Toniolo and Linder, Equation (2)

Percentage Accurate: 83.2% → 98.7%

Time: 16.7s

Alternatives: 10

Speedup: 1.3×

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: 83.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ t_2 := 1 - t_1\\ \mathbf{if}\;\frac{t_2}{1 + 2 \cdot {\left(\frac{t_m}{\ell}\right)}^{2}} \leq 0:\\ \;\;\;\;\sin^{-1} \left(\left(1 + t_1 \cdot -0.5\right) \cdot \frac{\left|\ell \cdot \sqrt{0.5}\right|}{t_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_2}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (pow (/ Om Omc) 2.0)) (t_2 (- 1.0 t_1)))
   (if (<= (/ t_2 (+ 1.0 (* 2.0 (pow (/ t_m l) 2.0)))) 0.0)
     (asin (* (+ 1.0 (* t_1 -0.5)) (/ (fabs (* l (sqrt 0.5))) t_m)))
     (asin (sqrt (/ t_2 (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m))))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = pow((Om / Omc), 2.0);
	double t_2 = 1.0 - t_1;
	double tmp;
	if ((t_2 / (1.0 + (2.0 * pow((t_m / l), 2.0)))) <= 0.0) {
		tmp = asin(((1.0 + (t_1 * -0.5)) * (fabs((l * sqrt(0.5))) / t_m)));
	} else {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (om / omc) ** 2.0d0
    t_2 = 1.0d0 - t_1
    if ((t_2 / (1.0d0 + (2.0d0 * ((t_m / l) ** 2.0d0)))) <= 0.0d0) then
        tmp = asin(((1.0d0 + (t_1 * (-0.5d0))) * (abs((l * sqrt(0.5d0))) / t_m)))
    else
        tmp = asin(sqrt((t_2 / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = Math.pow((Om / Omc), 2.0);
	double t_2 = 1.0 - t_1;
	double tmp;
	if ((t_2 / (1.0 + (2.0 * Math.pow((t_m / l), 2.0)))) <= 0.0) {
		tmp = Math.asin(((1.0 + (t_1 * -0.5)) * (Math.abs((l * Math.sqrt(0.5))) / t_m)));
	} else {
		tmp = Math.asin(Math.sqrt((t_2 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = math.pow((Om / Omc), 2.0)
	t_2 = 1.0 - t_1
	tmp = 0
	if (t_2 / (1.0 + (2.0 * math.pow((t_m / l), 2.0)))) <= 0.0:
		tmp = math.asin(((1.0 + (t_1 * -0.5)) * (math.fabs((l * math.sqrt(0.5))) / t_m)))
	else:
		tmp = math.asin(math.sqrt((t_2 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(Om / Omc) ^ 2.0
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (Float64(t_2 / Float64(1.0 + Float64(2.0 * (Float64(t_m / l) ^ 2.0)))) <= 0.0)
		tmp = asin(Float64(Float64(1.0 + Float64(t_1 * -0.5)) * Float64(abs(Float64(l * sqrt(0.5))) / t_m)));
	else
		tmp = asin(sqrt(Float64(t_2 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l) / Float64(l / t_m)))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = (Om / Omc) ^ 2.0;
	t_2 = 1.0 - t_1;
	tmp = 0.0;
	if ((t_2 / (1.0 + (2.0 * ((t_m / l) ^ 2.0)))) <= 0.0)
		tmp = asin(((1.0 + (t_1 * -0.5)) * (abs((l * sqrt(0.5))) / t_m)));
	else
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[N[(t$95$2 / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[ArcSin[N[(N[(1.0 + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2)) (+.f64 1 (*.f64 2 (pow.f64 (/.f64 t l) 2)))) < 0.0

   1. Initial program 38.6%

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

   2. Taylor expanded in Om around 0 35.1%

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

      1. associate-*r* 35.1%

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

      2. distribute-rgt1-in 35.1%

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

      3. unpow2 35.1%

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

      4. unpow2 35.1%

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

      5. times-frac 38.6%

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

      6. unpow2 38.6%

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

      7. +-commutative 38.6%

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

      8. fma-def 38.6%

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

      9. unpow2 38.6%

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

      10. unpow2 38.6%

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

      11. times-frac 38.6%

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

      12. unpow2 38.6%

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

   4. Simplified 38.6%

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

   5. Taylor expanded in t around inf 66.0%

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

      1. add-sqr-sqrt 38.4%

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

      2. sqrt-unprod 54.6%

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

      3. pow2 54.6%

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

   7. Applied egg-rr 54.6%

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

      1. unpow2 54.6%

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

      2. rem-sqrt-square 64.5%

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

   9. Simplified 64.5%

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

   if 0.0 < (/.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2)) (+.f64 1 (*.f64 2 (pow.f64 (/.f64 t l) 2))))

   1. Initial program 99.0%

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

      1. unpow2 99.0%

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

      2. clear-num 99.0%

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

      3. un-div-inv 99.0%

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

   3. Applied egg-rr 99.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 0:\\ \;\;\;\;\sin^{-1} \left(\left(1 + {\left(\frac{Om}{Omc}\right)}^{2} \cdot -0.5\right) \cdot \frac{\left|\ell \cdot \sqrt{0.5}\right|}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.8× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
   (hypot 1.0 (/ (* t_m (sqrt 2.0)) l)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, 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$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. sqrt-div 84.1%

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

   2. div-inv 84.1%

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

   3. add-sqr-sqrt 84.1%

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

   4. hypot-1-def 84.1%

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

   5. *-commutative 84.1%

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

   6. sqrt-prod 84.0%

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

   7. unpow2 84.0%

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

   8. sqrt-prod 55.4%

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

   9. add-sqr-sqrt 98.8%

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

3. Applied egg-rr 98.8%

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

   1. associate-*r/ 98.8%

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

   2. *-rgt-identity 98.8%

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

   3. associate-*l/ 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 3: 83.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t_m}{\ell \cdot \frac{\ell}{t_m}}}}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(t_m \cdot \left(\frac{t_m}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ Om (* Omc (/ Omc Om))))))
   (if (<= l -3.5e-305)
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (/ t_m (* l (/ l t_m))))))))
     (if (<= l 2.1e-57)
       (asin (* (sqrt t_1) (/ l (* t_m (sqrt 2.0)))))
       (asin
        (sqrt (/ t_1 (+ 1.0 (* 2.0 (* t_m (* (/ t_m l) (/ 1.0 l))))))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	double tmp;
	if (l <= -3.5e-305) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (t_m / (l * (l / t_m))))))));
	} else if (l <= 2.1e-57) {
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - (om / (omc * (omc / om)))
    if (l <= (-3.5d-305)) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * (t_m / (l * (l / t_m))))))))
    else if (l <= 2.1d-57) then
        tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0d0)))))
    else
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (t_m * ((t_m / l) * (1.0d0 / l))))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	double tmp;
	if (l <= -3.5e-305) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (t_m / (l * (l / t_m))))))));
	} else if (l <= 2.1e-57) {
		tmp = Math.asin((Math.sqrt(t_1) * (l / (t_m * Math.sqrt(2.0)))));
	} else {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - (Om / (Omc * (Omc / Om)))
	tmp = 0
	if l <= -3.5e-305:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (t_m / (l * (l / t_m))))))))
	elif l <= 2.1e-57:
		tmp = math.asin((math.sqrt(t_1) * (l / (t_m * math.sqrt(2.0)))))
	else:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om))))
	tmp = 0.0
	if (l <= -3.5e-305)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(t_m / Float64(l * Float64(l / t_m))))))));
	elseif (l <= 2.1e-57)
		tmp = asin(Float64(sqrt(t_1) * Float64(l / Float64(t_m * sqrt(2.0)))));
	else
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(t_m * Float64(Float64(t_m / l) * Float64(1.0 / l))))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	tmp = 0.0;
	if (l <= -3.5e-305)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * (t_m / (l * (l / t_m))))))));
	elseif (l <= 2.1e-57)
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	else
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.5e-305], 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[(t$95$m / N[(l * N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.1e-57], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4999999999999998e-305

   1. Initial program 86.1%

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

      1. unpow2 86.1%

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

      2. clear-num 86.1%

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

      3. frac-times 84.4%

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

      4. *-un-lft-identity 84.4%

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

   3. Applied egg-rr 84.4%

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

   if -3.4999999999999998e-305 < l < 2.0999999999999999e-57

   1. Initial program 66.4%

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

      1. sqrt-div 66.2%

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

      2. div-inv 66.2%

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

      3. add-sqr-sqrt 66.2%

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

      4. hypot-1-def 66.2%

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

      5. *-commutative 66.2%

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

      6. sqrt-prod 66.3%

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

      7. unpow2 66.3%

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

      8. sqrt-prod 43.3%

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

      9. add-sqr-sqrt 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. associate-*r/ 98.1%

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

      2. *-rgt-identity 98.1%

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

      3. associate-*l/ 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in t around inf 51.3%

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

      1. *-commutative 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

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

      4. times-frac 58.3%

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

      5. unpow2 58.3%

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

   8. Simplified 58.3%

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

      1. unpow2 16.4%

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

      2. clear-num 16.4%

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

      3. frac-times 16.4%

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

      4. *-un-lft-identity 16.4%

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

   10. Applied egg-rr 58.3%

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

   if 2.0999999999999999e-57 < l

   1. Initial program 89.9%

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

      1. unpow2 89.9%

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

      2. div-inv 89.9%

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

      3. associate-*l* 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. unpow2 68.9%

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

      2. clear-num 68.9%

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

      3. frac-times 68.9%

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

      4. *-un-lft-identity 68.9%

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

   5. Applied egg-rr 89.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \]

Alternative 4: 84.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t_m}{\ell}}{\frac{\ell}{t_m}}}}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(t_m \cdot \left(\frac{t_m}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ Om (* Omc (/ Omc Om))))))
   (if (<= l -3.5e-305)
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (/ (/ t_m l) (/ l t_m)))))))
     (if (<= l 2.5e-57)
       (asin (* (sqrt t_1) (/ l (* t_m (sqrt 2.0)))))
       (asin
        (sqrt (/ t_1 (+ 1.0 (* 2.0 (* t_m (* (/ t_m l) (/ 1.0 l))))))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	double tmp;
	if (l <= -3.5e-305) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else if (l <= 2.5e-57) {
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - (om / (omc * (omc / om)))
    if (l <= (-3.5d-305)) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l) / (l / t_m)))))))
    else if (l <= 2.5d-57) then
        tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0d0)))))
    else
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (t_m * ((t_m / l) * (1.0d0 / l))))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	double tmp;
	if (l <= -3.5e-305) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	} else if (l <= 2.5e-57) {
		tmp = Math.asin((Math.sqrt(t_1) * (l / (t_m * Math.sqrt(2.0)))));
	} else {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - (Om / (Omc * (Omc / Om)))
	tmp = 0
	if l <= -3.5e-305:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))))
	elif l <= 2.5e-57:
		tmp = math.asin((math.sqrt(t_1) * (l / (t_m * math.sqrt(2.0)))))
	else:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om))))
	tmp = 0.0
	if (l <= -3.5e-305)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l) / Float64(l / t_m)))))));
	elseif (l <= 2.5e-57)
		tmp = asin(Float64(sqrt(t_1) * Float64(l / Float64(t_m * sqrt(2.0)))));
	else
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(t_m * Float64(Float64(t_m / l) * Float64(1.0 / l))))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	tmp = 0.0;
	if (l <= -3.5e-305)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l) / (l / t_m)))))));
	elseif (l <= 2.5e-57)
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	else
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.5e-305], 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[(N[(t$95$m / l), $MachinePrecision] / N[(l / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.5e-57], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4999999999999998e-305

   1. Initial program 86.1%

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

      1. unpow2 86.1%

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

      2. clear-num 86.1%

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

      3. un-div-inv 86.1%

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

   3. Applied egg-rr 86.1%

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

   if -3.4999999999999998e-305 < l < 2.5000000000000001e-57

   1. Initial program 66.4%

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

      1. sqrt-div 66.2%

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

      2. div-inv 66.2%

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

      3. add-sqr-sqrt 66.2%

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

      4. hypot-1-def 66.2%

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

      5. *-commutative 66.2%

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

      6. sqrt-prod 66.3%

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

      7. unpow2 66.3%

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

      8. sqrt-prod 43.3%

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

      9. add-sqr-sqrt 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. associate-*r/ 98.1%

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

      2. *-rgt-identity 98.1%

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

      3. associate-*l/ 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in t around inf 51.3%

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

      1. *-commutative 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

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

      4. times-frac 58.3%

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

      5. unpow2 58.3%

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

   8. Simplified 58.3%

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

      1. unpow2 16.4%

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

      2. clear-num 16.4%

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

      3. frac-times 16.4%

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

      4. *-un-lft-identity 16.4%

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

   10. Applied egg-rr 58.3%

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

   if 2.5000000000000001e-57 < l

   1. Initial program 89.9%

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

      1. unpow2 89.9%

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

      2. div-inv 89.9%

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

      3. associate-*l* 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. unpow2 68.9%

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

      2. clear-num 68.9%

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

      3. frac-times 68.9%

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

      4. *-un-lft-identity 68.9%

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

   5. Applied egg-rr 89.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \]

Alternative 5: 83.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{t_m \cdot \frac{t_m}{\ell}}{\ell}}}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t_1} \cdot \frac{\ell}{t_m \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \left(t_m \cdot \left(\frac{t_m}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ Om (* Omc (/ Omc Om))))))
   (if (<= l -3.5e-305)
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (* t_m (/ t_m l)) l))))))
     (if (<= l 2.7e-57)
       (asin (* (sqrt t_1) (/ l (* t_m (sqrt 2.0)))))
       (asin
        (sqrt (/ t_1 (+ 1.0 (* 2.0 (* t_m (* (/ t_m l) (/ 1.0 l))))))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	double tmp;
	if (l <= -3.5e-305) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	} else if (l <= 2.7e-57) {
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - (om / (omc * (omc / om)))
    if (l <= (-3.5d-305)) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m * (t_m / l)) / l))))))
    else if (l <= 2.7d-57) then
        tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0d0)))))
    else
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (t_m * ((t_m / l) * (1.0d0 / l))))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	double tmp;
	if (l <= -3.5e-305) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	} else if (l <= 2.7e-57) {
		tmp = Math.asin((Math.sqrt(t_1) * (l / (t_m * Math.sqrt(2.0)))));
	} else {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 - (Om / (Omc * (Omc / Om)))
	tmp = 0
	if l <= -3.5e-305:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))))
	elif l <= 2.7e-57:
		tmp = math.asin((math.sqrt(t_1) * (l / (t_m * math.sqrt(2.0)))))
	else:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om))))
	tmp = 0.0
	if (l <= -3.5e-305)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l)) / l))))));
	elseif (l <= 2.7e-57)
		tmp = asin(Float64(sqrt(t_1) * Float64(l / Float64(t_m * sqrt(2.0)))));
	else
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(t_m * Float64(Float64(t_m / l) * Float64(1.0 / l))))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 - (Om / (Omc * (Omc / Om)));
	tmp = 0.0;
	if (l <= -3.5e-305)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	elseif (l <= 2.7e-57)
		tmp = asin((sqrt(t_1) * (l / (t_m * sqrt(2.0)))));
	else
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.5e-305], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.7e-57], N[ArcSin[N[(N[Sqrt[t$95$1], $MachinePrecision] * N[(l / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4999999999999998e-305

   1. Initial program 86.1%

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

      1. unpow2 86.1%

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

      2. associate-*r/ 83.8%

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

   3. Applied egg-rr 83.8%

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

      1. unpow2 61.2%

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

      2. clear-num 61.2%

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

      3. frac-times 61.2%

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

      4. *-un-lft-identity 61.2%

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

   5. Applied egg-rr 83.8%

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

   if -3.4999999999999998e-305 < l < 2.7000000000000002e-57

   1. Initial program 66.4%

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

      1. sqrt-div 66.2%

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

      2. div-inv 66.2%

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

      3. add-sqr-sqrt 66.2%

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

      4. hypot-1-def 66.2%

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

      5. *-commutative 66.2%

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

      6. sqrt-prod 66.3%

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

      7. unpow2 66.3%

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

      8. sqrt-prod 43.3%

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

      9. add-sqr-sqrt 98.1%

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

   3. Applied egg-rr 98.1%

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

      1. associate-*r/ 98.1%

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

      2. *-rgt-identity 98.1%

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

      3. associate-*l/ 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in t around inf 51.3%

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

      1. *-commutative 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

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

      4. times-frac 58.3%

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

      5. unpow2 58.3%

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

   8. Simplified 58.3%

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

      1. unpow2 16.4%

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

      2. clear-num 16.4%

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

      3. frac-times 16.4%

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

      4. *-un-lft-identity 16.4%

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

   10. Applied egg-rr 58.3%

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

   if 2.7000000000000002e-57 < l

   1. Initial program 89.9%

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

      1. unpow2 89.9%

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

      2. div-inv 89.9%

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

      3. associate-*l* 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. unpow2 68.9%

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

      2. clear-num 68.9%

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

      3. frac-times 68.9%

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

      4. *-un-lft-identity 68.9%

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

   5. Applied egg-rr 89.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{elif}\;\ell \leq 2.7 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}} \cdot \frac{\ell}{t \cdot \sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \]

Alternative 6: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{Om}{Omc \cdot \frac{Omc}{Om}}\\ t_2 := 1 - t_1\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_2}{1 + 2 \cdot \frac{t_m \cdot \frac{t_m}{\ell}}{\ell}}}\right)\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + -0.5 \cdot t_1\right) \cdot \frac{\ell \cdot \sqrt{0.5}}{t_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_2}{1 + 2 \cdot \left(t_m \cdot \left(\frac{t_m}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (/ Om (* Omc (/ Omc Om)))) (t_2 (- 1.0 t_1)))
   (if (<= l -3.5e-305)
     (asin (sqrt (/ t_2 (+ 1.0 (* 2.0 (/ (* t_m (/ t_m l)) l))))))
     (if (<= l 1.7e-57)
       (asin (* (+ 1.0 (* -0.5 t_1)) (/ (* l (sqrt 0.5)) t_m)))
       (asin
        (sqrt (/ t_2 (+ 1.0 (* 2.0 (* t_m (* (/ t_m l) (/ 1.0 l))))))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = Om / (Omc * (Omc / Om));
	double t_2 = 1.0 - t_1;
	double tmp;
	if (l <= -3.5e-305) {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	} else if (l <= 1.7e-57) {
		tmp = asin(((1.0 + (-0.5 * t_1)) * ((l * sqrt(0.5)) / t_m)));
	} else {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = om / (omc * (omc / om))
    t_2 = 1.0d0 - t_1
    if (l <= (-3.5d-305)) then
        tmp = asin(sqrt((t_2 / (1.0d0 + (2.0d0 * ((t_m * (t_m / l)) / l))))))
    else if (l <= 1.7d-57) then
        tmp = asin(((1.0d0 + ((-0.5d0) * t_1)) * ((l * sqrt(0.5d0)) / t_m)))
    else
        tmp = asin(sqrt((t_2 / (1.0d0 + (2.0d0 * (t_m * ((t_m / l) * (1.0d0 / l))))))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = Om / (Omc * (Omc / Om));
	double t_2 = 1.0 - t_1;
	double tmp;
	if (l <= -3.5e-305) {
		tmp = Math.asin(Math.sqrt((t_2 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	} else if (l <= 1.7e-57) {
		tmp = Math.asin(((1.0 + (-0.5 * t_1)) * ((l * Math.sqrt(0.5)) / t_m)));
	} else {
		tmp = Math.asin(Math.sqrt((t_2 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = Om / (Omc * (Omc / Om))
	t_2 = 1.0 - t_1
	tmp = 0
	if l <= -3.5e-305:
		tmp = math.asin(math.sqrt((t_2 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))))
	elif l <= 1.7e-57:
		tmp = math.asin(((1.0 + (-0.5 * t_1)) * ((l * math.sqrt(0.5)) / t_m)))
	else:
		tmp = math.asin(math.sqrt((t_2 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(Om / Float64(Omc * Float64(Omc / Om)))
	t_2 = Float64(1.0 - t_1)
	tmp = 0.0
	if (l <= -3.5e-305)
		tmp = asin(sqrt(Float64(t_2 / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l)) / l))))));
	elseif (l <= 1.7e-57)
		tmp = asin(Float64(Float64(1.0 + Float64(-0.5 * t_1)) * Float64(Float64(l * sqrt(0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(t_2 / Float64(1.0 + Float64(2.0 * Float64(t_m * Float64(Float64(t_m / l) * Float64(1.0 / l))))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = Om / (Omc * (Omc / Om));
	t_2 = 1.0 - t_1;
	tmp = 0.0;
	if (l <= -3.5e-305)
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	elseif (l <= 1.7e-57)
		tmp = asin(((1.0 + (-0.5 * t_1)) * ((l * sqrt(0.5)) / t_m)));
	else
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * (t_m * ((t_m / l) * (1.0 / l))))))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[l, -3.5e-305], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.7e-57], N[ArcSin[N[(N[(1.0 + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[(t$95$m * N[(N[(t$95$m / l), $MachinePrecision] * N[(1.0 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.4999999999999998e-305

   1. Initial program 86.1%

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

      1. unpow2 86.1%

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

      2. associate-*r/ 83.8%

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

   3. Applied egg-rr 83.8%

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

      1. unpow2 61.2%

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

      2. clear-num 61.2%

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

      3. frac-times 61.2%

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

      4. *-un-lft-identity 61.2%

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

   5. Applied egg-rr 83.8%

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

   if -3.4999999999999998e-305 < l < 1.70000000000000008e-57

   1. Initial program 66.4%

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

   2. Taylor expanded in Om around 0 41.6%

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

      1. associate-*r* 41.6%

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

      2. distribute-rgt1-in 41.6%

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

      3. unpow2 41.6%

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

      4. unpow2 41.6%

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

      5. times-frac 46.4%

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

      6. unpow2 46.4%

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

      7. +-commutative 46.4%

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

      8. fma-def 46.4%

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

      9. unpow2 46.4%

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

      10. unpow2 46.4%

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

      11. times-frac 66.4%

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

      12. unpow2 66.4%

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

   4. Simplified 66.4%

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

   5. Taylor expanded in t around inf 58.1%

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

      1. unpow2 16.4%

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

      2. clear-num 16.4%

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

      3. frac-times 16.4%

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

      4. *-un-lft-identity 16.4%

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

   7. Applied egg-rr 58.1%

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

   if 1.70000000000000008e-57 < l

   1. Initial program 89.9%

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

      1. unpow2 89.9%

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

      2. div-inv 89.9%

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

      3. associate-*l* 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. unpow2 68.9%

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

      2. clear-num 68.9%

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

      3. frac-times 68.9%

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

      4. *-un-lft-identity 68.9%

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

   5. Applied egg-rr 89.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{-305}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-57}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right) \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \left(t \cdot \left(\frac{t}{\ell} \cdot \frac{1}{\ell}\right)\right)}}\right)\\ \end{array} \]

Alternative 7: 81.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{Om}{Omc \cdot \frac{Omc}{Om}}\\ \mathbf{if}\;t_m \leq 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - t_1}{1 + 2 \cdot \frac{t_m \cdot \frac{t_m}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + -0.5 \cdot t_1\right) \cdot \frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (/ Om (* Omc (/ Omc Om)))))
   (if (<= t_m 1e+222)
     (asin (sqrt (/ (- 1.0 t_1) (+ 1.0 (* 2.0 (/ (* t_m (/ t_m l)) l))))))
     (asin (* (+ 1.0 (* -0.5 t_1)) (/ l (/ t_m (sqrt 0.5))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = Om / (Omc * (Omc / Om));
	double tmp;
	if (t_m <= 1e+222) {
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	} else {
		tmp = asin(((1.0 + (-0.5 * t_1)) * (l / (t_m / sqrt(0.5)))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = om / (omc * (omc / om))
    if (t_m <= 1d+222) then
        tmp = asin(sqrt(((1.0d0 - t_1) / (1.0d0 + (2.0d0 * ((t_m * (t_m / l)) / l))))))
    else
        tmp = asin(((1.0d0 + ((-0.5d0) * t_1)) * (l / (t_m / sqrt(0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = Om / (Omc * (Omc / Om));
	double tmp;
	if (t_m <= 1e+222) {
		tmp = Math.asin(Math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	} else {
		tmp = Math.asin(((1.0 + (-0.5 * t_1)) * (l / (t_m / Math.sqrt(0.5)))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = Om / (Omc * (Omc / Om))
	tmp = 0
	if t_m <= 1e+222:
		tmp = math.asin(math.sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))))
	else:
		tmp = math.asin(((1.0 + (-0.5 * t_1)) * (l / (t_m / math.sqrt(0.5)))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(Om / Float64(Omc * Float64(Omc / Om)))
	tmp = 0.0
	if (t_m <= 1e+222)
		tmp = asin(sqrt(Float64(Float64(1.0 - t_1) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l)) / l))))));
	else
		tmp = asin(Float64(Float64(1.0 + Float64(-0.5 * t_1)) * Float64(l / Float64(t_m / sqrt(0.5)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = Om / (Omc * (Omc / Om));
	tmp = 0.0;
	if (t_m <= 1e+222)
		tmp = asin(sqrt(((1.0 - t_1) / (1.0 + (2.0 * ((t_m * (t_m / l)) / l))))));
	else
		tmp = asin(((1.0 + (-0.5 * t_1)) * (l / (t_m / sqrt(0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$m, 1e+222], N[ArcSin[N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1e222

   1. Initial program 83.8%

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

      1. unpow2 83.8%

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

      2. associate-*r/ 83.0%

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

   3. Applied egg-rr 83.0%

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

      1. unpow2 60.2%

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

      2. clear-num 60.2%

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

      3. frac-times 60.2%

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

      4. *-un-lft-identity 60.2%

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

   5. Applied egg-rr 83.0%

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

   if 1e222 < t

   1. Initial program 86.8%

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

   2. Taylor expanded in Om around 0 46.6%

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

      1. associate-*r* 46.6%

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

      2. distribute-rgt1-in 46.6%

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

      3. unpow2 46.6%

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

      4. unpow2 46.6%

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

      5. times-frac 46.6%

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

      6. unpow2 46.6%

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

      7. +-commutative 46.6%

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

      8. fma-def 46.6%

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

      9. unpow2 46.6%

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

      10. unpow2 46.6%

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

      11. times-frac 86.9%

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

      12. unpow2 86.9%

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

   4. Simplified 86.9%

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

   5. Taylor expanded in t around inf 79.2%

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

      1. associate-/l* 79.1%

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

   7. Simplified 79.1%

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

      1. unpow2 19.2%

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

      2. clear-num 19.2%

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

      3. frac-times 19.2%

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

      4. *-un-lft-identity 19.2%

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

   9. Applied egg-rr 79.1%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right) \cdot \frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 8: 61.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-300} \lor \neg \left(\ell \leq 6.4 \cdot 10^{+60}\right):\\ \;\;\;\;\sin^{-1} t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\ell}{\frac{t_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (+ 1.0 (* -0.5 (/ Om (* Omc (/ Omc Om)))))))
   (if (or (<= l -1.45e-300) (not (<= l 6.4e+60)))
     (asin t_1)
     (asin (* t_1 (/ l (/ t_m (sqrt 0.5))))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))));
	double tmp;
	if ((l <= -1.45e-300) || !(l <= 6.4e+60)) {
		tmp = asin(t_1);
	} else {
		tmp = asin((t_1 * (l / (t_m / sqrt(0.5)))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + ((-0.5d0) * (om / (omc * (omc / om))))
    if ((l <= (-1.45d-300)) .or. (.not. (l <= 6.4d+60))) then
        tmp = asin(t_1)
    else
        tmp = asin((t_1 * (l / (t_m / sqrt(0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))));
	double tmp;
	if ((l <= -1.45e-300) || !(l <= 6.4e+60)) {
		tmp = Math.asin(t_1);
	} else {
		tmp = Math.asin((t_1 * (l / (t_m / Math.sqrt(0.5)))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))))
	tmp = 0
	if (l <= -1.45e-300) or not (l <= 6.4e+60):
		tmp = math.asin(t_1)
	else:
		tmp = math.asin((t_1 * (l / (t_m / math.sqrt(0.5)))))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 + Float64(-0.5 * Float64(Om / Float64(Omc * Float64(Omc / Om)))))
	tmp = 0.0
	if ((l <= -1.45e-300) || !(l <= 6.4e+60))
		tmp = asin(t_1);
	else
		tmp = asin(Float64(t_1 * Float64(l / Float64(t_m / sqrt(0.5)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))));
	tmp = 0.0;
	if ((l <= -1.45e-300) || ~((l <= 6.4e+60)))
		tmp = asin(t_1);
	else
		tmp = asin((t_1 * (l / (t_m / sqrt(0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[l, -1.45e-300], N[Not[LessEqual[l, 6.4e+60]], $MachinePrecision]], N[ArcSin[t$95$1], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[(l / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.44999999999999996e-300 or 6.39999999999999982e60 < l

   1. Initial program 89.7%

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

   2. Taylor expanded in Om around 0 57.9%

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

      1. associate-*r* 57.9%

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

      2. distribute-rgt1-in 57.9%

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

      3. unpow2 57.9%

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

      4. unpow2 57.9%

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

      5. times-frac 67.8%

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

      6. unpow2 67.8%

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

      7. +-commutative 67.8%

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

      8. fma-def 67.8%

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

      9. unpow2 67.8%

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

      10. unpow2 67.8%

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

      11. times-frac 89.8%

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

      12. unpow2 89.8%

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

   4. Simplified 89.8%

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

   5. Taylor expanded in t around 0 68.1%

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

      1. unpow2 68.1%

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

      2. clear-num 68.1%

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

      3. frac-times 68.1%

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

      4. *-un-lft-identity 68.1%

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

   7. Applied egg-rr 68.1%

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

   if -1.44999999999999996e-300 < l < 6.39999999999999982e60

   1. Initial program 70.6%

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

   2. Taylor expanded in Om around 0 52.3%

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

      1. associate-*r* 52.3%

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

      2. distribute-rgt1-in 52.3%

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

      3. unpow2 52.3%

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

      4. unpow2 52.3%

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

      5. times-frac 57.7%

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

      6. unpow2 57.7%

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

      7. +-commutative 57.7%

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

      8. fma-def 57.7%

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

      9. unpow2 57.7%

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

      10. unpow2 57.7%

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

      11. times-frac 70.6%

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

      12. unpow2 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in t around inf 49.6%

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

      1. associate-/l* 49.7%

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

   7. Simplified 49.7%

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

      1. unpow2 28.0%

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

      2. clear-num 28.0%

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

      3. frac-times 28.0%

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

      4. *-un-lft-identity 28.0%

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

   9. Applied egg-rr 49.7%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-300} \lor \neg \left(\ell \leq 6.4 \cdot 10^{+60}\right):\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right) \cdot \frac{\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]

Alternative 9: 61.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\\ \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-300} \lor \neg \left(\ell \leq 5 \cdot 10^{+60}\right):\\ \;\;\;\;\sin^{-1} t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \frac{\ell \cdot \sqrt{0.5}}{t_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (let* ((t_1 (+ 1.0 (* -0.5 (/ Om (* Omc (/ Omc Om)))))))
   (if (or (<= l -1.45e-300) (not (<= l 5e+60)))
     (asin t_1)
     (asin (* t_1 (/ (* l (sqrt 0.5)) t_m))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))));
	double tmp;
	if ((l <= -1.45e-300) || !(l <= 5e+60)) {
		tmp = asin(t_1);
	} else {
		tmp = asin((t_1 * ((l * sqrt(0.5)) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 + ((-0.5d0) * (om / (omc * (omc / om))))
    if ((l <= (-1.45d-300)) .or. (.not. (l <= 5d+60))) then
        tmp = asin(t_1)
    else
        tmp = asin((t_1 * ((l * sqrt(0.5d0)) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	double t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))));
	double tmp;
	if ((l <= -1.45e-300) || !(l <= 5e+60)) {
		tmp = Math.asin(t_1);
	} else {
		tmp = Math.asin((t_1 * ((l * Math.sqrt(0.5)) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))))
	tmp = 0
	if (l <= -1.45e-300) or not (l <= 5e+60):
		tmp = math.asin(t_1)
	else:
		tmp = math.asin((t_1 * ((l * math.sqrt(0.5)) / t_m)))
	return tmp

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	t_1 = Float64(1.0 + Float64(-0.5 * Float64(Om / Float64(Omc * Float64(Omc / Om)))))
	tmp = 0.0
	if ((l <= -1.45e-300) || !(l <= 5e+60))
		tmp = asin(t_1);
	else
		tmp = asin(Float64(t_1 * Float64(Float64(l * sqrt(0.5)) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
function tmp_2 = code(t_m, l, Om, Omc)
	t_1 = 1.0 + (-0.5 * (Om / (Omc * (Omc / Om))));
	tmp = 0.0;
	if ((l <= -1.45e-300) || ~((l <= 5e+60)))
		tmp = asin(t_1);
	else
		tmp = asin((t_1 * ((l * sqrt(0.5)) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 + N[(-0.5 * N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[l, -1.45e-300], N[Not[LessEqual[l, 5e+60]], $MachinePrecision]], N[ArcSin[t$95$1], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.44999999999999996e-300 or 4.99999999999999975e60 < l

   1. Initial program 89.7%

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

   2. Taylor expanded in Om around 0 57.9%

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

      1. associate-*r* 57.9%

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

      2. distribute-rgt1-in 57.9%

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

      3. unpow2 57.9%

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

      4. unpow2 57.9%

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

      5. times-frac 67.8%

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

      6. unpow2 67.8%

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

      7. +-commutative 67.8%

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

      8. fma-def 67.8%

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

      9. unpow2 67.8%

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

      10. unpow2 67.8%

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

      11. times-frac 89.8%

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

      12. unpow2 89.8%

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

   4. Simplified 89.8%

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

   5. Taylor expanded in t around 0 68.1%

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

      1. unpow2 68.1%

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

      2. clear-num 68.1%

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

      3. frac-times 68.1%

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

      4. *-un-lft-identity 68.1%

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

   7. Applied egg-rr 68.1%

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

   if -1.44999999999999996e-300 < l < 4.99999999999999975e60

   1. Initial program 70.6%

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

   2. Taylor expanded in Om around 0 52.3%

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

      1. associate-*r* 52.3%

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

      2. distribute-rgt1-in 52.3%

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

      3. unpow2 52.3%

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

      4. unpow2 52.3%

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

      5. times-frac 57.7%

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

      6. unpow2 57.7%

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

      7. +-commutative 57.7%

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

      8. fma-def 57.7%

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

      9. unpow2 57.7%

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

      10. unpow2 57.7%

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

      11. times-frac 70.6%

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

      12. unpow2 70.6%

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

   4. Simplified 70.6%

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

   5. Taylor expanded in t around inf 49.6%

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

      1. unpow2 28.0%

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

      2. clear-num 28.0%

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

      3. frac-times 28.0%

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

      4. *-un-lft-identity 28.0%

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

   7. Applied egg-rr 49.6%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{-300} \lor \neg \left(\ell \leq 5 \cdot 10^{+60}\right):\\ \;\;\;\;\sin^{-1} \left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right) \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Alternative 10: 50.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ \sin^{-1} \left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
(FPCore (t_m l Om Omc)
 :precision binary64
 (asin (+ 1.0 (* -0.5 (/ Om (* Omc (/ Omc Om)))))))

C

t_m = fabs(t);
double code(double t_m, double l, double Om, double Omc) {
	return asin((1.0 + (-0.5 * (Om / (Omc * (Omc / Om))))));
}

Fortran

t_m = abs(t)
real(8) function code(t_m, l, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin((1.0d0 + ((-0.5d0) * (om / (omc * (omc / om))))))
end function

Java

t_m = Math.abs(t);
public static double code(double t_m, double l, double Om, double Omc) {
	return Math.asin((1.0 + (-0.5 * (Om / (Omc * (Omc / Om))))));
}

Python

t_m = math.fabs(t)
def code(t_m, l, Om, Omc):
	return math.asin((1.0 + (-0.5 * (Om / (Omc * (Omc / Om))))))

Julia

t_m = abs(t)
function code(t_m, l, Om, Omc)
	return asin(Float64(1.0 + Float64(-0.5 * Float64(Om / Float64(Omc * Float64(Omc / Om))))))
end

MATLAB

t_m = abs(t);
function tmp = code(t_m, l, Om, Omc)
	tmp = asin((1.0 + (-0.5 * (Om / (Omc * (Omc / Om))))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l_, Om_, Omc_] := N[ArcSin[N[(1.0 + N[(-0.5 * N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in Om around 0 56.3%

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

   1. associate-*r* 56.3%

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

   2. distribute-rgt1-in 56.3%

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

   3. unpow2 56.3%

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

   4. unpow2 56.3%

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

   5. times-frac 64.9%

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

   6. unpow2 64.9%

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

   7. +-commutative 64.9%

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

   8. fma-def 64.9%

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

   9. unpow2 64.9%

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

   10. unpow2 64.9%

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

   11. times-frac 84.1%

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

   12. unpow2 84.1%

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

4. Simplified 84.1%

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

5. Taylor expanded in t around 0 56.4%

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

   1. unpow2 56.4%

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

   2. clear-num 56.4%

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

   3. frac-times 56.4%

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

   4. *-un-lft-identity 56.4%

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

7. Applied egg-rr 56.4%

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

8. Final simplification 56.4%

   \[\leadsto \sin^{-1} \left(1 + -0.5 \cdot \frac{Om}{Omc \cdot \frac{Omc}{Om}}\right) \]

Reproduce

herbie shell --seed 2023337 
(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)))))))