Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.7%

Time: 11.8s

Alternatives: 6

Speedup: 2.3×

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: 84.0% 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, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \mathsf{fma}\left(\frac{l\_m}{{t\_m}^{3}} \cdot \frac{l\_m}{\sqrt{0.5}}, -0.125, \frac{\sqrt{0.5}}{t\_m}\right)\right) \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1
         (/
          (- 1.0 (pow (/ Om Omc) 2.0))
          (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
   (if (<= t_1 0.0)
     (asin
      (*
       (*
        (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))
        (fma
         (* (/ l_m (pow t_m 3.0)) (/ l_m (sqrt 0.5)))
         -0.125
         (/ (sqrt 0.5) t_m)))
       l_m))
     (asin (sqrt t_1)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = (1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * fma(((l_m / pow(t_m, 3.0)) * (l_m / sqrt(0.5))), -0.125, (sqrt(0.5) / t_m))) * l_m));
	} else {
		tmp = asin(sqrt(t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = asin(Float64(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * fma(Float64(Float64(l_m / (t_m ^ 3.0)) * Float64(l_m / sqrt(0.5))), -0.125, Float64(sqrt(0.5) / t_m))) * l_m));
	else
		tmp = asin(sqrt(t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[ArcSin[N[(N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(l$95$m / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[t$95$1], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 0.0

   1. Initial program 40.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

   5. Applied rewrites 67.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \mathsf{fma}\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\sqrt{0.5}}, -0.125, \frac{\sqrt{0.5}}{t}\right)\right) \cdot \ell\right)} \]

   if 0.0 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   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. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
   (if (<=
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
        0.1)
     (asin (* (/ (* (sqrt 0.5) l_m) t_m) t_1))
     (asin t_1))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 0.1) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * t_1));
	} else {
		tmp = asin(t_1);
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((1.0d0 - ((om / omc) * (om / omc))))
    if (((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))) <= 0.1d0) then
        tmp = asin((((sqrt(0.5d0) * l_m) / t_m) * t_1))
    else
        tmp = asin(t_1)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	double tmp;
	if (((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))) <= 0.1) {
		tmp = Math.asin((((Math.sqrt(0.5) * l_m) / t_m) * t_1));
	} else {
		tmp = Math.asin(t_1);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	t_1 = math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))
	tmp = 0
	if ((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))) <= 0.1:
		tmp = math.asin((((math.sqrt(0.5) * l_m) / t_m) * t_1))
	else:
		tmp = math.asin(t_1)
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))
	tmp = 0.0
	if (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 0.1)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * t_1));
	else
		tmp = asin(t_1);
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	tmp = 0.0;
	if (((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))) <= 0.1)
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * t_1));
	else
		tmp = asin(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.1], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[ArcSin[t$95$1], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 67.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if 0.10000000000000001 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 98.8%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 3: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
      0.1)
   (asin (* l_m (/ (sqrt 0.5) t_m)))
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 0.1) {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))) <= 0.1d0) then
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    else
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))) <= 0.1) {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if ((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))) <= 0.1:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 0.1)
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))) <= 0.1)
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	else
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.1], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 67.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 57.7%

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

   10. Applied rewrites 57.7%

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

   if 0.10000000000000001 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 98.8%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.0)
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))
   (asin (* (* (/ (sin (acos (/ Om Omc))) t_m) (sqrt 0.5)) l_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin((((sin(acos((Om / Omc))) / t_m) * sqrt(0.5)) * l_m));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 1.0d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    else
        tmp = asin((((sin(acos((om / omc))) / t_m) * sqrt(0.5d0)) * l_m))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin((((Math.sin(Math.acos((Om / Omc))) / t_m) * Math.sqrt(0.5)) * l_m));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1.0:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin((((math.sin(math.acos((Om / Omc))) / t_m) * math.sqrt(0.5)) * l_m))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.0)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(Float64(Float64(Float64(sin(acos(Float64(Om / Omc))) / t_m) * sqrt(0.5)) * l_m));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1.0)
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin((((sin(acos((Om / Omc))) / t_m) * sqrt(0.5)) * l_m));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.0], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[Sin[N[ArcCos[N[(Om / Omc), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.0%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 1 < (/.f64 t l)

   1. Initial program 67.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.1%

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

Alternative 5: 91.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = asin(sqrt((1.0 - ((Om * Om) / (Omc * Omc)))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.0], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.0%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 37.2%

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

   9. Taylor expanded in t around 0

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 58.7

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

   11. Applied rewrites 58.7%

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

   if 1 < (/.f64 t l)

   1. Initial program 67.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 98.1%

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

   10. Applied rewrites 98.2%

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

Alternative 6: 48.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc) :precision binary64 (asin (* l_m (/ (sqrt 0.5) t_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(l_m * Float64(sqrt(0.5) / t_m)))
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((l_m * (sqrt(0.5) / t_m)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in t around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower--.f64 N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 29.1

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

5. Applied rewrites 29.1%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 29.1%

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

10. Applied rewrites 29.1%

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

Reproduce

herbie shell --seed 2024332 
(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)))))))