Toniolo and Linder, Equation (2)

Percentage Accurate: 84.2% → 98.7%

Time: 15.1s

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: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -4e+156)
   (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (* l (- (sqrt 0.5))) t)))
   (if (<= (/ t l) 5e+142)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
        (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
     (asin (/ l (/ t (sqrt 0.5)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * ((l * -sqrt(0.5)) / t)));
	} else if ((t / l) <= 5e+142) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin((l / (t / sqrt(0.5))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t / l) <= (-4d+156)) then
        tmp = asin((sqrt((1.0d0 - ((om / omc) ** 2.0d0))) * ((l * -sqrt(0.5d0)) / t)))
    else if ((t / l) <= 5d+142) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin((l / (t / sqrt(0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = Math.asin((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) * ((l * -Math.sqrt(0.5)) / t)));
	} else if ((t / l) <= 5e+142) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin((l / (t / Math.sqrt(0.5))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -4e+156:
		tmp = math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) * ((l * -math.sqrt(0.5)) / t)))
	elif (t / l) <= 5e+142:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin((l / (t / math.sqrt(0.5))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -4e+156)
		tmp = asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * Float64(Float64(l * Float64(-sqrt(0.5))) / t)));
	elseif (Float64(t / l) <= 5e+142)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(l / Float64(t / sqrt(0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -4e+156)
		tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) * ((l * -sqrt(0.5)) / t)));
	elseif ((t / l) <= 5e+142)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin((l / (t / sqrt(0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -4e+156], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l * (-N[Sqrt[0.5], $MachinePrecision])), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+142], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.9999999999999999e156

   1. Initial program 51.1%

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

   2. Taylor expanded in t around -inf 66.4%

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

      1. mul-1-neg 66.4%

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

      2. *-commutative 66.4%

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

      3. unpow2 66.4%

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

      4. unpow2 66.4%

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

      5. times-frac 99.7%

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

      6. unpow2 99.7%

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

   4. Simplified 99.7%

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

   if -3.9999999999999999e156 < (/.f64 t l) < 5.0000000000000001e142

   1. Initial program 98.1%

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

      1. unpow2 98.1%

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

      2. clear-num 98.1%

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

      3. un-div-inv 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 5.0000000000000001e142 < (/.f64 t l)

   1. Initial program 48.0%

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

   2. Taylor expanded in Om around 0 44.3%

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

      1. unpow2 44.3%

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

      2. unpow2 44.3%

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

   4. Simplified 44.3%

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

   5. Taylor expanded in t around inf 45.6%

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

      1. unpow2 45.6%

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

      2. unpow2 45.6%

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

      3. times-frac 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in l around 0 99.5%

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

      1. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \left(-\sqrt{0.5}\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t / l), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. sqrt-div 84.0%

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

   2. div-inv 84.0%

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

   3. add-sqr-sqrt 84.0%

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

   4. hypot-1-def 84.0%

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

   5. *-commutative 84.0%

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

   6. sqrt-prod 83.9%

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

   7. unpow2 83.9%

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

   8. sqrt-prod 59.9%

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

   9. add-sqr-sqrt 98.1%

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

3. Applied egg-rr 98.1%

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

   1. unpow2 98.1%

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

   2. times-frac 83.7%

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

   3. unpow2 83.7%

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

   4. unpow2 83.7%

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

   5. associate-*r/ 83.7%

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

   6. *-rgt-identity 83.7%

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

   7. unpow2 83.7%

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

   8. unpow2 83.7%

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

   9. times-frac 98.1%

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

   10. unpow2 98.1%

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

5. Simplified 98.1%

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

   1. unpow2 84.1%

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

   2. clear-num 84.1%

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

   3. un-div-inv 84.1%

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

7. Applied egg-rr 98.1%

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

8. Final simplification 98.1%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{0.5}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ t (sqrt 0.5))))
   (if (<= (/ t l) -4e+156)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 5e+142)
       (asin
        (sqrt
         (/
          (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
          (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
       (asin (/ l t_1))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = t / sqrt(0.5);
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 5e+142) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin((l / t_1));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / sqrt(0.5d0)
    if ((t / l) <= (-4d+156)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 5d+142) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t / l) ** 2.0d0))))))
    else
        tmp = asin((l / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = t / Math.sqrt(0.5);
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 5e+142) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * Math.pow((t / l), 2.0))))));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = t / math.sqrt(0.5)
	tmp = 0
	if (t / l) <= -4e+156:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 5e+142:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * math.pow((t / l), 2.0))))))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(t / sqrt(0.5))
	tmp = 0.0
	if (Float64(t / l) <= -4e+156)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 5e+142)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0))))));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = t / sqrt(0.5);
	tmp = 0.0;
	if ((t / l) <= -4e+156)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 5e+142)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t / l) ^ 2.0))))));
	else
		tmp = asin((l / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -4e+156], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+142], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.9999999999999999e156

   1. Initial program 51.1%

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

   2. Taylor expanded in Om around 0 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

   4. Simplified 51.1%

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

   5. Taylor expanded in t around inf 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

      3. times-frac 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in l around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. associate-/l* 99.3%

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

      3. distribute-neg-frac 99.3%

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

   10. Simplified 99.3%

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

   if -3.9999999999999999e156 < (/.f64 t l) < 5.0000000000000001e142

   1. Initial program 98.1%

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

      1. unpow2 98.1%

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

      2. clear-num 98.1%

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

      3. un-div-inv 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 5.0000000000000001e142 < (/.f64 t l)

   1. Initial program 48.0%

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

   2. Taylor expanded in Om around 0 44.3%

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

      1. unpow2 44.3%

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

      2. unpow2 44.3%

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

   4. Simplified 44.3%

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

   5. Taylor expanded in t around inf 45.6%

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

      1. unpow2 45.6%

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

      2. unpow2 45.6%

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

      3. times-frac 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in l around 0 99.5%

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

      1. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{0.5}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{{\left(\frac{\ell}{t}\right)}^{2}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ t (sqrt 0.5))))
   (if (<= (/ t l) -4e+156)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 5e+142)
       (asin (sqrt (/ 1.0 (+ 1.0 (/ 2.0 (pow (/ l t) 2.0))))))
       (asin (/ l t_1))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = t / sqrt(0.5);
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 5e+142) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 / pow((l / t), 2.0))))));
	} else {
		tmp = asin((l / t_1));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / sqrt(0.5d0)
    if ((t / l) <= (-4d+156)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 5d+142) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 / ((l / t) ** 2.0d0))))))
    else
        tmp = asin((l / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = t / Math.sqrt(0.5);
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 5e+142) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 / Math.pow((l / t), 2.0))))));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = t / math.sqrt(0.5)
	tmp = 0
	if (t / l) <= -4e+156:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 5e+142:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 / math.pow((l / t), 2.0))))))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(t / sqrt(0.5))
	tmp = 0.0
	if (Float64(t / l) <= -4e+156)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 5e+142)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 / (Float64(l / t) ^ 2.0))))));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = t / sqrt(0.5);
	tmp = 0.0;
	if ((t / l) <= -4e+156)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 5e+142)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 / ((l / t) ^ 2.0))))));
	else
		tmp = asin((l / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -4e+156], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+142], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 / N[Power[N[(l / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.9999999999999999e156

   1. Initial program 51.1%

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

   2. Taylor expanded in Om around 0 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

   4. Simplified 51.1%

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

   5. Taylor expanded in t around inf 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

      3. times-frac 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in l around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. associate-/l* 99.3%

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

      3. distribute-neg-frac 99.3%

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

   10. Simplified 99.3%

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

   if -3.9999999999999999e156 < (/.f64 t l) < 5.0000000000000001e142

   1. Initial program 98.1%

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

      1. sqrt-div 98.0%

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

      2. div-inv 98.0%

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

      3. add-sqr-sqrt 98.0%

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

      4. hypot-1-def 98.0%

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

      5. *-commutative 98.0%

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

      6. sqrt-prod 97.9%

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

      7. unpow2 97.9%

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

      8. sqrt-prod 55.9%

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

      9. add-sqr-sqrt 97.9%

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

   3. Applied egg-rr 97.9%

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

      1. unpow2 97.9%

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

      2. times-frac 83.8%

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

      3. unpow2 83.8%

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

      4. unpow2 83.8%

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

      5. associate-*r/ 83.8%

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

      6. *-rgt-identity 83.8%

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

      7. unpow2 83.8%

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

      8. unpow2 83.8%

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

      9. times-frac 97.9%

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

      10. unpow2 97.9%

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

   5. Simplified 97.9%

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

      1. unpow2 98.1%

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

      2. clear-num 98.1%

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

      3. un-div-inv 98.1%

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in Om around 0 69.0%

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

      1. associate-/l* 69.0%

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

      2. unpow2 69.0%

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

      3. rem-square-sqrt 69.1%

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

      4. associate-/l* 69.1%

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

      5. *-commutative 69.1%

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

      6. associate-/l* 69.1%

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

      7. unpow2 69.1%

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

      8. unpow2 69.1%

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

      9. times-frac 97.1%

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

      10. unpow2 97.1%

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

   10. Simplified 97.1%

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

   if 5.0000000000000001e142 < (/.f64 t l)

   1. Initial program 48.0%

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

   2. Taylor expanded in Om around 0 44.3%

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

      1. unpow2 44.3%

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

      2. unpow2 44.3%

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

   4. Simplified 44.3%

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

   5. Taylor expanded in t around inf 45.6%

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

      1. unpow2 45.6%

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

      2. unpow2 45.6%

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

      3. times-frac 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in l around 0 99.5%

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

      1. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{{\left(\frac{\ell}{t}\right)}^{2}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 5: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\sqrt{0.5}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ t (sqrt 0.5))))
   (if (<= (/ t l) -4e+156)
     (asin (/ (- l) t_1))
     (if (<= (/ t l) 5e+142)
       (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* (/ t l) (/ t l)))))))
       (asin (/ l t_1))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = t / sqrt(0.5);
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = asin((-l / t_1));
	} else if ((t / l) <= 5e+142) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = asin((l / t_1));
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / sqrt(0.5d0)
    if ((t / l) <= (-4d+156)) then
        tmp = asin((-l / t_1))
    else if ((t / l) <= 5d+142) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) * (t / l)))))))
    else
        tmp = asin((l / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = t / Math.sqrt(0.5);
	double tmp;
	if ((t / l) <= -4e+156) {
		tmp = Math.asin((-l / t_1));
	} else if ((t / l) <= 5e+142) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	} else {
		tmp = Math.asin((l / t_1));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = t / math.sqrt(0.5)
	tmp = 0
	if (t / l) <= -4e+156:
		tmp = math.asin((-l / t_1))
	elif (t / l) <= 5e+142:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))))
	else:
		tmp = math.asin((l / t_1))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(t / sqrt(0.5))
	tmp = 0.0
	if (Float64(t / l) <= -4e+156)
		tmp = asin(Float64(Float64(-l) / t_1));
	elseif (Float64(t / l) <= 5e+142)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) * Float64(t / l)))))));
	else
		tmp = asin(Float64(l / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = t / sqrt(0.5);
	tmp = 0.0;
	if ((t / l) <= -4e+156)
		tmp = asin((-l / t_1));
	elseif ((t / l) <= 5e+142)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) * (t / l)))))));
	else
		tmp = asin((l / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -4e+156], N[ArcSin[N[((-l) / t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+142], 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[(l / t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -3.9999999999999999e156

   1. Initial program 51.1%

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

   2. Taylor expanded in Om around 0 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

   4. Simplified 51.1%

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

   5. Taylor expanded in t around inf 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

      3. times-frac 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in l around -inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. associate-/l* 99.3%

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

      3. distribute-neg-frac 99.3%

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

   10. Simplified 99.3%

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

   if -3.9999999999999999e156 < (/.f64 t l) < 5.0000000000000001e142

   1. Initial program 98.1%

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

   2. Taylor expanded in Om around 0 69.1%

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

      1. unpow2 69.1%

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

      2. unpow2 69.1%

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

   4. Simplified 69.1%

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

      1. times-frac 97.1%

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

   6. Applied egg-rr 97.1%

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

   if 5.0000000000000001e142 < (/.f64 t l)

   1. Initial program 48.0%

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

   2. Taylor expanded in Om around 0 44.3%

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

      1. unpow2 44.3%

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

      2. unpow2 44.3%

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

   4. Simplified 44.3%

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

   5. Taylor expanded in t around inf 45.6%

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

      1. unpow2 45.6%

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

      2. unpow2 45.6%

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

      3. times-frac 50.2%

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

   7. Simplified 50.2%

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

   8. Taylor expanded in l around 0 99.5%

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

      1. associate-/l* 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 6: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+105}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-296}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (asin (* l (/ (- (sqrt 0.5)) t)))))
   (if (<= l -1.15e+105)
     (asin 1.0)
     (if (<= l -1.15e-38)
       t_1
       (if (<= l -1e-73)
         (asin 1.0)
         (if (<= l -1.15e-296)
           t_1
           (if (<= l 2.9e+26) (asin (* l (/ (sqrt 0.5) t))) (asin 1.0))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = asin((l * (-sqrt(0.5) / t)));
	double tmp;
	if (l <= -1.15e+105) {
		tmp = asin(1.0);
	} else if (l <= -1.15e-38) {
		tmp = t_1;
	} else if (l <= -1e-73) {
		tmp = asin(1.0);
	} else if (l <= -1.15e-296) {
		tmp = t_1;
	} else if (l <= 2.9e+26) {
		tmp = asin((l * (sqrt(0.5) / t)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = asin((l * (-sqrt(0.5d0) / t)))
    if (l <= (-1.15d+105)) then
        tmp = asin(1.0d0)
    else if (l <= (-1.15d-38)) then
        tmp = t_1
    else if (l <= (-1d-73)) then
        tmp = asin(1.0d0)
    else if (l <= (-1.15d-296)) then
        tmp = t_1
    else if (l <= 2.9d+26) then
        tmp = asin((l * (sqrt(0.5d0) / t)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.asin((l * (-Math.sqrt(0.5) / t)));
	double tmp;
	if (l <= -1.15e+105) {
		tmp = Math.asin(1.0);
	} else if (l <= -1.15e-38) {
		tmp = t_1;
	} else if (l <= -1e-73) {
		tmp = Math.asin(1.0);
	} else if (l <= -1.15e-296) {
		tmp = t_1;
	} else if (l <= 2.9e+26) {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = math.asin((l * (-math.sqrt(0.5) / t)))
	tmp = 0
	if l <= -1.15e+105:
		tmp = math.asin(1.0)
	elif l <= -1.15e-38:
		tmp = t_1
	elif l <= -1e-73:
		tmp = math.asin(1.0)
	elif l <= -1.15e-296:
		tmp = t_1
	elif l <= 2.9e+26:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = asin(Float64(l * Float64(Float64(-sqrt(0.5)) / t)))
	tmp = 0.0
	if (l <= -1.15e+105)
		tmp = asin(1.0);
	elseif (l <= -1.15e-38)
		tmp = t_1;
	elseif (l <= -1e-73)
		tmp = asin(1.0);
	elseif (l <= -1.15e-296)
		tmp = t_1;
	elseif (l <= 2.9e+26)
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = asin((l * (-sqrt(0.5) / t)));
	tmp = 0.0;
	if (l <= -1.15e+105)
		tmp = asin(1.0);
	elseif (l <= -1.15e-38)
		tmp = t_1;
	elseif (l <= -1e-73)
		tmp = asin(1.0);
	elseif (l <= -1.15e-296)
		tmp = t_1;
	elseif (l <= 2.9e+26)
		tmp = asin((l * (sqrt(0.5) / t)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[ArcSin[N[(l * N[((-N[Sqrt[0.5], $MachinePrecision]) / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.15e+105], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -1.15e-38], t$95$1, If[LessEqual[l, -1e-73], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -1.15e-296], t$95$1, If[LessEqual[l, 2.9e+26], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1499999999999999e105 or -1.15000000000000001e-38 < l < -9.99999999999999997e-74 or 2.9e26 < l

   1. Initial program 95.1%

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

   2. Taylor expanded in Om around 0 68.8%

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

      1. unpow2 68.8%

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

      2. unpow2 68.8%

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

   4. Simplified 68.8%

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

   5. Taylor expanded in t around 0 80.8%

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

   if -1.1499999999999999e105 < l < -1.15000000000000001e-38 or -9.99999999999999997e-74 < l < -1.15000000000000002e-296

   1. Initial program 71.7%

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

   2. Taylor expanded in Om around 0 49.8%

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

      1. unpow2 49.8%

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

      2. unpow2 49.8%

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

   4. Simplified 49.8%

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

   5. Taylor expanded in t around -inf 34.7%

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

      1. mul-1-neg 34.7%

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

      2. associate-*r/ 34.8%

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

      3. distribute-rgt-neg-in 34.8%

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

   7. Simplified 34.8%

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

   if -1.15000000000000002e-296 < l < 2.9e26

   1. Initial program 81.6%

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

   2. Taylor expanded in Om around 0 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in t around inf 52.4%

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

      1. associate-*r/ 52.4%

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

   7. Simplified 52.4%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+105}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq -1 \cdot 10^{-73}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-296}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 7: 58.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sqrt{0.5}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+105}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{t_1}{t}\right)\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-296}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot t_1\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+29}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- (sqrt 0.5))))
   (if (<= l -1.2e+105)
     (asin 1.0)
     (if (<= l -2.1e-38)
       (asin (* l (/ t_1 t)))
       (if (<= l -3.8e-75)
         (asin 1.0)
         (if (<= l -1.15e-296)
           (asin (* (/ l t) t_1))
           (if (<= l 8e+29) (asin (* l (/ (sqrt 0.5) t))) (asin 1.0))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = -sqrt(0.5);
	double tmp;
	if (l <= -1.2e+105) {
		tmp = asin(1.0);
	} else if (l <= -2.1e-38) {
		tmp = asin((l * (t_1 / t)));
	} else if (l <= -3.8e-75) {
		tmp = asin(1.0);
	} else if (l <= -1.15e-296) {
		tmp = asin(((l / t) * t_1));
	} else if (l <= 8e+29) {
		tmp = asin((l * (sqrt(0.5) / t)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sqrt(0.5d0)
    if (l <= (-1.2d+105)) then
        tmp = asin(1.0d0)
    else if (l <= (-2.1d-38)) then
        tmp = asin((l * (t_1 / t)))
    else if (l <= (-3.8d-75)) then
        tmp = asin(1.0d0)
    else if (l <= (-1.15d-296)) then
        tmp = asin(((l / t) * t_1))
    else if (l <= 8d+29) then
        tmp = asin((l * (sqrt(0.5d0) / t)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = -Math.sqrt(0.5);
	double tmp;
	if (l <= -1.2e+105) {
		tmp = Math.asin(1.0);
	} else if (l <= -2.1e-38) {
		tmp = Math.asin((l * (t_1 / t)));
	} else if (l <= -3.8e-75) {
		tmp = Math.asin(1.0);
	} else if (l <= -1.15e-296) {
		tmp = Math.asin(((l / t) * t_1));
	} else if (l <= 8e+29) {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = -math.sqrt(0.5)
	tmp = 0
	if l <= -1.2e+105:
		tmp = math.asin(1.0)
	elif l <= -2.1e-38:
		tmp = math.asin((l * (t_1 / t)))
	elif l <= -3.8e-75:
		tmp = math.asin(1.0)
	elif l <= -1.15e-296:
		tmp = math.asin(((l / t) * t_1))
	elif l <= 8e+29:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(-sqrt(0.5))
	tmp = 0.0
	if (l <= -1.2e+105)
		tmp = asin(1.0);
	elseif (l <= -2.1e-38)
		tmp = asin(Float64(l * Float64(t_1 / t)));
	elseif (l <= -3.8e-75)
		tmp = asin(1.0);
	elseif (l <= -1.15e-296)
		tmp = asin(Float64(Float64(l / t) * t_1));
	elseif (l <= 8e+29)
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = -sqrt(0.5);
	tmp = 0.0;
	if (l <= -1.2e+105)
		tmp = asin(1.0);
	elseif (l <= -2.1e-38)
		tmp = asin((l * (t_1 / t)));
	elseif (l <= -3.8e-75)
		tmp = asin(1.0);
	elseif (l <= -1.15e-296)
		tmp = asin(((l / t) * t_1));
	elseif (l <= 8e+29)
		tmp = asin((l * (sqrt(0.5) / t)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = (-N[Sqrt[0.5], $MachinePrecision])}, If[LessEqual[l, -1.2e+105], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -2.1e-38], N[ArcSin[N[(l * N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -3.8e-75], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, -1.15e-296], N[ArcSin[N[(N[(l / t), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 8e+29], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.19999999999999987e105 or -2.10000000000000013e-38 < l < -3.79999999999999994e-75 or 7.99999999999999931e29 < l

   1. Initial program 95.1%

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

   2. Taylor expanded in Om around 0 68.8%

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

      1. unpow2 68.8%

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

      2. unpow2 68.8%

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

   4. Simplified 68.8%

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

   5. Taylor expanded in t around 0 80.8%

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

   if -1.19999999999999987e105 < l < -2.10000000000000013e-38

   1. Initial program 73.5%

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

   2. Taylor expanded in Om around 0 56.7%

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

      1. unpow2 56.7%

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

      2. unpow2 56.7%

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

   4. Simplified 56.7%

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

   5. Taylor expanded in t around -inf 15.8%

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

      1. mul-1-neg 15.8%

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

      2. associate-*r/ 15.9%

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

      3. distribute-rgt-neg-in 15.9%

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

   7. Simplified 15.9%

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

   if -3.79999999999999994e-75 < l < -1.15000000000000002e-296

   1. Initial program 70.3%

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

   2. Taylor expanded in Om around 0 44.4%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

   4. Simplified 44.4%

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

   5. Taylor expanded in t around inf 39.0%

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

      1. unpow2 39.0%

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

      2. unpow2 39.0%

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

      3. times-frac 55.7%

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

   7. Simplified 55.7%

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

   8. Taylor expanded in l around -inf 49.7%

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

      1. mul-1-neg 49.7%

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

      2. associate-/l* 49.6%

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

      3. associate-/r/ 49.7%

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

      4. distribute-lft-neg-in 49.7%

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

   10. Simplified 49.7%

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

   if -1.15000000000000002e-296 < l < 7.99999999999999931e29

   1. Initial program 81.6%

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

   2. Taylor expanded in Om around 0 67.0%

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

      1. unpow2 67.0%

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

      2. unpow2 67.0%

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

   4. Simplified 67.0%

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

   5. Taylor expanded in t around inf 52.4%

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

      1. associate-*r/ 52.4%

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

   7. Simplified 52.4%

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+105}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{-38}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\ell \leq -3.8 \cdot 10^{-75}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq -1.15 \cdot 10^{-296}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \left(-\sqrt{0.5}\right)\right)\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+29}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 8: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5.0)
   (asin (* l (/ (- (sqrt 0.5)) t)))
   (if (<= (/ t l) 0.002)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (* l (/ (sqrt 0.5) t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5.0) {
		tmp = asin((l * (-sqrt(0.5) / t)));
	} else if ((t / l) <= 0.002) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t / l) <= (-5.0d0)) then
        tmp = asin((l * (-sqrt(0.5d0) / t)))
    else if ((t / l) <= 0.002d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5.0) {
		tmp = Math.asin((l * (-Math.sqrt(0.5) / t)));
	} else if ((t / l) <= 0.002) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5.0:
		tmp = math.asin((l * (-math.sqrt(0.5) / t)))
	elif (t / l) <= 0.002:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5.0)
		tmp = asin(Float64(l * Float64(Float64(-sqrt(0.5)) / t)));
	elseif (Float64(t / l) <= 0.002)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5.0)
		tmp = asin((l * (-sqrt(0.5) / t)));
	elseif ((t / l) <= 0.002)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5.0], N[ArcSin[N[(l * N[((-N[Sqrt[0.5], $MachinePrecision]) / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.002], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5

   1. Initial program 77.0%

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

   2. Taylor expanded in Om around 0 40.0%

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

      1. unpow2 40.0%

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

      2. unpow2 40.0%

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

   4. Simplified 40.0%

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

   5. Taylor expanded in t around -inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. associate-*r/ 96.5%

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

      3. distribute-rgt-neg-in 96.5%

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

   7. Simplified 96.5%

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

   if -5 < (/.f64 t l) < 2e-3

   1. Initial program 97.4%

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

   2. Taylor expanded in Om around 0 79.8%

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

      1. unpow2 79.8%

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

      2. unpow2 79.8%

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

   4. Simplified 79.8%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. mul-1-neg 79.8%

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

      2. unpow2 79.8%

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

      3. unpow2 79.8%

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

      4. times-frac 96.0%

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

      5. unpow2 96.0%

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

      6. unsub-neg 96.0%

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

   7. Simplified 96.0%

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

   if 2e-3 < (/.f64 t l)

   1. Initial program 68.5%

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

   2. Taylor expanded in Om around 0 49.3%

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

      1. unpow2 49.3%

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

      2. unpow2 49.3%

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

   4. Simplified 49.3%

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

   5. Taylor expanded in t around inf 97.2%

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

      1. associate-*r/ 97.2%

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

   7. Simplified 97.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 9: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5.0)
   (asin (/ (- l) (/ t (sqrt 0.5))))
   (if (<= (/ t l) 0.002)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (* l (/ (sqrt 0.5) t))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5.0) {
		tmp = asin((-l / (t / sqrt(0.5))));
	} else if ((t / l) <= 0.002) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t / l) <= (-5.0d0)) then
        tmp = asin((-l / (t / sqrt(0.5d0))))
    else if ((t / l) <= 0.002d0) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5.0) {
		tmp = Math.asin((-l / (t / Math.sqrt(0.5))));
	} else if ((t / l) <= 0.002) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5.0:
		tmp = math.asin((-l / (t / math.sqrt(0.5))))
	elif (t / l) <= 0.002:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5.0)
		tmp = asin(Float64(Float64(-l) / Float64(t / sqrt(0.5))));
	elseif (Float64(t / l) <= 0.002)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5.0)
		tmp = asin((-l / (t / sqrt(0.5))));
	elseif ((t / l) <= 0.002)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5.0], N[ArcSin[N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 0.002], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5

   1. Initial program 77.0%

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

   2. Taylor expanded in Om around 0 40.0%

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

      1. unpow2 40.0%

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

      2. unpow2 40.0%

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

   4. Simplified 40.0%

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

   5. Taylor expanded in t around inf 40.0%

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

      1. unpow2 40.0%

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

      2. unpow2 40.0%

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

      3. times-frac 74.2%

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

   7. Simplified 74.2%

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

   8. Taylor expanded in l around -inf 96.5%

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

      1. mul-1-neg 96.5%

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

      2. associate-/l* 96.4%

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

      3. distribute-neg-frac 96.4%

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

   10. Simplified 96.4%

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

   if -5 < (/.f64 t l) < 2e-3

   1. Initial program 97.4%

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

   2. Taylor expanded in Om around 0 79.8%

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

      1. unpow2 79.8%

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

      2. unpow2 79.8%

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

   4. Simplified 79.8%

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

   5. Taylor expanded in t around 0 79.8%

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

      1. mul-1-neg 79.8%

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

      2. unpow2 79.8%

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

      3. unpow2 79.8%

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

      4. times-frac 96.0%

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

      5. unpow2 96.0%

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

      6. unsub-neg 96.0%

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

   7. Simplified 96.0%

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

   if 2e-3 < (/.f64 t l)

   1. Initial program 68.5%

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

   2. Taylor expanded in Om around 0 49.3%

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

      1. unpow2 49.3%

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

      2. unpow2 49.3%

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

   4. Simplified 49.3%

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

   5. Taylor expanded in t around inf 97.2%

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

      1. associate-*r/ 97.2%

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

   7. Simplified 97.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 0.002:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10: 56.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6000000000 \lor \neg \left(t \leq 2.5 \cdot 10^{+35}\right) \land t \leq 3.1 \cdot 10^{+89}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (or (<= t 6000000000.0) (and (not (<= t 2.5e+35)) (<= t 3.1e+89)))
   (asin 1.0)
   (asin (* l (/ (sqrt 0.5) t)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t <= 6000000000.0) || (!(t <= 2.5e+35) && (t <= 3.1e+89))) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l * (sqrt(0.5) / t)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((t <= 6000000000.0d0) .or. (.not. (t <= 2.5d+35)) .and. (t <= 3.1d+89)) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l * (sqrt(0.5d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t <= 6000000000.0) || (!(t <= 2.5e+35) && (t <= 3.1e+89))) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l * (Math.sqrt(0.5) / t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (t <= 6000000000.0) or (not (t <= 2.5e+35) and (t <= 3.1e+89)):
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l * (math.sqrt(0.5) / t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if ((t <= 6000000000.0) || (!(t <= 2.5e+35) && (t <= 3.1e+89)))
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l * Float64(sqrt(0.5) / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t <= 6000000000.0) || (~((t <= 2.5e+35)) && (t <= 3.1e+89)))
		tmp = asin(1.0);
	else
		tmp = asin((l * (sqrt(0.5) / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[Or[LessEqual[t, 6000000000.0], And[N[Not[LessEqual[t, 2.5e+35]], $MachinePrecision], LessEqual[t, 3.1e+89]]], 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 < 6e9 or 2.50000000000000011e35 < t < 3.1e89

   1. Initial program 87.0%

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

   2. Taylor expanded in Om around 0 66.1%

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

      1. unpow2 66.1%

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

      2. unpow2 66.1%

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

   4. Simplified 66.1%

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

   5. Taylor expanded in t around 0 58.8%

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

   if 6e9 < t < 2.50000000000000011e35 or 3.1e89 < t

   1. Initial program 73.8%

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

   2. Taylor expanded in Om around 0 50.2%

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

      1. unpow2 50.2%

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

      2. unpow2 50.2%

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

   4. Simplified 50.2%

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

   5. Taylor expanded in t around inf 71.4%

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

      1. associate-*r/ 71.5%

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

   7. Simplified 71.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6000000000 \lor \neg \left(t \leq 2.5 \cdot 10^{+35}\right) \land t \leq 3.1 \cdot 10^{+89}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 11: 51.1% accurate, 4.1× speedup

Math

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

FPCore

(FPCore (t l Om Omc) :precision binary64 (asin 1.0))

C

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

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(1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in Om around 0 62.6%

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

   1. unpow2 62.6%

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

   2. unpow2 62.6%

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

4. Simplified 62.6%

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

5. Taylor expanded in t around 0 49.2%

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

6. Final simplification 49.2%

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

Reproduce

herbie shell --seed 2023292 
(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)))))))