Toniolo and Linder, Equation (2)

Percentage Accurate: 83.9% → 98.0%

Time: 13.7s

Alternatives: 9

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.9% 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: 98.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t} \cdot \frac{\ell}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5e+101)
   (asin (/ (- l) (* t (sqrt 2.0))))
   (if (<= (/ t l) 10000000.0)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (/ t (* l (/ l t))))))))
     (asin (* (/ 1.0 t) (/ l (sqrt 2.0)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+101) {
		tmp = asin((-l / (t * sqrt(2.0))));
	} else if ((t / l) <= 10000000.0) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t / (l * (l / t))))))));
	} else {
		tmp = asin(((1.0 / t) * (l / 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 / l) <= (-5d+101)) then
        tmp = asin((-l / (t * sqrt(2.0d0))))
    else if ((t / l) <= 10000000.0d0) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * (t / (l * (l / t))))))))
    else
        tmp = asin(((1.0d0 / t) * (l / 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 / l) <= -5e+101) {
		tmp = Math.asin((-l / (t * Math.sqrt(2.0))));
	} else if ((t / l) <= 10000000.0) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t / (l * (l / t))))))));
	} else {
		tmp = Math.asin(((1.0 / t) * (l / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5e+101:
		tmp = math.asin((-l / (t * math.sqrt(2.0))))
	elif (t / l) <= 10000000.0:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t / (l * (l / t))))))))
	else:
		tmp = math.asin(((1.0 / t) * (l / math.sqrt(2.0))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5e+101)
		tmp = asin(Float64(Float64(-l) / Float64(t * sqrt(2.0))));
	elseif (Float64(t / l) <= 10000000.0)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(t / Float64(l * Float64(l / t))))))));
	else
		tmp = asin(Float64(Float64(1.0 / t) * Float64(l / sqrt(2.0))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5e+101)
		tmp = asin((-l / (t * sqrt(2.0))));
	elseif ((t / l) <= 10000000.0)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * (t / (l * (l / t))))))));
	else
		tmp = asin(((1.0 / t) * (l / 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[N[(t / l), $MachinePrecision], -5e+101], N[ArcSin[N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 10000000.0], 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[(t / N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 / t), $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.99999999999999989e101

   1. Initial program 53.8%

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

      1. sqrt-div 53.8%

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

      2. div-inv 53.8%

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

      3. add-sqr-sqrt 53.8%

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

      4. hypot-1-def 53.8%

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

      5. *-commutative 53.8%

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

      6. sqrt-prod 53.7%

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

      7. unpow2 53.7%

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

      8. sqrt-prod 0.0%

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

      9. add-sqr-sqrt 98.6%

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

   3. Applied egg-rr 98.6%

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

      1. associate-*r/ 98.6%

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

      2. *-rgt-identity 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in Om around 0 98.6%

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

   7. Taylor expanded in t around -inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. neg-mul-1 99.5%

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

   9. Simplified 99.5%

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

   if -4.99999999999999989e101 < (/.f64 t l) < 1e7

   1. Initial program 97.9%

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

      1. unpow2 97.9%

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

      2. clear-num 97.9%

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

      3. frac-times 97.9%

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

      4. *-un-lft-identity 97.9%

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

   3. Applied egg-rr 97.9%

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

      1. unpow2 97.9%

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

      2. clear-num 97.9%

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

      3. un-div-inv 97.9%

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

   5. Applied egg-rr 97.9%

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

   if 1e7 < (/.f64 t l)

   1. Initial program 70.5%

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

      1. sqrt-div 70.6%

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

      2. div-inv 70.6%

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

      3. add-sqr-sqrt 70.6%

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

      4. hypot-1-def 70.6%

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

      5. *-commutative 70.6%

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

      6. sqrt-prod 70.4%

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

      7. unpow2 70.4%

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

      8. sqrt-prod 98.8%

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

      9. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in Om around 0 97.7%

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

   7. Taylor expanded in t around inf 98.2%

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

      1. *-un-lft-identity 98.2%

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

      2. times-frac 98.2%

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

   9. Applied egg-rr 98.2%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+101}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t} \cdot \frac{\ell}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 2: 98.3% accurate, 0.8× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (asin
  (/ (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (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 - pow((Om / Omc), 2.0))) / 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 - Math.pow((Om / Omc), 2.0))) / 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 - math.pow((Om / Omc), 2.0))) / 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(Om / Omc) ^ 2.0))) / hypot(1.0, Float64(Float64(t / l) * sqrt(2.0)))))
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 / 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[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.9%

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

   1. sqrt-div 83.9%

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

   2. div-inv 83.9%

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

   3. add-sqr-sqrt 83.8%

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

   4. hypot-1-def 83.9%

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

   5. *-commutative 83.9%

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

   6. sqrt-prod 83.8%

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

   7. unpow2 83.8%

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

   8. sqrt-prod 54.1%

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

   9. add-sqr-sqrt 98.3%

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

3. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

5. Simplified 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 78.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = pow((t / l), 2.0);
	double tmp;
	if (t_1 <= 1.0) {
		tmp = asin((1.0 - t_1));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (t / l) ** 2.0d0
    if (t_1 <= 1.0d0) then
        tmp = asin((1.0d0 - t_1))
    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 t_1 = Math.pow((t / l), 2.0);
	double tmp;
	if (t_1 <= 1.0) {
		tmp = Math.asin((1.0 - t_1));
	} else {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

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

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(t / l) ^ 2.0
	tmp = 0.0
	if (t_1 <= 1.0)
		tmp = asin(Float64(1.0 - t_1));
	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)
	t_1 = (t / l) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= 1.0)
		tmp = asin((1.0 - t_1));
	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_] := Block[{t$95$1 = N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 1.0], N[ArcSin[N[(1.0 - t$95$1), $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 (pow.f64 (/.f64 t l) 2) < 1

   1. Initial program 97.7%

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

      1. sqrt-div 97.6%

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

      2. div-inv 97.6%

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

      3. add-sqr-sqrt 97.6%

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

      4. hypot-1-def 97.6%

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

      5. *-commutative 97.6%

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

      6. sqrt-prod 97.6%

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

      7. unpow2 97.6%

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

      8. sqrt-prod 57.9%

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

      9. add-sqr-sqrt 97.6%

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

   3. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.6%

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

      2. *-rgt-identity 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in Om around 0 96.7%

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

   7. Taylor expanded in t around 0 96.7%

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

      1. associate-/l* 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in t around 0 80.0%

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

       1. associate-*r/ 80.0%

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

       2. *-commutative 80.0%

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

       3. unpow2 80.0%

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

       4. rem-square-sqrt 80.0%

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

       5. associate-*r* 80.0%

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

       6. metadata-eval 80.0%

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

       7. associate-*r/ 80.0%

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

       8. mul-1-neg 80.0%

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

       9. unsub-neg 80.0%

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

       10. unpow2 80.0%

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

       11. unpow2 80.0%

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

       12. times-frac 95.5%

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

       13. unpow2 95.5%

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

   12. Simplified 95.5%

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

   if 1 < (pow.f64 (/.f64 t l) 2)

   1. Initial program 69.7%

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

      1. sqrt-div 69.6%

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

      2. div-inv 69.6%

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

      3. add-sqr-sqrt 69.6%

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

      4. hypot-1-def 69.6%

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

      5. *-commutative 69.6%

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

      6. sqrt-prod 69.5%

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

      7. unpow2 69.5%

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

      8. sqrt-prod 50.2%

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

      9. add-sqr-sqrt 98.9%

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

   3. Applied egg-rr 98.9%

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

      1. associate-*r/ 98.9%

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

      2. *-rgt-identity 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in Om around 0 98.2%

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

   7. Taylor expanded in t around inf 60.3%

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

4. Final simplification 78.2%

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

Alternative 4: 97.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 83.9%

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

   1. sqrt-div 83.9%

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

   2. div-inv 83.9%

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

   3. add-sqr-sqrt 83.8%

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

   4. hypot-1-def 83.9%

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

   5. *-commutative 83.9%

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

   6. sqrt-prod 83.8%

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

   7. unpow2 83.8%

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

   8. sqrt-prod 54.1%

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

   9. add-sqr-sqrt 98.3%

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

3. Applied egg-rr 98.3%

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

   1. associate-*r/ 98.3%

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

   2. *-rgt-identity 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in Om around 0 97.4%

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

7. Final simplification 97.4%

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

Alternative 5: 97.4% accurate, 1.9× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -0.5)
   (asin (/ (- l) (* t (sqrt 2.0))))
   (if (<= (/ t l) 0.01)
     (asin (sqrt (- 1.0 (/ (/ Om Omc) (/ Omc Om)))))
     (asin (* (/ 1.0 t) (/ l (sqrt 2.0)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -0.5) {
		tmp = asin((-l / (t * sqrt(2.0))));
	} else if ((t / l) <= 0.01) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = asin(((1.0 / t) * (l / 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 / l) <= (-0.5d0)) then
        tmp = asin((-l / (t * sqrt(2.0d0))))
    else if ((t / l) <= 0.01d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) / (omc / om)))))
    else
        tmp = asin(((1.0d0 / t) * (l / 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 / l) <= -0.5) {
		tmp = Math.asin((-l / (t * Math.sqrt(2.0))));
	} else if ((t / l) <= 0.01) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	} else {
		tmp = Math.asin(((1.0 / t) * (l / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -0.5)
		tmp = asin((-l / (t * sqrt(2.0))));
	elseif ((t / l) <= 0.01)
		tmp = asin(sqrt((1.0 - ((Om / Omc) / (Omc / Om)))));
	else
		tmp = asin(((1.0 / t) * (l / 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[N[(t / l), $MachinePrecision], -0.5], N[ArcSin[N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.01], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 / t), $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -0.5

   1. Initial program 68.3%

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

      1. sqrt-div 68.3%

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

      2. div-inv 68.3%

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

      3. add-sqr-sqrt 68.3%

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

      4. hypot-1-def 68.3%

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

      5. *-commutative 68.3%

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

      6. sqrt-prod 68.1%

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

      7. unpow2 68.1%

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

      8. sqrt-prod 0.0%

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

      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%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in Om around 0 98.7%

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

   7. Taylor expanded in t around -inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. neg-mul-1 95.9%

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

   9. Simplified 95.9%

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

   if -0.5 < (/.f64 t l) < 0.0100000000000000002

   1. Initial program 97.7%

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

   2. Taylor expanded in t around 0 83.9%

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

      1. unpow2 83.9%

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

      2. unpow2 83.9%

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

      3. times-frac 96.4%

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

      4. unpow2 96.4%

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

   4. Simplified 96.4%

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

      1. unpow2 97.7%

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

      2. clear-num 97.7%

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

      3. un-div-inv 97.7%

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

   6. Applied egg-rr 96.4%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 71.4%

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

      1. sqrt-div 71.5%

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

      2. div-inv 71.5%

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

      3. add-sqr-sqrt 71.4%

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

      4. hypot-1-def 71.4%

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

      5. *-commutative 71.4%

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

      6. sqrt-prod 71.3%

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

      7. unpow2 71.3%

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

      8. sqrt-prod 98.8%

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

      9. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in Om around 0 97.7%

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

   7. Taylor expanded in t around inf 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. times-frac 97.7%

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

   9. Applied egg-rr 97.7%

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

4. Final simplification 96.6%

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

Alternative 6: 97.0% accurate, 1.9× speedup

Math

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

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -1.0)
   (asin (/ (- l) (* t (sqrt 2.0))))
   (if (<= (/ t l) 0.01)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (* (/ 1.0 t) (/ l (sqrt 2.0)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -1.0) {
		tmp = asin((-l / (t * sqrt(2.0))));
	} else if ((t / l) <= 0.01) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin(((1.0 / t) * (l / 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 / l) <= (-1.0d0)) then
        tmp = asin((-l / (t * sqrt(2.0d0))))
    else if ((t / l) <= 0.01d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin(((1.0d0 / t) * (l / 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 / l) <= -1.0) {
		tmp = Math.asin((-l / (t * Math.sqrt(2.0))));
	} else if ((t / l) <= 0.01) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin(((1.0 / t) * (l / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -1.0)
		tmp = asin((-l / (t * sqrt(2.0))));
	elseif ((t / l) <= 0.01)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin(((1.0 / t) * (l / 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[N[(t / l), $MachinePrecision], -1.0], N[ArcSin[N[((-l) / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.01], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 / t), $MachinePrecision] * N[(l / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.9%

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

      1. sqrt-div 67.8%

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

      2. div-inv 67.8%

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

      3. add-sqr-sqrt 67.8%

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

      4. hypot-1-def 67.8%

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

      5. *-commutative 67.8%

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

      6. sqrt-prod 67.6%

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

      7. unpow2 67.6%

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

      8. sqrt-prod 0.0%

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

      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%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in Om around 0 98.7%

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

   7. Taylor expanded in t around -inf 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   9. Simplified 97.2%

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

   if -1 < (/.f64 t l) < 0.0100000000000000002

   1. Initial program 97.7%

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

      1. sqrt-div 97.6%

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

      2. div-inv 97.6%

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

      3. add-sqr-sqrt 97.6%

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

      4. hypot-1-def 97.6%

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

      5. *-commutative 97.6%

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

      6. sqrt-prod 97.6%

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

      7. unpow2 97.6%

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

      8. sqrt-prod 57.9%

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

      9. add-sqr-sqrt 97.6%

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

   3. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.6%

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

      2. *-rgt-identity 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in Om around 0 96.7%

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

   7. Taylor expanded in t around 0 96.7%

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

      1. associate-/l* 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in t around 0 80.0%

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

       1. associate-*r/ 80.0%

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

       2. *-commutative 80.0%

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

       3. unpow2 80.0%

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

       4. rem-square-sqrt 80.0%

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

       5. associate-*r* 80.0%

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

       6. metadata-eval 80.0%

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

       7. associate-*r/ 80.0%

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

       8. mul-1-neg 80.0%

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

       9. unsub-neg 80.0%

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

       10. unpow2 80.0%

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

       11. unpow2 80.0%

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

       12. times-frac 95.5%

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

       13. unpow2 95.5%

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

   12. Simplified 95.5%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 71.4%

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

      1. sqrt-div 71.5%

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

      2. div-inv 71.5%

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

      3. add-sqr-sqrt 71.4%

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

      4. hypot-1-def 71.4%

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

      5. *-commutative 71.4%

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

      6. sqrt-prod 71.3%

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

      7. unpow2 71.3%

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

      8. sqrt-prod 98.8%

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

      9. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in Om around 0 97.7%

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

   7. Taylor expanded in t around inf 97.7%

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

      1. *-un-lft-identity 97.7%

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

      2. times-frac 97.7%

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

   9. Applied egg-rr 97.7%

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

4. Final simplification 96.5%

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

Alternative 7: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (* t (sqrt 2.0))))
   (if (<= (/ t l) -1.0)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 0.01)
       (asin (- 1.0 (pow (/ t l) 2.0)))
       (asin (/ l t_1))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = t * sqrt(2.0);
	double tmp;
	if ((t / l) <= -1.0) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 0.01) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((l / t_1));
	}
	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 = t * sqrt(2.0d0)
    if ((t / l) <= (-1.0d0)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 0.01d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((l / t_1))
    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 = t * Math.sqrt(2.0);
	double tmp;
	if ((t / l) <= -1.0) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 0.01) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = t * math.sqrt(2.0)
	tmp = 0
	if (t / l) <= -1.0:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 0.01:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(t * sqrt(2.0))
	tmp = 0.0
	if (Float64(t / l) <= -1.0)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 0.01)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = t * sqrt(2.0);
	tmp = 0.0;
	if ((t / l) <= -1.0)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 0.01)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l / t_1));
	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[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1.0], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.01], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 67.9%

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

      1. sqrt-div 67.8%

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

      2. div-inv 67.8%

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

      3. add-sqr-sqrt 67.8%

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

      4. hypot-1-def 67.8%

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

      5. *-commutative 67.8%

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

      6. sqrt-prod 67.6%

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

      7. unpow2 67.6%

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

      8. sqrt-prod 0.0%

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

      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%

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

   5. Simplified 98.7%

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

   6. Taylor expanded in Om around 0 98.7%

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

   7. Taylor expanded in t around -inf 97.2%

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

      1. associate-*r/ 97.2%

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

      2. neg-mul-1 97.2%

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

   9. Simplified 97.2%

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

   if -1 < (/.f64 t l) < 0.0100000000000000002

   1. Initial program 97.7%

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

      1. sqrt-div 97.6%

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

      2. div-inv 97.6%

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

      3. add-sqr-sqrt 97.6%

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

      4. hypot-1-def 97.6%

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

      5. *-commutative 97.6%

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

      6. sqrt-prod 97.6%

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

      7. unpow2 97.6%

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

      8. sqrt-prod 57.9%

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

      9. add-sqr-sqrt 97.6%

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

   3. Applied egg-rr 97.6%

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

      1. associate-*r/ 97.6%

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

      2. *-rgt-identity 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in Om around 0 96.7%

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

   7. Taylor expanded in t around 0 96.7%

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

      1. associate-/l* 96.7%

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

   9. Simplified 96.7%

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

   10. Taylor expanded in t around 0 80.0%

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

       1. associate-*r/ 80.0%

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

       2. *-commutative 80.0%

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

       3. unpow2 80.0%

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

       4. rem-square-sqrt 80.0%

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

       5. associate-*r* 80.0%

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

       6. metadata-eval 80.0%

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

       7. associate-*r/ 80.0%

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

       8. mul-1-neg 80.0%

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

       9. unsub-neg 80.0%

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

       10. unpow2 80.0%

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

       11. unpow2 80.0%

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

       12. times-frac 95.5%

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

       13. unpow2 95.5%

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

   12. Simplified 95.5%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 71.4%

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

      1. sqrt-div 71.5%

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

      2. div-inv 71.5%

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

      3. add-sqr-sqrt 71.4%

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

      4. hypot-1-def 71.4%

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

      5. *-commutative 71.4%

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

      6. sqrt-prod 71.3%

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

      7. unpow2 71.3%

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

      8. sqrt-prod 98.8%

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

      9. add-sqr-sqrt 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-*r/ 99.1%

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

      2. *-rgt-identity 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in Om around 0 97.7%

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

   7. Taylor expanded in t around inf 97.7%

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

4. Final simplification 96.5%

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

Alternative 8: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-109}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 27:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -1.1e-109)
   (asin 1.0)
   (if (<= l 27.0) (asin (/ l (* t (sqrt 2.0)))) (asin 1.0))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -1.1e-109) {
		tmp = asin(1.0);
	} else if (l <= 27.0) {
		tmp = asin((l / (t * sqrt(2.0))));
	} else {
		tmp = asin(1.0);
	}
	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 (l <= (-1.1d-109)) then
        tmp = asin(1.0d0)
    else if (l <= 27.0d0) then
        tmp = asin((l / (t * sqrt(2.0d0))))
    else
        tmp = asin(1.0d0)
    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 (l <= -1.1e-109) {
		tmp = Math.asin(1.0);
	} else if (l <= 27.0) {
		tmp = Math.asin((l / (t * Math.sqrt(2.0))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if l <= -1.1e-109:
		tmp = math.asin(1.0)
	elif l <= 27.0:
		tmp = math.asin((l / (t * math.sqrt(2.0))))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -1.1e-109)
		tmp = asin(1.0);
	elseif (l <= 27.0)
		tmp = asin(Float64(l / Float64(t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -1.1e-109)
		tmp = asin(1.0);
	elseif (l <= 27.0)
		tmp = asin((l / (t * sqrt(2.0))));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[l, -1.1e-109], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 27.0], N[ArcSin[N[(l / N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.1e-109 or 27 < l

   1. Initial program 88.3%

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

   2. Taylor expanded in t around 0 61.3%

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

      1. unpow2 61.3%

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

      2. unpow2 61.3%

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

      3. times-frac 70.6%

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

      4. unpow2 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in Om around 0 70.1%

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

   if -1.1e-109 < l < 27

   1. Initial program 76.4%

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

      1. sqrt-div 76.3%

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

      2. div-inv 76.3%

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

      3. add-sqr-sqrt 76.3%

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

      4. hypot-1-def 76.3%

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

      5. *-commutative 76.3%

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

      6. sqrt-prod 76.3%

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

      7. unpow2 76.3%

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

      8. sqrt-prod 46.7%

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

      9. add-sqr-sqrt 97.8%

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

   3. Applied egg-rr 97.8%

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

      1. associate-*r/ 97.8%

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

      2. *-rgt-identity 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in Om around 0 96.5%

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

   7. Taylor expanded in t around inf 54.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{-109}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 27:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 9: 50.2% 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 83.9%

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

2. Taylor expanded in t around 0 44.6%

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

   1. unpow2 44.6%

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

   2. unpow2 44.6%

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

   3. times-frac 51.2%

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

   4. unpow2 51.2%

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

4. Simplified 51.2%

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

5. Taylor expanded in Om around 0 50.7%

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

6. Final simplification 50.7%

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

Reproduce

herbie shell --seed 2023320 
(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)))))))