Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 98.5%

Time: 18.5s

Alternatives: 6

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.8% 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: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+89}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
   (if (<= (/ t l) -2e+158)
     (asin (* (/ (- l) t) (sqrt 0.5)))
     (if (<= (/ t l) 1e+89)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (* (sqrt t_1) (* l (/ (sqrt 0.5) t))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -2e+158) {
		tmp = asin(((-l / t) * sqrt(0.5)));
	} else if ((t / l) <= 1e+89) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt(t_1) * (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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) / (omc / om))
    if ((t / l) <= (-2d+158)) then
        tmp = asin(((-l / t) * sqrt(0.5d0)))
    else if ((t / l) <= 1d+89) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt(t_1) * (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 t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	double tmp;
	if ((t / l) <= -2e+158) {
		tmp = Math.asin(((-l / t) * Math.sqrt(0.5)));
	} else if ((t / l) <= 1e+89) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt(t_1) * (l * (Math.sqrt(0.5) / t))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om))
	tmp = 0
	if (t / l) <= -2e+158:
		tmp = math.asin(((-l / t) * math.sqrt(0.5)))
	elif (t / l) <= 1e+89:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt(t_1) * (l * (math.sqrt(0.5) / t))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om)))
	tmp = 0.0
	if (Float64(t / l) <= -2e+158)
		tmp = asin(Float64(Float64(Float64(-l) / t) * sqrt(0.5)));
	elseif (Float64(t / l) <= 1e+89)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(sqrt(t_1) * Float64(l * Float64(sqrt(0.5) / t))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) / (Omc / Om));
	tmp = 0.0;
	if ((t / l) <= -2e+158)
		tmp = asin(((-l / t) * sqrt(0.5)));
	elseif ((t / l) <= 1e+89)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt(t_1) * (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_] := Block[{t$95$1 = N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+158], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+89], N[ArcSin[N[Sqrt[N[(t$95$1 / 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[t$95$1], $MachinePrecision] * N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.99999999999999991e158

   1. Initial program 52.7%

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

      1. unpow2 52.7%

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

   3. Applied egg-rr 52.7%

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

   4. Taylor expanded in t around -inf 87.5%

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

      1. associate-*r* 87.5%

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

      2. associate-*r/ 87.5%

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

      3. neg-mul-1 87.5%

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

      4. distribute-rgt-neg-in 87.5%

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

      5. associate-*l/ 87.7%

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

      6. unpow2 87.7%

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

      7. associate-/l/ 91.4%

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

      8. unpow2 91.4%

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

      9. associate-*r/ 99.7%

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

      10. associate-*l/ 99.7%

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

      11. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in Om around 0 97.8%

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

       1. associate-*l/ 98.0%

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

       2. associate-*r* 98.0%

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

       3. associate-*r/ 98.0%

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

       4. neg-mul-1 98.0%

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

   11. Simplified 98.0%

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

   if -1.99999999999999991e158 < (/.f64 t l) < 9.99999999999999995e88

   1. Initial program 97.8%

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

      1. unpow2 97.8%

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

   3. Applied egg-rr 97.8%

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

      1. unpow2 14.2%

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

      2. clear-num 14.2%

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

      3. un-div-inv 14.2%

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

   5. Applied egg-rr 97.8%

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

      1. associate-*r/ 95.1%

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

      2. associate-/l* 97.8%

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

   7. Applied egg-rr 97.8%

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

   if 9.99999999999999995e88 < (/.f64 t l)

   1. Initial program 59.8%

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

      1. unpow2 59.8%

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

   3. Applied egg-rr 59.8%

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

   4. Taylor expanded in t around inf 87.0%

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

      1. *-commutative 87.0%

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

      2. unpow2 87.0%

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

      3. associate-/l/ 92.0%

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

      4. unpow2 92.0%

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

      5. associate-*r/ 99.5%

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

      6. associate-*l/ 99.5%

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

      7. unpow2 99.5%

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

      8. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 44.6%

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

      2. clear-num 44.6%

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

      3. un-div-inv 44.6%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+89}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \sin^{-1} \left(\frac{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\mathsf{hypot}\left(1, \frac{t}{\frac{\ell}{\sqrt{2}}}\right)}}{\sqrt{1 + {\left(\frac{Om}{Omc}\right)}^{2}}}\right) \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/
   (/ (sqrt (- 1.0 (pow (/ Om Omc) 4.0))) (hypot 1.0 (/ t (/ l (sqrt 2.0)))))
   (sqrt (+ 1.0 (pow (/ Om Omc) 2.0))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	return asin(((sqrt((1.0 - pow((Om / Omc), 4.0))) / hypot(1.0, (t / (l / sqrt(2.0))))) / sqrt((1.0 + pow((Om / Omc), 2.0)))));
}

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), 4.0))) / Math.hypot(1.0, (t / (l / Math.sqrt(2.0))))) / Math.sqrt((1.0 + Math.pow((Om / Omc), 2.0)))));
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	return math.asin(((math.sqrt((1.0 - math.pow((Om / Omc), 4.0))) / math.hypot(1.0, (t / (l / math.sqrt(2.0))))) / math.sqrt((1.0 + math.pow((Om / Omc), 2.0)))))

Julia

t = abs(t)
function code(t, l, Om, Omc)
	return asin(Float64(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 4.0))) / hypot(1.0, Float64(t / Float64(l / sqrt(2.0))))) / sqrt(Float64(1.0 + (Float64(Om / Omc) ^ 2.0)))))
end

MATLAB

t = abs(t)
function tmp = code(t, l, Om, Omc)
	tmp = asin(((sqrt((1.0 - ((Om / Omc) ^ 4.0))) / hypot(1.0, (t / (l / sqrt(2.0))))) / sqrt((1.0 + ((Om / Omc) ^ 2.0)))));
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(t / N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.6%

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

   1. sqrt-div 84.6%

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

   2. flip-- 84.6%

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

   3. sqrt-div 84.6%

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

   4. associate-/l/ 84.6%

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

   5. metadata-eval 84.6%

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

   6. pow-sqr 84.6%

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

   7. metadata-eval 84.6%

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

3. Applied egg-rr 98.1%

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

   1. associate-/r* 98.1%

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

   2. associate-*l/ 98.1%

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

   3. associate-/l* 98.2%

      \[\leadsto \sin^{-1} \left(\frac{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\mathsf{hypot}\left(1, \color{blue}{\frac{t}{\frac{\ell}{\sqrt{2}}}}\right)}}{\sqrt{1 + {\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

5. Simplified 98.2%

   \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\mathsf{hypot}\left(1, \frac{t}{\frac{\ell}{\sqrt{2}}}\right)}}{\sqrt{1 + {\left(\frac{Om}{Omc}\right)}^{2}}}\right)} \]

6. Final simplification 98.2%

   \[\leadsto \sin^{-1} \left(\frac{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{4}}}{\mathsf{hypot}\left(1, \frac{t}{\frac{\ell}{\sqrt{2}}}\right)}}{\sqrt{1 + {\left(\frac{Om}{Omc}\right)}^{2}}}\right) \]

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	return Math.asin((Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))) / Math.hypot(1.0, (t / (l / Math.sqrt(2.0))))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 84.6%

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

   1. sqrt-div 84.6%

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

   2. add-sqr-sqrt 84.6%

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

   3. hypot-1-def 84.6%

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

   4. *-commutative 84.6%

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

   5. sqrt-prod 84.5%

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

   6. unpow2 84.5%

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

   7. sqrt-prod 50.9%

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

   8. add-sqr-sqrt 98.1%

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

3. Applied egg-rr 98.1%

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

   1. associate-*l/ 98.1%

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

   2. associate-/l* 98.1%

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

5. Simplified 98.1%

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

   1. unpow2 32.6%

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

   2. clear-num 32.6%

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

   3. un-div-inv 32.6%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 4: 90.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+158)
   (asin (* (/ (- l) t) (sqrt 0.5)))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+158) {
		tmp = asin(((-l / t) * sqrt(0.5)));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (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) <= (-2d+158)) then
        tmp = asin(((-l / t) * sqrt(0.5d0)))
    else
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) * (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) <= -2e+158) {
		tmp = Math.asin(((-l / t) * Math.sqrt(0.5)));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+158:
		tmp = math.asin(((-l / t) * math.sqrt(0.5)))
	else:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+158)
		tmp = asin(Float64(Float64(Float64(-l) / t) * sqrt(0.5)));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * 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) <= -2e+158)
		tmp = asin(((-l / t) * sqrt(0.5)));
	else
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) * (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], -2e+158], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < -1.99999999999999991e158

   1. Initial program 52.7%

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

      1. unpow2 52.7%

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

   3. Applied egg-rr 52.7%

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

   4. Taylor expanded in t around -inf 87.5%

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

      1. associate-*r* 87.5%

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

      2. associate-*r/ 87.5%

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

      3. neg-mul-1 87.5%

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

      4. distribute-rgt-neg-in 87.5%

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

      5. associate-*l/ 87.7%

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

      6. unpow2 87.7%

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

      7. associate-/l/ 91.4%

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

      8. unpow2 91.4%

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

      9. associate-*r/ 99.7%

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

      10. associate-*l/ 99.7%

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

      11. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in Om around 0 97.8%

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

       1. associate-*l/ 98.0%

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

       2. associate-*r* 98.0%

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

       3. associate-*r/ 98.0%

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

       4. neg-mul-1 98.0%

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

   11. Simplified 98.0%

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

   if -1.99999999999999991e158 < (/.f64 t l)

   1. Initial program 90.7%

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

      1. unpow2 90.7%

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

   3. Applied egg-rr 90.7%

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

      1. unpow2 19.8%

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

      2. clear-num 19.8%

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

      3. un-div-inv 19.8%

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

   5. Applied egg-rr 90.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \end{array} \]

Alternative 5: 90.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+158)
   (asin (* (/ (- l) t) (sqrt 0.5)))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+158) {
		tmp = asin(((-l / t) * sqrt(0.5)));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / 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 / l) <= (-2d+158)) then
        tmp = asin(((-l / t) * sqrt(0.5d0)))
    else
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) / (l / 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 / l) <= -2e+158) {
		tmp = Math.asin(((-l / t) * Math.sqrt(0.5)));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+158:
		tmp = math.asin(((-l / t) * math.sqrt(0.5)))
	else:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+158)
		tmp = asin(Float64(Float64(Float64(-l) / t) * sqrt(0.5)));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2e+158)
		tmp = asin(((-l / t) * sqrt(0.5)));
	else
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	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], -2e+158], N[ArcSin[N[(N[((-l) / t), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < -1.99999999999999991e158

   1. Initial program 52.7%

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

      1. unpow2 52.7%

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

   3. Applied egg-rr 52.7%

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

   4. Taylor expanded in t around -inf 87.5%

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

      1. associate-*r* 87.5%

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

      2. associate-*r/ 87.5%

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

      3. neg-mul-1 87.5%

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

      4. distribute-rgt-neg-in 87.5%

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

      5. associate-*l/ 87.7%

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

      6. unpow2 87.7%

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

      7. associate-/l/ 91.4%

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

      8. unpow2 91.4%

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

      9. associate-*r/ 99.7%

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

      10. associate-*l/ 99.7%

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

      11. unpow2 99.7%

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

   6. Simplified 99.7%

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

      1. unpow2 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in Om around 0 97.8%

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

       1. associate-*l/ 98.0%

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

       2. associate-*r* 98.0%

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

       3. associate-*r/ 98.0%

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

       4. neg-mul-1 98.0%

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

   11. Simplified 98.0%

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

   if -1.99999999999999991e158 < (/.f64 t l)

   1. Initial program 90.7%

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

      1. unpow2 90.7%

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

   3. Applied egg-rr 90.7%

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

      1. unpow2 19.8%

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

      2. clear-num 19.8%

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

      3. un-div-inv 19.8%

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

   5. Applied egg-rr 90.7%

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

      1. associate-*r/ 88.1%

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

      2. associate-/l* 90.7%

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

   7. Applied egg-rr 90.7%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t} \cdot \sqrt{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \end{array} \]

Alternative 6: 30.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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(((-l / t) * sqrt(0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

   1. unpow2 84.6%

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

3. Applied egg-rr 84.6%

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

4. Taylor expanded in t around -inf 27.5%

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

   1. associate-*r* 27.5%

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

   2. associate-*r/ 27.5%

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

   3. neg-mul-1 27.5%

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

   4. distribute-rgt-neg-in 27.5%

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

   5. associate-*l/ 27.5%

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

   6. unpow2 27.5%

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

   7. associate-/l/ 30.1%

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

   8. unpow2 30.1%

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

   9. associate-*r/ 32.6%

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

   10. associate-*l/ 32.6%

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

   11. unpow2 32.6%

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

6. Simplified 32.6%

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

   1. unpow2 32.6%

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

   2. clear-num 32.6%

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

   3. un-div-inv 32.6%

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

8. Applied egg-rr 32.6%

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

9. Taylor expanded in Om around 0 32.3%

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

    1. associate-*l/ 32.3%

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

    2. associate-*r* 32.3%

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

    3. associate-*r/ 32.3%

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

    4. neg-mul-1 32.3%

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

11. Simplified 32.3%

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

12. Final simplification 32.3%

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

Reproduce

herbie shell --seed 2023301 
(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)))))))