Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 98.9%

Time: 15.6s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} t_1 := \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))))
   (if (<= (/ t l) -2e+48)
     (asin (* t_1 (/ (- l) (/ t (sqrt 0.5)))))
     (if (<= (/ t l) 2e+152)
       (asin
        (sqrt
         (/
          (- 1.0 (* (/ Om Omc) (/ Om Omc)))
          (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (* t_1 (/ (/ l t) (sqrt 2.0))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double t_1 = sqrt((1.0 - pow((Om / Omc), 2.0)));
	double tmp;
	if ((t / l) <= -2e+48) {
		tmp = asin((t_1 * (-l / (t / sqrt(0.5)))));
	} else if ((t / l) <= 2e+152) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((t_1 * ((l / t) / sqrt(2.0))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 - ((om / omc) ** 2.0d0)))
    if ((t / l) <= (-2d+48)) then
        tmp = asin((t_1 * (-l / (t / sqrt(0.5d0)))))
    else if ((t / l) <= 2d+152) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) * (om / omc))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((t_1 * ((l / t) / sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0)));
	double tmp;
	if ((t / l) <= -2e+48) {
		tmp = Math.asin((t_1 * (-l / (t / Math.sqrt(0.5)))));
	} else if ((t / l) <= 2e+152) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((t_1 * ((l / t) / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	t_1 = math.sqrt((1.0 - math.pow((Om / Omc), 2.0)))
	tmp = 0
	if (t / l) <= -2e+48:
		tmp = math.asin((t_1 * (-l / (t / math.sqrt(0.5)))))
	elif (t / l) <= 2e+152:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((t_1 * ((l / t) / math.sqrt(2.0))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	t_1 = sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)))
	tmp = 0.0
	if (Float64(t / l) <= -2e+48)
		tmp = asin(Float64(t_1 * Float64(Float64(-l) / Float64(t / sqrt(0.5)))));
	elseif (Float64(t / l) <= 2e+152)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(t_1 * Float64(Float64(l / t) / sqrt(2.0))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	t_1 = sqrt((1.0 - ((Om / Omc) ^ 2.0)));
	tmp = 0.0;
	if ((t / l) <= -2e+48)
		tmp = asin((t_1 * (-l / (t / sqrt(0.5)))));
	elseif ((t / l) <= 2e+152)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((t_1 * ((l / t) / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+48], N[ArcSin[N[(t$95$1 * N[((-l) / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+152], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[(N[(l / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2.00000000000000009e48

   1. Initial program 59.1%

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

   2. Taylor expanded in t around -inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. *-commutative 88.5%

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

      3. distribute-rgt-neg-in 88.5%

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

      4. unpow2 88.5%

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

      5. unpow2 88.5%

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

      6. times-frac 99.4%

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

      7. unpow2 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/l* 99.7%

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

   4. Simplified 99.7%

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

   if -2.00000000000000009e48 < (/.f64 t l) < 2.0000000000000001e152

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

   5. Applied egg-rr 99.3%

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

   if 2.0000000000000001e152 < (/.f64 t l)

   1. Initial program 22.2%

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

      1. sqrt-div 22.2%

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

      2. add-sqr-sqrt 22.2%

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

      3. hypot-1-def 22.2%

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

      4. *-commutative 22.2%

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

      5. sqrt-prod 22.2%

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

      6. unpow2 22.2%

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

      7. sqrt-prod 95.9%

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

      8. add-sqr-sqrt 96.0%

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

   3. Applied egg-rr 96.0%

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

   4. Taylor expanded in t around inf 88.0%

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

      1. *-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. unpow2 88.0%

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

      4. times-frac 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+48}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. sqrt-div 80.3%

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

   2. add-sqr-sqrt 80.3%

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

   3. hypot-1-def 80.3%

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

   4. *-commutative 80.3%

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

   5. sqrt-prod 80.2%

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

   6. unpow2 80.2%

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

   7. sqrt-prod 56.5%

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

   8. add-sqr-sqrt 98.2%

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

3. Applied egg-rr 98.2%

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

4. Final simplification 98.2%

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

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+154)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 5e+69)
     (asin
      (sqrt
       (/
        (- 1.0 (* (/ Om Omc) (/ Om Omc)))
        (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ l (/ t (sqrt 0.5))))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 5e+69) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * (l / (t / sqrt(0.5)))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-2d+154)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 5d+69) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) * (om / omc))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) ** 2.0d0))) * (l / (t / sqrt(0.5d0)))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 5e+69) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) * (l / (t / Math.sqrt(0.5)))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+154:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 5e+69:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) * (l / (t / math.sqrt(0.5)))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+154)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 5e+69)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * Float64(l / Float64(t / sqrt(0.5)))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2e+154)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 5e+69)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) * (l / (t / sqrt(0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+154], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+69], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l / N[(t / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2.00000000000000007e154

   1. Initial program 37.3%

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

      1. unpow2 37.3%

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

      2. clear-num 37.3%

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

      3. un-div-inv 37.3%

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

   3. Applied egg-rr 37.3%

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

   4. Taylor expanded in Om around 0 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l/ 37.3%

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

      3. rem-square-sqrt 37.3%

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

      4. unpow2 37.3%

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

      5. *-commutative 37.3%

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

      6. *-commutative 37.3%

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

      7. unpow2 37.3%

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

      8. rem-square-sqrt 37.3%

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

      9. associate-*l/ 37.3%

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

      10. *-commutative 37.3%

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

      11. unpow2 37.3%

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

      12. unpow2 37.3%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in t around -inf 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.6%

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

      3. neg-mul-1 97.6%

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

      4. distribute-rgt-neg-in 97.6%

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

   9. Simplified 97.6%

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

   if -2.00000000000000007e154 < (/.f64 t l) < 5.00000000000000036e69

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

   5. Applied egg-rr 99.3%

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

   if 5.00000000000000036e69 < (/.f64 t l)

   1. Initial program 50.2%

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

   2. Taylor expanded in t around inf 92.2%

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

      1. *-commutative 92.2%

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

      2. unpow2 92.2%

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

      3. unpow2 92.2%

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

      4. times-frac 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-/l* 99.6%

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

   4. Simplified 99.6%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 4: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+154)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 2e+152)
     (asin
      (sqrt
       (/
        (- 1.0 (* (/ Om Omc) (/ Om Omc)))
        (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (/ l t) (sqrt 2.0)))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 2e+152) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * ((l / t) / sqrt(2.0))));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-2d+154)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 2d+152) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) * (om / omc))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) ** 2.0d0))) * ((l / t) / sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 2e+152) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) * ((l / t) / Math.sqrt(2.0))));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+154:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 2e+152:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) * ((l / t) / math.sqrt(2.0))))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+154)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 2e+152)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * Float64(Float64(l / t) / sqrt(2.0))));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2e+154)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 2e+152)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) * ((l / t) / sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+154], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+152], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2.00000000000000007e154

   1. Initial program 37.3%

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

      1. unpow2 37.3%

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

      2. clear-num 37.3%

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

      3. un-div-inv 37.3%

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

   3. Applied egg-rr 37.3%

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

   4. Taylor expanded in Om around 0 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l/ 37.3%

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

      3. rem-square-sqrt 37.3%

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

      4. unpow2 37.3%

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

      5. *-commutative 37.3%

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

      6. *-commutative 37.3%

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

      7. unpow2 37.3%

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

      8. rem-square-sqrt 37.3%

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

      9. associate-*l/ 37.3%

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

      10. *-commutative 37.3%

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

      11. unpow2 37.3%

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

      12. unpow2 37.3%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in t around -inf 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.6%

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

      3. neg-mul-1 97.6%

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

      4. distribute-rgt-neg-in 97.6%

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

   9. Simplified 97.6%

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

   if -2.00000000000000007e154 < (/.f64 t l) < 2.0000000000000001e152

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

   5. Applied egg-rr 99.3%

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

   if 2.0000000000000001e152 < (/.f64 t l)

   1. Initial program 22.2%

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

      1. sqrt-div 22.2%

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

      2. add-sqr-sqrt 22.2%

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

      3. hypot-1-def 22.2%

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

      4. *-commutative 22.2%

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

      5. sqrt-prod 22.2%

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

      6. unpow2 22.2%

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

      7. sqrt-prod 95.9%

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

      8. add-sqr-sqrt 96.0%

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

   3. Applied egg-rr 96.0%

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

   4. Taylor expanded in t around inf 88.0%

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

      1. *-commutative 88.0%

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

      2. unpow2 88.0%

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

      3. unpow2 88.0%

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

      4. times-frac 99.4%

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

      5. unpow2 99.4%

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

      6. *-commutative 99.4%

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

      7. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \end{array} \]

Alternative 5: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2e+154)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 1e+120)
     (asin
      (sqrt
       (/
        (- 1.0 (* (/ Om Omc) (/ Om Omc)))
        (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin (/ (* l (sqrt 0.5)) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 1e+120) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-2d+154)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 1d+120) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) * (om / omc))) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2e+154) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 1e+120) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2e+154:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 1e+120:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2e+154)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 1e+120)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2e+154)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 1e+120)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) * (Om / Omc))) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+154], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+120], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2.00000000000000007e154

   1. Initial program 37.3%

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

      1. unpow2 37.3%

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

      2. clear-num 37.3%

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

      3. un-div-inv 37.3%

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

   3. Applied egg-rr 37.3%

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

   4. Taylor expanded in Om around 0 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l/ 37.3%

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

      3. rem-square-sqrt 37.3%

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

      4. unpow2 37.3%

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

      5. *-commutative 37.3%

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

      6. *-commutative 37.3%

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

      7. unpow2 37.3%

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

      8. rem-square-sqrt 37.3%

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

      9. associate-*l/ 37.3%

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

      10. *-commutative 37.3%

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

      11. unpow2 37.3%

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

      12. unpow2 37.3%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in t around -inf 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.6%

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

      3. neg-mul-1 97.6%

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

      4. distribute-rgt-neg-in 97.6%

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

   9. Simplified 97.6%

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

   if -2.00000000000000007e154 < (/.f64 t l) < 9.9999999999999998e119

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

      1. unpow2 99.3%

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

   5. Applied egg-rr 99.3%

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

   if 9.9999999999999998e119 < (/.f64 t l)

   1. Initial program 39.2%

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

      1. unpow2 39.2%

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

      2. clear-num 39.2%

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

      3. un-div-inv 39.3%

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

   3. Applied egg-rr 39.3%

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

   4. Taylor expanded in Om around 0 22.8%

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

      1. *-commutative 22.8%

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

      2. associate-*l/ 22.8%

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

      3. rem-square-sqrt 22.8%

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

      4. unpow2 22.8%

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

      5. *-commutative 22.8%

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

      6. *-commutative 22.8%

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

      7. unpow2 22.8%

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

      8. rem-square-sqrt 22.8%

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

      9. associate-*l/ 22.8%

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

      10. *-commutative 22.8%

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

      11. unpow2 22.8%

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

      12. unpow2 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in t around inf 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 50000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5e+48)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 50000.0)
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (* t (/ (/ t l) l)))))))
     (asin (* (sqrt 0.5) (/ l t))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+48) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 50000.0) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * (t * ((t / l) / l)))))));
	} else {
		tmp = asin((sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-5d+48)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 50000.0d0) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * (t * ((t / l) / l)))))))
    else
        tmp = asin((sqrt(0.5d0) * (l / t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+48) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 50000.0) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * (t * ((t / l) / l)))))));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5e+48:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 50000.0:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * (t * ((t / l) / l)))))))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5e+48)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 50000.0)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(t * Float64(Float64(t / l) / l)))))));
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5e+48)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 50000.0)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * (t * ((t / l) / l)))))));
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+48], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 50000.0], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(t * N[(N[(t / l), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -4.99999999999999973e48

   1. Initial program 59.1%

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

      1. unpow2 59.1%

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

      2. clear-num 59.2%

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

      3. un-div-inv 59.2%

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

   3. Applied egg-rr 59.2%

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

   4. Taylor expanded in Om around 0 43.6%

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

      1. *-commutative 43.6%

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

      2. associate-*l/ 43.6%

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

      3. rem-square-sqrt 43.7%

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

      4. unpow2 43.7%

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

      5. *-commutative 43.7%

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

      6. *-commutative 43.7%

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

      7. unpow2 43.7%

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

      8. rem-square-sqrt 43.6%

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

      9. associate-*l/ 43.6%

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

      10. *-commutative 43.6%

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

      11. unpow2 43.6%

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

      12. unpow2 43.6%

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

   6. Simplified 43.6%

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

   7. Taylor expanded in t around -inf 97.6%

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

      1. *-commutative 97.6%

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

      2. associate-*l/ 97.5%

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

      3. neg-mul-1 97.5%

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

      4. distribute-rgt-neg-in 97.5%

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

   9. Simplified 97.5%

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

   if -4.99999999999999973e48 < (/.f64 t l) < 5e4

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in Om around 0 83.8%

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

      1. *-commutative 83.8%

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

      2. associate-*l/ 83.8%

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

      3. rem-square-sqrt 83.7%

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

      4. unpow2 83.7%

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

      5. *-commutative 83.7%

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

      6. *-commutative 83.7%

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

      7. unpow2 83.7%

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

      8. rem-square-sqrt 83.8%

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

      9. associate-*l/ 83.8%

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

      10. *-commutative 83.8%

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

      11. unpow2 83.8%

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

      12. unpow2 83.8%

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

   6. Simplified 83.8%

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

      1. associate-/r* 88.8%

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

      2. associate-*l/ 98.2%

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

      3. associate-/l* 98.3%

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

      4. associate-/r/ 98.3%

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

   8. Applied egg-rr 98.3%

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

   if 5e4 < (/.f64 t l)

   1. Initial program 60.2%

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

      1. unpow2 60.2%

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

      2. clear-num 60.2%

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

      3. un-div-inv 60.3%

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

   3. Applied egg-rr 60.3%

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

   4. Taylor expanded in Om around 0 29.7%

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

      1. *-commutative 29.7%

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

      2. associate-*l/ 29.7%

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

      3. rem-square-sqrt 29.6%

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

      4. unpow2 29.6%

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

      5. *-commutative 29.6%

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

      6. *-commutative 29.6%

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

      7. unpow2 29.6%

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

      8. rem-square-sqrt 29.7%

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

      9. associate-*l/ 29.7%

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

      10. *-commutative 29.7%

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

      11. unpow2 29.7%

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

      12. unpow2 29.7%

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

   6. Simplified 29.7%

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

   7. Taylor expanded in t around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

   9. Simplified 99.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 50000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 7: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+117}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -1e+117)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 1e+120)
     (asin (sqrt (/ 1.0 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
     (asin (/ (* l (sqrt 0.5)) t)))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -1e+117) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 1e+120) {
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin(((l * sqrt(0.5)) / t));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-1d+117)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 1d+120) then
        tmp = asin(sqrt((1.0d0 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin(((l * sqrt(0.5d0)) / t))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -1e+117) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 1e+120) {
		tmp = Math.asin(Math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin(((l * Math.sqrt(0.5)) / t));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -1e+117:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 1e+120:
		tmp = math.asin(math.sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin(((l * math.sqrt(0.5)) / t))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -1e+117)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 1e+120)
		tmp = asin(sqrt(Float64(1.0 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(l * sqrt(0.5)) / t));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -1e+117)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 1e+120)
		tmp = asin(sqrt((1.0 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin(((l * sqrt(0.5)) / t));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -1e+117], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+120], N[ArcSin[N[Sqrt[N[(1.0 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1.00000000000000005e117

   1. Initial program 46.4%

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

      1. unpow2 46.4%

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

      2. clear-num 46.4%

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

      3. un-div-inv 46.4%

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

   3. Applied egg-rr 46.4%

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

   4. Taylor expanded in Om around 0 40.9%

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

      1. *-commutative 40.9%

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

      2. associate-*l/ 40.9%

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

      3. rem-square-sqrt 40.9%

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

      4. unpow2 40.9%

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

      5. *-commutative 40.9%

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

      6. *-commutative 40.9%

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

      7. unpow2 40.9%

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

      8. rem-square-sqrt 40.9%

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

      9. associate-*l/ 40.9%

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

      10. *-commutative 40.9%

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

      11. unpow2 40.9%

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

      12. unpow2 40.9%

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

   6. Simplified 40.9%

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

   7. Taylor expanded in t around -inf 97.1%

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

      1. *-commutative 97.1%

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

      2. associate-*l/ 97.0%

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

      3. neg-mul-1 97.0%

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

      4. distribute-rgt-neg-in 97.0%

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

   9. Simplified 97.0%

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

   if -1.00000000000000005e117 < (/.f64 t l) < 9.9999999999999998e119

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in Om around 0 75.5%

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

      1. *-commutative 75.5%

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

      2. associate-*l/ 75.5%

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

      3. rem-square-sqrt 75.5%

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

      4. unpow2 75.5%

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

      5. *-commutative 75.5%

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

      6. *-commutative 75.5%

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

      7. unpow2 75.5%

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

      8. rem-square-sqrt 75.5%

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

      9. associate-*l/ 75.5%

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

      10. *-commutative 75.5%

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

      11. unpow2 75.5%

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

      12. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. associate-/r* 82.7%

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

      2. associate-*l/ 95.7%

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

      3. associate-/l* 98.5%

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

      4. associate-/r/ 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. associate-/r/ 98.5%

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

   10. Applied egg-rr 98.5%

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

   if 9.9999999999999998e119 < (/.f64 t l)

   1. Initial program 39.2%

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

      1. unpow2 39.2%

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

      2. clear-num 39.2%

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

      3. un-div-inv 39.3%

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

   3. Applied egg-rr 39.3%

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

   4. Taylor expanded in Om around 0 22.8%

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

      1. *-commutative 22.8%

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

      2. associate-*l/ 22.8%

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

      3. rem-square-sqrt 22.8%

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

      4. unpow2 22.8%

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

      5. *-commutative 22.8%

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

      6. *-commutative 22.8%

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

      7. unpow2 22.8%

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

      8. rem-square-sqrt 22.8%

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

      9. associate-*l/ 22.8%

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

      10. *-commutative 22.8%

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

      11. unpow2 22.8%

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

      12. unpow2 22.8%

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

   6. Simplified 22.8%

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

   7. Taylor expanded in t around inf 99.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+117}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 8: 96.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2000000000000.0)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 4e-8)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (* (sqrt 0.5) (/ l t))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2000000000000.0) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 4e-8) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-2000000000000.0d0)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 4d-8) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((sqrt(0.5d0) * (l / t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2000000000000.0) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 4e-8) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2000000000000.0:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 4e-8:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2000000000000.0)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 4e-8)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2000000000000.0)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 4e-8)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2000000000000.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 4e-8], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e12

   1. Initial program 63.2%

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

      1. unpow2 63.2%

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

      2. clear-num 63.2%

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

      3. un-div-inv 63.2%

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

   3. Applied egg-rr 63.2%

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

   4. Taylor expanded in Om around 0 45.9%

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

      1. *-commutative 45.9%

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

      2. associate-*l/ 45.9%

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

      3. rem-square-sqrt 45.9%

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

      4. unpow2 45.9%

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

      5. *-commutative 45.9%

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

      6. *-commutative 45.9%

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

      7. unpow2 45.9%

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

      8. rem-square-sqrt 45.9%

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

      9. associate-*l/ 45.9%

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

      10. *-commutative 45.9%

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

      11. unpow2 45.9%

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

      12. unpow2 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in t around -inf 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.6%

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

      3. neg-mul-1 97.6%

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

      4. distribute-rgt-neg-in 97.6%

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

   9. Simplified 97.6%

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

   if -2e12 < (/.f64 t l) < 4.0000000000000001e-8

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in Om around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-*l/ 84.4%

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

      3. rem-square-sqrt 84.4%

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

      4. unpow2 84.4%

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

      5. *-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. unpow2 84.4%

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

      8. rem-square-sqrt 84.4%

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

      9. associate-*l/ 84.4%

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

      10. *-commutative 84.4%

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

      11. unpow2 84.4%

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

      12. unpow2 84.4%

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

   6. Simplified 84.4%

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

      1. associate-/r* 89.7%

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

      2. associate-*l/ 98.2%

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

      3. associate-/l* 98.2%

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

      4. associate-/r/ 98.2%

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

   8. Applied egg-rr 98.2%

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

   9. Taylor expanded in t around 0 84.4%

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

       1. mul-1-neg 84.4%

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

       2. unsub-neg 84.4%

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

       3. unpow2 84.4%

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

       4. unpow2 84.4%

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

       5. times-frac 96.6%

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

       6. unpow2 96.6%

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

   11. Simplified 96.6%

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

   if 4.0000000000000001e-8 < (/.f64 t l)

   1. Initial program 61.3%

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

      1. unpow2 61.3%

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

      2. clear-num 61.2%

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

      3. un-div-inv 61.3%

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

   3. Applied egg-rr 61.3%

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

   4. Taylor expanded in Om around 0 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-*l/ 31.6%

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

      3. rem-square-sqrt 31.5%

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

      4. unpow2 31.5%

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

      5. *-commutative 31.5%

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

      6. *-commutative 31.5%

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

      7. unpow2 31.5%

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

      8. rem-square-sqrt 31.6%

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

      9. associate-*l/ 31.6%

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

      10. *-commutative 31.6%

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

      11. unpow2 31.6%

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

      12. unpow2 31.6%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in t around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*l/ 97.1%

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

      3. *-commutative 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 9: 96.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2000000000000.0)
   (asin (/ (- (/ l t)) (sqrt 2.0)))
   (if (<= (/ t l) 4e-8)
     (asin (- 1.0 (pow (/ t l) 2.0)))
     (asin (* (sqrt 0.5) (/ l t))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2000000000000.0) {
		tmp = asin((-(l / t) / sqrt(2.0)));
	} else if ((t / l) <= 4e-8) {
		tmp = asin((1.0 - pow((t / l), 2.0)));
	} else {
		tmp = asin((sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-2000000000000.0d0)) then
        tmp = asin((-(l / t) / sqrt(2.0d0)))
    else if ((t / l) <= 4d-8) then
        tmp = asin((1.0d0 - ((t / l) ** 2.0d0)))
    else
        tmp = asin((sqrt(0.5d0) * (l / t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2000000000000.0) {
		tmp = Math.asin((-(l / t) / Math.sqrt(2.0)));
	} else if ((t / l) <= 4e-8) {
		tmp = Math.asin((1.0 - Math.pow((t / l), 2.0)));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2000000000000.0:
		tmp = math.asin((-(l / t) / math.sqrt(2.0)))
	elif (t / l) <= 4e-8:
		tmp = math.asin((1.0 - math.pow((t / l), 2.0)))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2000000000000.0)
		tmp = asin(Float64(Float64(-Float64(l / t)) / sqrt(2.0)));
	elseif (Float64(t / l) <= 4e-8)
		tmp = asin(Float64(1.0 - (Float64(t / l) ^ 2.0)));
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2000000000000.0)
		tmp = asin((-(l / t) / sqrt(2.0)));
	elseif ((t / l) <= 4e-8)
		tmp = asin((1.0 - ((t / l) ^ 2.0)));
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2000000000000.0], N[ArcSin[N[((-N[(l / t), $MachinePrecision]) / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 4e-8], N[ArcSin[N[(1.0 - N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e12

   1. Initial program 63.2%

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

      1. sqrt-div 63.3%

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

      2. add-sqr-sqrt 63.3%

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

      3. hypot-1-def 63.3%

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

      4. *-commutative 63.3%

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

      5. sqrt-prod 63.1%

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

      6. unpow2 63.1%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 96.9%

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

   3. Applied egg-rr 96.9%

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

   4. Taylor expanded in t around -inf 89.5%

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

      1. mul-1-neg 89.5%

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

      2. *-commutative 89.5%

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

      3. distribute-rgt-neg-in 89.5%

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

      4. unpow2 89.5%

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

      5. unpow2 89.5%

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

      6. times-frac 99.4%

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

      7. unpow2 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-/r* 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in Om around 0 97.7%

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

      1. *-lft-identity 97.7%

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

      2. times-frac 97.6%

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

      3. *-commutative 97.6%

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

      4. associate-*r/ 97.8%

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

      5. *-rgt-identity 97.8%

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

      6. neg-mul-1 97.8%

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

      7. distribute-frac-neg 97.8%

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

   9. Simplified 97.8%

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

   if -2e12 < (/.f64 t l) < 4.0000000000000001e-8

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in Om around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-*l/ 84.4%

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

      3. rem-square-sqrt 84.4%

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

      4. unpow2 84.4%

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

      5. *-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. unpow2 84.4%

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

      8. rem-square-sqrt 84.4%

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

      9. associate-*l/ 84.4%

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

      10. *-commutative 84.4%

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

      11. unpow2 84.4%

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

      12. unpow2 84.4%

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

   6. Simplified 84.4%

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

      1. associate-/r* 89.7%

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

      2. associate-*l/ 98.2%

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

      3. associate-/l* 98.2%

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

      4. associate-/r/ 98.2%

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

   8. Applied egg-rr 98.2%

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

   9. Taylor expanded in t around 0 84.4%

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

       1. mul-1-neg 84.4%

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

       2. unsub-neg 84.4%

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

       3. unpow2 84.4%

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

       4. unpow2 84.4%

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

       5. times-frac 96.6%

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

       6. unpow2 96.6%

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

   11. Simplified 96.6%

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

   if 4.0000000000000001e-8 < (/.f64 t l)

   1. Initial program 61.3%

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

      1. unpow2 61.3%

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

      2. clear-num 61.2%

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

      3. un-div-inv 61.3%

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

   3. Applied egg-rr 61.3%

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

   4. Taylor expanded in Om around 0 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-*l/ 31.6%

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

      3. rem-square-sqrt 31.5%

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

      4. unpow2 31.5%

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

      5. *-commutative 31.5%

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

      6. *-commutative 31.5%

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

      7. unpow2 31.5%

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

      8. rem-square-sqrt 31.6%

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

      9. associate-*l/ 31.6%

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

      10. *-commutative 31.6%

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

      11. unpow2 31.6%

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

      12. unpow2 31.6%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in t around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*l/ 97.1%

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

      3. *-commutative 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000000:\\ \;\;\;\;\sin^{-1} \left(\frac{-\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{t}{\ell}\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 10: 79.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -5e+206)
   (asin (/ (sqrt 0.5) (/ t l)))
   (if (<= (/ t l) 4e-8) (asin 1.0) (asin (* (sqrt 0.5) (/ l t))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+206) {
		tmp = asin((sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 4e-8) {
		tmp = asin(1.0);
	} else {
		tmp = asin((sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-5d+206)) then
        tmp = asin((sqrt(0.5d0) / (t / l)))
    else if ((t / l) <= 4d-8) then
        tmp = asin(1.0d0)
    else
        tmp = asin((sqrt(0.5d0) * (l / t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -5e+206) {
		tmp = Math.asin((Math.sqrt(0.5) / (t / l)));
	} else if ((t / l) <= 4e-8) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -5e+206:
		tmp = math.asin((math.sqrt(0.5) / (t / l)))
	elif (t / l) <= 4e-8:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -5e+206)
		tmp = asin(Float64(sqrt(0.5) / Float64(t / l)));
	elseif (Float64(t / l) <= 4e-8)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -5e+206)
		tmp = asin((sqrt(0.5) / (t / l)));
	elseif ((t / l) <= 4e-8)
		tmp = asin(1.0);
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -5e+206], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 4e-8], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5.0000000000000002e206

   1. Initial program 45.4%

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

      1. unpow2 45.4%

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

      2. clear-num 45.4%

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

      3. un-div-inv 45.4%

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

   3. Applied egg-rr 45.4%

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

   4. Taylor expanded in Om around 0 45.4%

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

      1. *-commutative 45.4%

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

      2. associate-*l/ 45.4%

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

      3. rem-square-sqrt 45.4%

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

      4. unpow2 45.4%

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

      5. *-commutative 45.4%

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

      6. *-commutative 45.4%

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

      7. unpow2 45.4%

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

      8. rem-square-sqrt 45.4%

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

      9. associate-*l/ 45.4%

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

      10. *-commutative 45.4%

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

      11. unpow2 45.4%

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

      12. unpow2 45.4%

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

   6. Simplified 45.4%

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

      1. associate-/r* 45.4%

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

      2. associate-*l/ 45.4%

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

      3. associate-/l* 45.4%

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

      4. associate-/r/ 45.4%

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

   8. Applied egg-rr 45.4%

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

   9. Taylor expanded in t around inf 44.4%

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

       1. associate-/l* 44.5%

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

   11. Simplified 44.5%

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

   if -5.0000000000000002e206 < (/.f64 t l) < 4.0000000000000001e-8

   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. Step-by-step derivation

      1. unpow2 95.1%

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

      2. clear-num 95.1%

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

      3. un-div-inv 95.1%

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

   3. Applied egg-rr 95.1%

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

   4. Taylor expanded in Om around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l/ 76.6%

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

      3. rem-square-sqrt 76.6%

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

      4. unpow2 76.6%

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

      5. *-commutative 76.6%

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

      6. *-commutative 76.6%

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

      7. unpow2 76.6%

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

      8. rem-square-sqrt 76.6%

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

      9. associate-*l/ 76.6%

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

      10. *-commutative 76.6%

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

      11. unpow2 76.6%

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

      12. unpow2 76.6%

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

   6. Simplified 76.6%

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

   7. Taylor expanded in t around 0 78.2%

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

   if 4.0000000000000001e-8 < (/.f64 t l)

   1. Initial program 61.3%

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

      1. unpow2 61.3%

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

      2. clear-num 61.2%

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

      3. un-div-inv 61.3%

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

   3. Applied egg-rr 61.3%

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

   4. Taylor expanded in Om around 0 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-*l/ 31.6%

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

      3. rem-square-sqrt 31.5%

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

      4. unpow2 31.5%

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

      5. *-commutative 31.5%

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

      6. *-commutative 31.5%

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

      7. unpow2 31.5%

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

      8. rem-square-sqrt 31.6%

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

      9. associate-*l/ 31.6%

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

      10. *-commutative 31.6%

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

      11. unpow2 31.6%

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

      12. unpow2 31.6%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in t around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*l/ 97.1%

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

      3. *-commutative 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 11: 96.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ t l) -2000000000000.0)
   (asin (* (sqrt 0.5) (- (/ l t))))
   (if (<= (/ t l) 4e-8) (asin 1.0) (asin (* (sqrt 0.5) (/ l t))))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2000000000000.0) {
		tmp = asin((sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 4e-8) {
		tmp = asin(1.0);
	} else {
		tmp = asin((sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t / l) <= (-2000000000000.0d0)) then
        tmp = asin((sqrt(0.5d0) * -(l / t)))
    else if ((t / l) <= 4d-8) then
        tmp = asin(1.0d0)
    else
        tmp = asin((sqrt(0.5d0) * (l / t)))
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((t / l) <= -2000000000000.0) {
		tmp = Math.asin((Math.sqrt(0.5) * -(l / t)));
	} else if ((t / l) <= 4e-8) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if (t / l) <= -2000000000000.0:
		tmp = math.asin((math.sqrt(0.5) * -(l / t)))
	elif (t / l) <= 4e-8:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(t / l) <= -2000000000000.0)
		tmp = asin(Float64(sqrt(0.5) * Float64(-Float64(l / t))));
	elseif (Float64(t / l) <= 4e-8)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((t / l) <= -2000000000000.0)
		tmp = asin((sqrt(0.5) * -(l / t)));
	elseif ((t / l) <= 4e-8)
		tmp = asin(1.0);
	else
		tmp = asin((sqrt(0.5) * (l / t)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[N[(t / l), $MachinePrecision], -2000000000000.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * (-N[(l / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 4e-8], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -2e12

   1. Initial program 63.2%

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

      1. unpow2 63.2%

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

      2. clear-num 63.2%

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

      3. un-div-inv 63.2%

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

   3. Applied egg-rr 63.2%

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

   4. Taylor expanded in Om around 0 45.9%

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

      1. *-commutative 45.9%

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

      2. associate-*l/ 45.9%

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

      3. rem-square-sqrt 45.9%

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

      4. unpow2 45.9%

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

      5. *-commutative 45.9%

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

      6. *-commutative 45.9%

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

      7. unpow2 45.9%

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

      8. rem-square-sqrt 45.9%

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

      9. associate-*l/ 45.9%

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

      10. *-commutative 45.9%

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

      11. unpow2 45.9%

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

      12. unpow2 45.9%

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

   6. Simplified 45.9%

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

   7. Taylor expanded in t around -inf 97.7%

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

      1. *-commutative 97.7%

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

      2. associate-*l/ 97.6%

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

      3. neg-mul-1 97.6%

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

      4. distribute-rgt-neg-in 97.6%

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

   9. Simplified 97.6%

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

   if -2e12 < (/.f64 t l) < 4.0000000000000001e-8

   1. Initial program 99.3%

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

      1. unpow2 99.3%

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

      2. clear-num 99.3%

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

      3. un-div-inv 99.3%

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

   3. Applied egg-rr 99.3%

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

   4. Taylor expanded in Om around 0 84.4%

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

      1. *-commutative 84.4%

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

      2. associate-*l/ 84.4%

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

      3. rem-square-sqrt 84.4%

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

      4. unpow2 84.4%

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

      5. *-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. unpow2 84.4%

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

      8. rem-square-sqrt 84.4%

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

      9. associate-*l/ 84.4%

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

      10. *-commutative 84.4%

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

      11. unpow2 84.4%

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

      12. unpow2 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in t around 0 96.7%

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

   if 4.0000000000000001e-8 < (/.f64 t l)

   1. Initial program 61.3%

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

      1. unpow2 61.3%

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

      2. clear-num 61.2%

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

      3. un-div-inv 61.3%

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

   3. Applied egg-rr 61.3%

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

   4. Taylor expanded in Om around 0 31.6%

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

      1. *-commutative 31.6%

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

      2. associate-*l/ 31.6%

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

      3. rem-square-sqrt 31.5%

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

      4. unpow2 31.5%

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

      5. *-commutative 31.5%

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

      6. *-commutative 31.5%

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

      7. unpow2 31.5%

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

      8. rem-square-sqrt 31.6%

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

      9. associate-*l/ 31.6%

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

      10. *-commutative 31.6%

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

      11. unpow2 31.6%

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

      12. unpow2 31.6%

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

   6. Simplified 31.6%

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

   7. Taylor expanded in t around inf 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*l/ 97.1%

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

      3. *-commutative 97.1%

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

   9. Simplified 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \left(-\frac{\ell}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 12: 65.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} t = |t|\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-204}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 300:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

NOTE: t should be positive before calling this function
(FPCore (t l Om Omc)
 :precision binary64
 (if (<= l -2.4e-204)
   (asin 1.0)
   (if (<= l 300.0) (asin (* (sqrt 0.5) (/ l t))) (asin 1.0))))

C

t = abs(t);
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.4e-204) {
		tmp = asin(1.0);
	} else if (l <= 300.0) {
		tmp = asin((sqrt(0.5) * (l / t)));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (l <= (-2.4d-204)) then
        tmp = asin(1.0d0)
    else if (l <= 300.0d0) then
        tmp = asin((sqrt(0.5d0) * (l / t)))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

t = Math.abs(t);
public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (l <= -2.4e-204) {
		tmp = Math.asin(1.0);
	} else if (l <= 300.0) {
		tmp = Math.asin((Math.sqrt(0.5) * (l / t)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t = abs(t)
def code(t, l, Om, Omc):
	tmp = 0
	if l <= -2.4e-204:
		tmp = math.asin(1.0)
	elif l <= 300.0:
		tmp = math.asin((math.sqrt(0.5) * (l / t)))
	else:
		tmp = math.asin(1.0)
	return tmp

Julia

t = abs(t)
function code(t, l, Om, Omc)
	tmp = 0.0
	if (l <= -2.4e-204)
		tmp = asin(1.0);
	elseif (l <= 300.0)
		tmp = asin(Float64(sqrt(0.5) * Float64(l / t)));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

t = abs(t)
function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (l <= -2.4e-204)
		tmp = asin(1.0);
	elseif (l <= 300.0)
		tmp = asin((sqrt(0.5) * (l / t)));
	else
		tmp = asin(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: t should be positive before calling this function
code[t_, l_, Om_, Omc_] := If[LessEqual[l, -2.4e-204], N[ArcSin[1.0], $MachinePrecision], If[LessEqual[l, 300.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.4e-204 or 300 < l

   1. Initial program 87.2%

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

      1. unpow2 87.2%

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

      2. clear-num 87.2%

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

      3. un-div-inv 87.2%

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

   3. Applied egg-rr 87.2%

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

   4. Taylor expanded in Om around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. associate-*l/ 67.9%

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

      3. rem-square-sqrt 67.9%

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

      4. unpow2 67.9%

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

      5. *-commutative 67.9%

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

      6. *-commutative 67.9%

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

      7. unpow2 67.9%

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

      8. rem-square-sqrt 67.9%

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

      9. associate-*l/ 67.9%

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

      10. *-commutative 67.9%

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

      11. unpow2 67.9%

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

      12. unpow2 67.9%

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

   6. Simplified 67.9%

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

   7. Taylor expanded in t around 0 64.6%

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

   if -2.4e-204 < l < 300

   1. Initial program 64.5%

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

      1. unpow2 64.5%

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

      2. clear-num 64.5%

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

      3. un-div-inv 64.5%

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

   3. Applied egg-rr 64.5%

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

   4. Taylor expanded in Om around 0 44.4%

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

      1. *-commutative 44.4%

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

      2. associate-*l/ 44.4%

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

      3. rem-square-sqrt 44.4%

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

      4. unpow2 44.4%

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

      5. *-commutative 44.4%

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

      6. *-commutative 44.4%

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

      7. unpow2 44.4%

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

      8. rem-square-sqrt 44.4%

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

      9. associate-*l/ 44.4%

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

      10. *-commutative 44.4%

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

      11. unpow2 44.4%

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

      12. unpow2 44.4%

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

   6. Simplified 44.4%

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

   7. Taylor expanded in t around inf 57.2%

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

      1. *-commutative 57.2%

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

      2. associate-*l/ 57.4%

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

      3. *-commutative 57.4%

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

   9. Simplified 57.4%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-204}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 300:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 13: 50.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: t should be positive before calling this function
real(8) function code(t, l, om, omc)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. unpow2 80.3%

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

   2. clear-num 80.3%

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

   3. un-div-inv 80.3%

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

3. Applied egg-rr 80.3%

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

4. Taylor expanded in Om around 0 60.7%

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

   1. *-commutative 60.7%

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

   2. associate-*l/ 60.7%

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

   3. rem-square-sqrt 60.7%

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

   4. unpow2 60.7%

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

   5. *-commutative 60.7%

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

   6. *-commutative 60.7%

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

   7. unpow2 60.7%

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

   8. rem-square-sqrt 60.7%

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

   9. associate-*l/ 60.7%

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

   10. *-commutative 60.7%

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

   11. unpow2 60.7%

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

   12. unpow2 60.7%

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

6. Simplified 60.7%

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

7. Taylor expanded in t around 0 49.9%

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

8. Final simplification 49.9%

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

Reproduce

herbie shell --seed 2023213 
(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)))))))