2 86.7%

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

      3. unpow2 86.7%

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

      4. associate-/l* 86.8%

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

   4. Simplified 86.8%

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

   5. Taylor expanded in Om around 0 96.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{-\ell}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 8: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -50.0)
   (asin (/ (- l) (/ t (sqrt 0.5))))
   (if (<= (/ t l) 2e-8)
     (asin (+ 1.0 (* -0.5 (* (/ Om Omc) (/ Om Omc)))))
     (asin (/ (sqrt 0.5) (/ t l))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin((sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: tmp
    if ((t / l) <= (-50.0d0)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 2d-8) then
        tmp = asin((1.0d0 + ((-0.5d0) * ((om / omc) * (om / omc)))))
    else
        tmp = asin((sqrt(0.5d0) / (t / l)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -50.0) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 2e-8) {
		tmp = Math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -50.0:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 2e-8:
		tmp = math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -50.0)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 2e-8)
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -50.0)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 2e-8)
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin((sqrt(0.5) / (t / l)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -50.0], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e-8], N[ArcSin[N[(1.0 + N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -50

   1. Initial program 71.5%

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

   2. Taylor expanded in t around -inf 87.2%

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

      1. mul-1-neg 87.2%

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

      2. *-commutative 87.2%

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

      3. distribute-rgt-neg-in 87.2%

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

      4. unpow2 87.2%

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

      5. unpow2 87.2%

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

      6. times-frac 99.4%

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

      7. unpow2 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.5%

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

   4. Simplified 99.5%

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

   5. Taylor expanded in Om around 0 99.5%

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

   if -50 < (/.f64 t l) < 2e-8

   1. Initial program 98.6%

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

   2. Taylor expanded in t around 0 84.9%

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

      1. unpow2 84.9%

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

      2. unpow2 84.9%

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

   4. Simplified 84.9%

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

   5. Taylor expanded in Om around 0 84.2%

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

      1. unpow2 84.2%

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

      2. unpow2 84.2%

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

      3. times-frac 97.7%

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

      4. unpow2 97.7%

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

   7. Simplified 97.7%

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

      1. unpow2 97.7%

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

   9. Applied egg-rr 97.7%

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

   if 2e-8 < (/.f64 t l)

   1. Initial program 72.9%

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

   2. Taylor expanded in t around inf 86.7%

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

      1. *-commutative 86.7%

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

      2. unpow2 86.7%

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

      3. unpow2 86.7%

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

      4. associate-/l* 86.8%

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

   4. Simplified 86.8%

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

   5. Taylor expanded in Om around 0 96.9%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array} \]

Alternative 9: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.34 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.34e+73)
   (asin (+ 1.0 (* -0.5 (* (/ Om Omc) (/ Om Omc)))))
   (asin (* l (/ (sqrt 0.5) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.34e+73) {
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: tmp
    if (t <= 1.34d+73) then
        tmp = asin((1.0d0 + ((-0.5d0) * ((om / omc) * (om / omc)))))
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.34e+73) {
		tmp = Math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if t <= 1.34e+73:
		tmp = math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1.34e+73)
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 1.34e+73)
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1.34e+73], N[ArcSin[N[(1.0 + N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.34e73

   1. Initial program 87.1%

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

   2. Taylor expanded in t around 0 46.8%

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

      1. unpow2 46.8%

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

      2. unpow2 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in Om around 0 46.4%

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

      1. unpow2 46.4%

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

      2. unpow2 46.4%

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

      3. times-frac 53.7%

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

      4. unpow2 53.7%

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

   7. Simplified 53.7%

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

      1. unpow2 53.7%

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

   9. Applied egg-rr 53.7%

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

   if 1.34e73 < t

   1. Initial program 72.0%

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

   2. Taylor expanded in t around inf 34.8%

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

      1. associate-/l* 34.8%

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

      2. unpow2 34.8%

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

      3. unpow2 34.8%

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

      4. unpow2 34.8%

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

      5. unpow2 34.8%

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

   4. Simplified 34.8%

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

   5. Taylor expanded in Om around 0 60.6%

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

      1. associate-*r/ 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in l around 0 60.6%

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

      1. associate-*l/ 60.6%

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

      2. *-commutative 60.6%

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

   10. Simplified 60.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.34 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 6.6e+72)
   (asin (+ 1.0 (* -0.5 (* (/ Om Omc) (/ Om Omc)))))
   (asin (/ (* l (sqrt 0.5)) t))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 6.6e+72) {
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: tmp
    if (t <= 6.6d+72) then
        tmp = asin((1.0d0 + ((-0.5d0) * ((om / omc) * (om / omc)))))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 6.6e+72) {
		tmp = Math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if t <= 6.6e+72:
		tmp = math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 6.6e+72)
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 6.6e+72)
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[t, 6.6e+72], N[ArcSin[N[(1.0 + N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.6e72

   1. Initial program 87.1%

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

   2. Taylor expanded in t around 0 46.8%

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

      1. unpow2 46.8%

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

      2. unpow2 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in Om around 0 46.4%

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

      1. unpow2 46.4%

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

      2. unpow2 46.4%

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

      3. times-frac 53.7%

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

      4. unpow2 53.7%

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

   7. Simplified 53.7%

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

      1. unpow2 53.7%

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

   9. Applied egg-rr 53.7%

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

   if 6.6e72 < t

   1. Initial program 72.0%

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

   2. Taylor expanded in t around inf 34.8%

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

      1. associate-/l* 34.8%

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

      2. unpow2 34.8%

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

      3. unpow2 34.8%

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

      4. unpow2 34.8%

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

      5. unpow2 34.8%

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

   4. Simplified 34.8%

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

   5. Taylor expanded in Om around 0 60.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.6 \cdot 10^{+72}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 11: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.18e+73)
   (asin (+ 1.0 (* -0.5 (* (/ Om Omc) (/ Om Omc)))))
   (asin (/ l (* t (sqrt 2.0))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.18e+73) {
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin((l / (t * sqrt(2.0))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
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
    real(8) :: tmp
    if (t <= 1.18d+73) then
        tmp = asin((1.0d0 + ((-0.5d0) * ((om / omc) * (om / omc)))))
    else
        tmp = asin((l / (t * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.18e+73) {
		tmp = Math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if t <= 1.18e+73:
		tmp = math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1.18e+73)
		tmp = asin(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 1.18e+73)
		tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin((l / (t * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1.18e+73], N[ArcSin[N[(1.0 + N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.18000000000000004e73

   1. Initial program 87.1%

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

   2. Taylor expanded in t around 0 46.8%

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

      1. unpow2 46.8%

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

      2. unpow2 46.8%

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

   4. Simplified 46.8%

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

   5. Taylor expanded in Om around 0 46.4%

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

      1. unpow2 46.4%

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

      2. unpow2 46.4%

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

      3. times-frac 53.7%

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

      4. unpow2 53.7%

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

   7. Simplified 53.7%

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

      1. unpow2 53.7%

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

   9. Applied egg-rr 53.7%

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

   if 1.18000000000000004e73 < t

   1. Initial program 72.0%

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

      1. sqrt-div 72.0%

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

      2. div-inv 72.0%

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

      3. add-sqr-sqrt 72.0%

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

      4. hypot-1-def 72.0%

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

      5. *-commutative 72.0%

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

      6. sqrt-prod 71.9%

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

      7. unpow2 71.9%

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

      8. sqrt-prod 58.3%

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

      9. add-sqr-sqrt 97.9%

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

   3. Applied egg-rr 97.9%

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

      1. associate-*r/ 97.9%

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

      2. *-rgt-identity 97.9%

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

      3. associate-*l/ 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in t around inf 50.9%

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

      1. *-commutative 50.9%

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

      2. unpow2 50.9%

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

      3. unpow2 50.9%

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

      4. times-frac 61.7%

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

      5. unpow2 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in Om around 0 60.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.18 \cdot 10^{+73}:\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]

Alternative 12: 51.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \sin^{-1} \left(1 + -0.5 \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (asin (+ 1.0 (* -0.5 (* (/ Om Omc) (/ Om Omc))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	return asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
}

Fortran

NOTE: t should be positive before calling this function
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((1.0d0 + ((-0.5d0) * ((om / omc) * (om / omc)))))
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	return Math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	return math.asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))))

Julia

t = abs(t)
function code(t, l, Om, Omc)
	return asin(Float64(1.0 + Float64(-0.5 * Float64(Float64(Om / Omc) * Float64(Om / Omc)))))
end

MATLAB

t = abs(t)
function tmp = code(t, l, Om, Omc)
	tmp = asin((1.0 + (-0.5 * ((Om / Omc) * (Om / Omc)))));
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := N[ArcSin[N[(1.0 + N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.4%

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

2. Taylor expanded in t around 0 41.7%

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

   1. unpow2 41.7%

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

   2. unpow2 41.7%

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

4. Simplified 41.7%

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

5. Taylor expanded in Om around 0 41.3%

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

   1. unpow2 41.3%

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

   2. unpow2 41.3%

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

   3. times-frac 47.8%

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

   4. unpow2 47.8%

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

7. Simplified 47.8%

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

   1. unpow2 47.8%

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

9. Applied egg-rr 47.8%

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

10. Final simplification 47.8%

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

Alternative 13: 50.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \sin^{-1} 1 \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc) :precision binary64 (asin 1.0))

C

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

Fortran

NOTE: t should be positive before calling this function
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(1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 84.4%

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

2. Taylor expanded in t around 0 41.7%

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

   1. unpow2 41.7%

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

   2. unpow2 41.7%

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

4. Simplified 41.7%

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

5. Taylor expanded in Om around 0 47.6%

   \[\leadsto \sin^{-1} \color{blue}{1} \]

6. Final simplification 47.6%

   \[\leadsto \sin^{-1} 1 \]

Reproduce

herbie shell --seed 2023217 
(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)))))))