Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 98.1%

Time: 12.0s

Alternatives: 11

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: 83.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+63}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -1e+57)
   (asin (/ (* (sqrt 0.5) (- l)) t))
   (if (<= (/ t l) 4e+63)
     (asin
      (sqrt
       (/
        (- 1.0 (* (/ Om Omc) (/ Om Omc)))
        (+ 1.0 (* 2.0 (/ (* t (/ t l)) l))))))
     (asin (/ (* l (sqrt 0.5)) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -1e+57) {
		tmp = asin(((sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 4e+63) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-1d+57)) then
        tmp = asin(((sqrt(0.5d0) * -l) / t))
    else if ((t / l) <= 4d+63) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) * (om / omc))) / (1.0d0 + (2.0d0 * ((t * (t / l)) / l))))))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -1e+57) {
		tmp = Math.asin(((Math.sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 4e+63) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -1e+57:
		tmp = math.asin(((math.sqrt(0.5) * -l) / t))
	elif (t / l) <= 4e+63:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t * (t / l)) / l))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -1e+57)
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(-l)) / t));
	elseif (Float64(t / l) <= 4e+63)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t * Float64(t / l)) / l))))));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -1e+57)
		tmp = asin(((sqrt(0.5) * -l) / t));
	elseif ((t / l) <= 4e+63)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t * (t / l)) / l))))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -1e+57], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * (-l)), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 4e+63], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.00000000000000005e57

   1. Initial program 68.1%

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

      1. unpow2 68.1%

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

      2. associate-*r/ 58.0%

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

   3. Applied egg-rr 58.0%

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

   4. Taylor expanded in Om around 0 45.7%

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

      1. *-commutative 45.7%

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

      2. associate-*l/ 45.7%

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

      3. rem-square-sqrt 45.7%

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

      4. unpow2 45.7%

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

      5. *-commutative 45.7%

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

      6. *-commutative 45.7%

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

      7. unpow2 45.7%

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

      8. rem-square-sqrt 45.7%

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

      9. associate-*l/ 45.7%

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

      10. *-commutative 45.7%

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

      11. unpow2 45.7%

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

      12. unpow2 45.7%

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

   6. Simplified 45.7%

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

   7. Taylor expanded in t around -inf 99.5%

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

   if -1.00000000000000005e57 < (/.f64 t l) < 4.00000000000000023e63

   1. Initial program 98.8%

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

      1. unpow2 98.8%

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

      2. associate-*r/ 98.2%

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

   3. Applied egg-rr 98.2%

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

      1. unpow2 98.2%

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

   5. Applied egg-rr 98.2%

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

   if 4.00000000000000023e63 < (/.f64 t l)

   1. Initial program 65.0%

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

      1. unpow2 65.0%

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

      2. associate-*r/ 60.2%

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

   3. Applied egg-rr 60.2%

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

   4. Taylor expanded in Om around 0 48.8%

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

      1. *-commutative 48.8%

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

      2. associate-*l/ 48.8%

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

      3. rem-square-sqrt 48.6%

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

      4. unpow2 48.6%

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

      5. *-commutative 48.6%

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

      6. *-commutative 48.6%

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

      7. unpow2 48.6%

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

      8. rem-square-sqrt 48.8%

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

      9. associate-*l/ 48.8%

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

      10. *-commutative 48.8%

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

      11. unpow2 48.8%

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

      12. unpow2 48.8%

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

   6. Simplified 48.8%

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

   7. Taylor expanded in t around inf 99.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+63}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 2: 98.4% 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 84.3%

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

   1. sqrt-div 84.3%

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

   2. add-sqr-sqrt 84.3%

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

   3. hypot-1-def 84.3%

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

   4. *-commutative 84.3%

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

   5. sqrt-prod 84.2%

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

   6. unpow2 84.2%

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

   7. sqrt-prod 53.9%

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

   8. add-sqr-sqrt 98.4%

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

3. Applied egg-rr 98.4%

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

4. Final simplification 98.4%

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

Alternative 3: 97.5% 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 84.3%

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

   1. unpow2 84.3%

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

   2. associate-*r/ 80.7%

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

3. Applied egg-rr 80.7%

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

4. Taylor expanded in Om around 0 64.9%

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

   1. *-commutative 64.9%

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

   2. associate-*l/ 64.9%

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

   3. rem-square-sqrt 64.8%

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

   4. unpow2 64.8%

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

   5. *-commutative 64.8%

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

   6. *-commutative 64.8%

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

   7. unpow2 64.8%

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

   8. rem-square-sqrt 64.9%

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

   9. associate-*l/ 64.9%

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

   10. *-commutative 64.9%

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

   11. unpow2 64.9%

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

   12. unpow2 64.9%

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

6. Simplified 64.9%

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

   1. sqrt-div 64.9%

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

   2. metadata-eval 64.9%

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

   3. add-sqr-sqrt 64.9%

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

   4. hypot-1-def 64.9%

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

   5. times-frac 83.8%

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

   6. unpow2 83.8%

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

   7. *-commutative 83.8%

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

   8. sqrt-prod 83.8%

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

   9. unpow2 83.8%

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

   10. sqrt-prod 53.8%

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

   11. add-sqr-sqrt 97.9%

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

8. Applied egg-rr 97.9%

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

9. Final simplification 97.9%

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

Alternative 4: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+76}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5e+143)
   (asin (/ (* (sqrt 0.5) (- l)) t))
   (if (<= (/ t l) 1e+76)
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
     (asin (/ (* l (sqrt 0.5)) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+143) {
		tmp = asin(((sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 1e+76) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-5d+143)) then
        tmp = asin(((sqrt(0.5d0) * -l) / t))
    else if ((t / l) <= 1d+76) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+143) {
		tmp = Math.asin(((Math.sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 1e+76) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5e+143:
		tmp = math.asin(((math.sqrt(0.5) * -l) / t))
	elif (t / l) <= 1e+76:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5e+143)
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(-l)) / t));
	elseif (Float64(t / l) <= 1e+76)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5e+143)
		tmp = asin(((sqrt(0.5) * -l) / t));
	elseif ((t / l) <= 1e+76)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+143], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * (-l)), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+76], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.00000000000000012e143

   1. Initial program 55.6%

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

      1. unpow2 55.6%

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

      2. associate-*r/ 50.9%

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

   3. Applied egg-rr 50.9%

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

   4. Taylor expanded in Om around 0 50.9%

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

      1. *-commutative 50.9%

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

      2. associate-*l/ 50.9%

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

      3. rem-square-sqrt 50.9%

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

      4. unpow2 50.9%

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

      5. *-commutative 50.9%

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

      6. *-commutative 50.9%

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

      7. unpow2 50.9%

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

      8. rem-square-sqrt 50.9%

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

      9. associate-*l/ 50.9%

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

      10. *-commutative 50.9%

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

      11. unpow2 50.9%

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

      12. unpow2 50.9%

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

   6. Simplified 50.9%

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

   7. Taylor expanded in t around -inf 99.7%

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

   if -5.00000000000000012e143 < (/.f64 t l) < 1e76

   1. Initial program 98.9%

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

      1. unpow2 98.9%

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

      2. associate-*r/ 96.1%

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

   3. Applied egg-rr 96.1%

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

   4. Taylor expanded in Om around 0 73.7%

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

      1. *-commutative 73.7%

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

      2. associate-*l/ 73.7%

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

      3. rem-square-sqrt 73.6%

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

      4. unpow2 73.6%

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

      5. *-commutative 73.6%

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

      6. *-commutative 73.6%

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

      7. unpow2 73.6%

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

      8. rem-square-sqrt 73.7%

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

      9. associate-*l/ 73.7%

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

      10. *-commutative 73.7%

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

      11. unpow2 73.7%

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

      12. unpow2 73.7%

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

   6. Simplified 73.7%

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

      1. times-frac 98.2%

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

   8. Applied egg-rr 98.2%

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

   if 1e76 < (/.f64 t l)

   1. Initial program 61.1%

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

      1. unpow2 61.1%

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

      2. associate-*r/ 55.7%

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

   3. Applied egg-rr 55.7%

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

   4. Taylor expanded in Om around 0 48.4%

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

      1. *-commutative 48.4%

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

      2. associate-*l/ 48.4%

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

      3. rem-square-sqrt 48.3%

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

      4. unpow2 48.3%

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

      5. *-commutative 48.3%

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

      6. *-commutative 48.3%

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

      7. unpow2 48.3%

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

      8. rem-square-sqrt 48.4%

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

      9. associate-*l/ 48.4%

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

      10. *-commutative 48.4%

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

      11. unpow2 48.4%

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

      12. unpow2 48.4%

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

   6. Simplified 48.4%

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

   7. Taylor expanded in t around inf 99.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+76}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -40.0)
   (asin (* (/ (sqrt 0.5) t) (- l)))
   (if (<= (/ t l) 0.04)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ (* l (sqrt 0.5)) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40.0) {
		tmp = asin(((sqrt(0.5) / t) * -l));
	} else if ((t / l) <= 0.04) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-40.0d0)) then
        tmp = asin(((sqrt(0.5d0) / t) * -l))
    else if ((t / l) <= 0.04d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40.0) {
		tmp = Math.asin(((Math.sqrt(0.5) / t) * -l));
	} else if ((t / l) <= 0.04) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -40.0:
		tmp = math.asin(((math.sqrt(0.5) / t) * -l))
	elif (t / l) <= 0.04:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -40.0)
		tmp = asin(Float64(Float64(sqrt(0.5) / t) * Float64(-l)));
	elseif (Float64(t / l) <= 0.04)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -40.0)
		tmp = asin(((sqrt(0.5) / t) * -l));
	elseif ((t / l) <= 0.04)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -40.0], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] * (-l)), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.04], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -40

   1. Initial program 74.1%

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

      1. unpow2 74.1%

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

      2. associate-*r/ 65.9%

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

   3. Applied egg-rr 65.9%

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

   4. Taylor expanded in Om around 0 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-*l/ 47.2%

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

      3. rem-square-sqrt 47.1%

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

      4. unpow2 47.1%

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

      5. *-commutative 47.1%

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

      6. *-commutative 47.1%

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

      7. unpow2 47.1%

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

      8. rem-square-sqrt 47.2%

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

      9. associate-*l/ 47.2%

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

      10. *-commutative 47.2%

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

      11. unpow2 47.2%

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

      12. unpow2 47.2%

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

   6. Simplified 47.2%

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

   7. Taylor expanded in t around -inf 98.2%

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

      1. mul-1-neg 98.2%

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

      2. associate-*l/ 98.0%

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

      3. distribute-rgt-neg-in 98.0%

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

   9. Simplified 98.0%

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

   if -40 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 98.7%

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

      1. unpow2 98.7%

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

      2. associate-*r/ 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in Om around 0 83.8%

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

      1. *-commutative 83.8%

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

      2. associate-*l/ 83.8%

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

      3. rem-square-sqrt 83.8%

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

      4. unpow2 83.8%

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

      5. *-commutative 83.8%

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

      6. *-commutative 83.8%

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

      7. unpow2 83.8%

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

      8. rem-square-sqrt 83.8%

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

      9. associate-*l/ 83.8%

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

      10. *-commutative 83.8%

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

      11. unpow2 83.8%

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

      12. unpow2 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in t around 0 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unpow2 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 97.3%

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

      5. unpow2 97.3%

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

      6. unsub-neg 97.3%

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

   9. Simplified 97.3%

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

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 69.6%

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

      1. unpow2 69.6%

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

      2. associate-*r/ 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in Om around 0 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l/ 49.7%

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

      3. rem-square-sqrt 49.4%

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

      4. unpow2 49.4%

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

      5. *-commutative 49.4%

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

      6. *-commutative 49.4%

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

      7. unpow2 49.4%

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

      8. rem-square-sqrt 49.7%

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

      9. associate-*l/ 49.7%

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

      10. *-commutative 49.7%

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

      11. unpow2 49.7%

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

      12. unpow2 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in t around inf 98.6%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -40.0)
   (asin (/ (* (sqrt 0.5) (- l)) t))
   (if (<= (/ t l) 0.04)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (/ (* l (sqrt 0.5)) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40.0) {
		tmp = asin(((sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 0.04) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-40.0d0)) then
        tmp = asin(((sqrt(0.5d0) * -l) / t))
    else if ((t / l) <= 0.04d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -40.0) {
		tmp = Math.asin(((Math.sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 0.04) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -40.0:
		tmp = math.asin(((math.sqrt(0.5) * -l) / t))
	elif (t / l) <= 0.04:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -40.0)
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(-l)) / t));
	elseif (Float64(t / l) <= 0.04)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -40.0)
		tmp = asin(((sqrt(0.5) * -l) / t));
	elseif ((t / l) <= 0.04)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -40.0], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * (-l)), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.04], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -40

   1. Initial program 74.1%

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

      1. unpow2 74.1%

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

      2. associate-*r/ 65.9%

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

   3. Applied egg-rr 65.9%

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

   4. Taylor expanded in Om around 0 47.2%

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

      1. *-commutative 47.2%

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

      2. associate-*l/ 47.2%

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

      3. rem-square-sqrt 47.1%

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

      4. unpow2 47.1%

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

      5. *-commutative 47.1%

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

      6. *-commutative 47.1%

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

      7. unpow2 47.1%

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

      8. rem-square-sqrt 47.2%

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

      9. associate-*l/ 47.2%

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

      10. *-commutative 47.2%

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

      11. unpow2 47.2%

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

      12. unpow2 47.2%

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

   6. Simplified 47.2%

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

   7. Taylor expanded in t around -inf 98.2%

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

   if -40 < (/.f64 t l) < 0.0400000000000000008

   1. Initial program 98.7%

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

      1. unpow2 98.7%

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

      2. associate-*r/ 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in Om around 0 83.8%

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

      1. *-commutative 83.8%

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

      2. associate-*l/ 83.8%

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

      3. rem-square-sqrt 83.8%

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

      4. unpow2 83.8%

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

      5. *-commutative 83.8%

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

      6. *-commutative 83.8%

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

      7. unpow2 83.8%

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

      8. rem-square-sqrt 83.8%

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

      9. associate-*l/ 83.8%

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

      10. *-commutative 83.8%

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

      11. unpow2 83.8%

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

      12. unpow2 83.8%

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

   6. Simplified 83.8%

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

   7. Taylor expanded in t around 0 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unpow2 83.4%

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

      3. unpow2 83.4%

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

      4. times-frac 97.3%

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

      5. unpow2 97.3%

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

      6. unsub-neg 97.3%

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

   9. Simplified 97.3%

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

   if 0.0400000000000000008 < (/.f64 t l)

   1. Initial program 69.6%

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

      1. unpow2 69.6%

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

      2. associate-*r/ 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in Om around 0 49.7%

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

      1. *-commutative 49.7%

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

      2. associate-*l/ 49.7%

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

      3. rem-square-sqrt 49.4%

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

      4. unpow2 49.4%

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

      5. *-commutative 49.4%

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

      6. *-commutative 49.4%

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

      7. unpow2 49.4%

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

      8. rem-square-sqrt 49.7%

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

      9. associate-*l/ 49.7%

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

      10. *-commutative 49.7%

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

      11. unpow2 49.7%

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

      12. unpow2 49.7%

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

   6. Simplified 49.7%

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

   7. Taylor expanded in t around inf 98.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -40:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.04:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 7: 72.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := \frac{\sqrt{0.5}}{t}\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+76}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-23}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (sqrt 0.5) t)))
   (if (<= l -1.4e+76)
     (asin 1.0)
     (if (<= l -1e-309)
       (asin (* t_1 (- l)))
       (if (<= l 3.6e-23) (asin (* l t_1)) (asin 1.0))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = sqrt(0.5) / t;
	double tmp;
	if (l <= -1.4e+76) {
		tmp = asin(1.0);
	} else if (l <= -1e-309) {
		tmp = asin((t_1 * -l));
	} else if (l <= 3.6e-23) {
		tmp = asin((l * t_1));
	} 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) :: t_1
    real(8) :: tmp
    t_1 = sqrt(0.5d0) / t
    if (l <= (-1.4d+76)) then
        tmp = asin(1.0d0)
    else if (l <= (-1d-309)) then
        tmp = asin((t_1 * -l))
    else if (l <= 3.6d-23) then
        tmp = asin((l * t_1))
    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 t_1 = Math.sqrt(0.5) / t;
	double tmp;
	if (l <= -1.4e+76) {
		tmp = Math.asin(1.0);
	} else if (l <= -1e-309) {
		tmp = Math.asin((t_1 * -l));
	} else if (l <= 3.6e-23) {
		tmp = Math.asin((l * t_1));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = math.sqrt(0.5) / t
	tmp = 0
	if l <= -1.4e+76:
		tmp = math.asin(1.0)
	elif l <= -1e-309:
		tmp = math.asin((t_1 * -l))
	elif l <= 3.6e-23:
		tmp = math.asin((l * t_1))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = Float64(sqrt(0.5) / t)
	tmp = 0.0
	if (l <= -1.4e+76)
		tmp = asin(1.0);
	elseif (l <= -1e-309)
		tmp = asin(Float64(t_1 * Float64(-l)));
	elseif (l <= 3.6e-23)
		tmp = asin(Float64(l * t_1));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = sqrt(0.5) / t;
	tmp = 0.0;
	if (l <= -1.4e+76)
		tmp = asin(1.0);
	elseif (l <= -1e-309)
		tmp = asin((t_1 * -l));
	elseif (l <= 3.6e-23)
		tmp = asin((l * t_1));
	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_] := Block[{t$95$1 = N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[l, -1.4e+76], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -1e-309], N[ArcSin[N[(t$95$1 * (-l)), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 3.6e-23], N[ArcSin[N[(l * t$95$1), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.3999999999999999e76 or 3.5999999999999998e-23 < l

   1. Initial program 96.8%

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

      1. unpow2 96.8%

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

      2. associate-*r/ 90.0%

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

   3. Applied egg-rr 90.0%

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

   4. Taylor expanded in Om around 0 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*l/ 75.5%

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

      3. rem-square-sqrt 75.4%

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

      4. unpow2 75.4%

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

      5. *-commutative 75.4%

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

      6. *-commutative 75.4%

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

      7. unpow2 75.4%

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

      8. rem-square-sqrt 75.5%

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

      9. associate-*l/ 75.5%

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

      10. *-commutative 75.5%

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

      11. unpow2 75.5%

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

      12. unpow2 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in t around 0 77.6%

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

   if -1.3999999999999999e76 < l < -1.000000000000002e-309

   1. Initial program 73.9%

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

      1. unpow2 73.9%

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

      2. associate-*r/ 71.4%

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

   3. Applied egg-rr 71.4%

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

   4. Taylor expanded in Om around 0 56.2%

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

      1. *-commutative 56.2%

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

      2. associate-*l/ 56.2%

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

      3. rem-square-sqrt 56.0%

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

      4. unpow2 56.0%

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

      5. *-commutative 56.0%

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

      6. *-commutative 56.0%

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

      7. unpow2 56.0%

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

      8. rem-square-sqrt 56.2%

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

      9. associate-*l/ 56.2%

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

      10. *-commutative 56.2%

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

      11. unpow2 56.2%

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

      12. unpow2 56.2%

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

   6. Simplified 56.2%

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

   7. Taylor expanded in t around -inf 47.1%

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

      1. mul-1-neg 47.1%

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

      2. associate-*l/ 47.1%

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

      3. distribute-rgt-neg-in 47.1%

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

   9. Simplified 47.1%

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

   if -1.000000000000002e-309 < l < 3.5999999999999998e-23

   1. Initial program 75.2%

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

      1. unpow2 75.2%

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

      2. associate-*r/ 75.2%

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

   3. Applied egg-rr 75.2%

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

   4. Taylor expanded in Om around 0 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*l/ 56.9%

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

      3. rem-square-sqrt 56.8%

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

      4. unpow2 56.8%

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

      5. *-commutative 56.8%

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

      6. *-commutative 56.8%

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

      7. unpow2 56.8%

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

      8. rem-square-sqrt 56.9%

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

      9. associate-*l/ 56.9%

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

      10. *-commutative 56.9%

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

      11. unpow2 56.9%

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

      12. unpow2 56.9%

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

   6. Simplified 56.9%

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

   7. Taylor expanded in t around inf 55.0%

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

      1. associate-*l/ 55.1%

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

   9. Simplified 55.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+76}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{-23}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 8: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 10^{+49}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1e+49) (asin 1.0) (asin (* l (/ (sqrt 0.5) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1e+49) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (t <= 1d+49) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1e+49) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if t <= 1e+49:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1e+49)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 1e+49)
		tmp = asin(1.0);
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1e+49], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 9.99999999999999946e48

   1. Initial program 85.9%

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

      1. unpow2 85.9%

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

      2. associate-*r/ 83.2%

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

   3. Applied egg-rr 83.2%

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

   4. Taylor expanded in Om around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-*l/ 67.8%

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

      3. rem-square-sqrt 67.7%

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

      4. unpow2 67.7%

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

      5. *-commutative 67.7%

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

      6. *-commutative 67.7%

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

      7. unpow2 67.7%

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

      8. rem-square-sqrt 67.8%

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

      9. associate-*l/ 67.8%

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

      10. *-commutative 67.8%

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

      11. unpow2 67.8%

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

      12. unpow2 67.8%

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

   6. Simplified 67.8%

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

   7. Taylor expanded in t around 0 54.5%

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

   if 9.99999999999999946e48 < t

   1. Initial program 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r/ 69.9%

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

   3. Applied egg-rr 69.9%

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

   4. Taylor expanded in Om around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l/ 52.8%

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

      3. rem-square-sqrt 52.7%

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

      4. unpow2 52.7%

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

      5. *-commutative 52.7%

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

      6. *-commutative 52.7%

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

      7. unpow2 52.7%

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

      8. rem-square-sqrt 52.8%

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

      9. associate-*l/ 52.8%

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

      10. *-commutative 52.8%

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

      11. unpow2 52.8%

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

      12. unpow2 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in t around inf 52.3%

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

      1. associate-*l/ 52.3%

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

   9. Simplified 52.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{+49}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9: 64.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{+48}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 5.4e+48) (asin 1.0) (asin (/ l (/ t (sqrt 0.5))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 5.4e+48) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l / (t / sqrt(0.5))));
	}
	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 <= 5.4d+48) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l / (t / sqrt(0.5d0))))
    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 <= 5.4e+48) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if t <= 5.4e+48:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 5.4e+48)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 5.4e+48)
		tmp = asin(1.0);
	else
		tmp = asin((l / (t / sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[t, 5.4e+48], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.40000000000000007e48

   1. Initial program 85.9%

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

      1. unpow2 85.9%

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

      2. associate-*r/ 83.2%

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

   3. Applied egg-rr 83.2%

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

   4. Taylor expanded in Om around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-*l/ 67.8%

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

      3. rem-square-sqrt 67.7%

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

      4. unpow2 67.7%

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

      5. *-commutative 67.7%

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

      6. *-commutative 67.7%

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

      7. unpow2 67.7%

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

      8. rem-square-sqrt 67.8%

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

      9. associate-*l/ 67.8%

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

      10. *-commutative 67.8%

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

      11. unpow2 67.8%

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

      12. unpow2 67.8%

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

   6. Simplified 67.8%

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

   7. Taylor expanded in t around 0 54.5%

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

   if 5.40000000000000007e48 < t

   1. Initial program 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r/ 69.9%

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

   3. Applied egg-rr 69.9%

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

   4. Taylor expanded in Om around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l/ 52.8%

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

      3. rem-square-sqrt 52.7%

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

      4. unpow2 52.7%

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

      5. *-commutative 52.7%

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

      6. *-commutative 52.7%

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

      7. unpow2 52.7%

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

      8. rem-square-sqrt 52.8%

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

      9. associate-*l/ 52.8%

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

      10. *-commutative 52.8%

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

      11. unpow2 52.8%

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

      12. unpow2 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in t around inf 52.3%

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

      1. *-commutative 52.3%

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

      2. associate-/l* 52.2%

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

   9. Simplified 52.2%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.4 \cdot 10^{+48}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 10: 63.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{+49}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= t 1.12e+49) (asin 1.0) (asin (/ (* l (sqrt 0.5)) t))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.12e+49) {
		tmp = asin(1.0);
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (t <= 1.12d+49) then
        tmp = asin(1.0d0)
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (t <= 1.12e+49) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if t <= 1.12e+49:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (t <= 1.12e+49)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (t <= 1.12e+49)
		tmp = asin(1.0);
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[t, 1.12e+49], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.12000000000000005e49

   1. Initial program 85.9%

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

      1. unpow2 85.9%

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

      2. associate-*r/ 83.2%

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

   3. Applied egg-rr 83.2%

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

   4. Taylor expanded in Om around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. associate-*l/ 67.8%

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

      3. rem-square-sqrt 67.7%

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

      4. unpow2 67.7%

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

      5. *-commutative 67.7%

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

      6. *-commutative 67.7%

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

      7. unpow2 67.7%

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

      8. rem-square-sqrt 67.8%

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

      9. associate-*l/ 67.8%

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

      10. *-commutative 67.8%

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

      11. unpow2 67.8%

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

      12. unpow2 67.8%

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

   6. Simplified 67.8%

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

   7. Taylor expanded in t around 0 54.5%

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

   if 1.12000000000000005e49 < t

   1. Initial program 77.6%

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

      1. unpow2 77.6%

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

      2. associate-*r/ 69.9%

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

   3. Applied egg-rr 69.9%

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

   4. Taylor expanded in Om around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. associate-*l/ 52.8%

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

      3. rem-square-sqrt 52.7%

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

      4. unpow2 52.7%

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

      5. *-commutative 52.7%

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

      6. *-commutative 52.7%

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

      7. unpow2 52.7%

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

      8. rem-square-sqrt 52.8%

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

      9. associate-*l/ 52.8%

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

      10. *-commutative 52.8%

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

      11. unpow2 52.8%

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

      12. unpow2 52.8%

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

   6. Simplified 52.8%

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

   7. Taylor expanded in t around inf 52.3%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.12 \cdot 10^{+49}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 11: 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 84.3%

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

   1. unpow2 84.3%

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

   2. associate-*r/ 80.7%

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

3. Applied egg-rr 80.7%

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

4. Taylor expanded in Om around 0 64.9%

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

   1. *-commutative 64.9%

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

   2. associate-*l/ 64.9%

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

   3. rem-square-sqrt 64.8%

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

   4. unpow2 64.8%

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

   5. *-commutative 64.8%

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

   6. *-commutative 64.8%

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

   7. unpow2 64.8%

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

   8. rem-square-sqrt 64.9%

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

   9. associate-*l/ 64.9%

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

   10. *-commutative 64.9%

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

   11. unpow2 64.9%

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

   12. unpow2 64.9%

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

6. Simplified 64.9%

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

7. Taylor expanded in t around 0 47.7%

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

8. Final simplification 47.7%

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

Reproduce

herbie shell --seed 2023244 
(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)))))))