Toniolo and Linder, Equation (2)

Percentage Accurate: 84.4% → 97.1%

Time: 17.3s

Alternatives: 16

Speedup: 2.0×

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: 84.4% 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: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{l\_m \cdot \left(l\_m \cdot l\_m\right)}{t\_m \cdot \left(t\_m \cdot \sqrt{0.5}\right)}, l\_m \cdot \sqrt{0.5}\right)}{t\_m}\right)\\ \end{array} \end{array} \]

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) 4e+142)
   (asin
    (sqrt
     (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
   (asin
    (/
     (*
      (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
      (fma
       -0.125
       (/ (* l_m (* l_m l_m)) (* t_m (* t_m (sqrt 0.5))))
       (* 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) <= 4e+142) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0))))));
	} else {
		tmp = asin(((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * fma(-0.125, ((l_m * (l_m * l_m)) / (t_m * (t_m * sqrt(0.5)))), (l_m * 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) <= 4e+142)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * fma(-0.125, Float64(Float64(l_m * Float64(l_m * l_m)) / Float64(t_m * Float64(t_m * sqrt(0.5)))), Float64(l_m * sqrt(0.5)))) / t_m));
	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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 4e+142], 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$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(l$95$m * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.0000000000000002e142

   1. Initial program 90.2%

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

   if 4.0000000000000002e142 < (/.f64 t l)

   1. Initial program 39.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 3.7%

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]

   9. Taylor expanded in t around inf

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

       1. lower-/.f64 N/A

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

   11. Applied rewrites 92.9%

       \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{\ell \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(t \cdot \sqrt{0.5}\right)}, \ell \cdot \sqrt{0.5}\right)}{t}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t\_1}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 2 \cdot 10^{-204}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot \frac{\frac{\frac{t\_m}{l\_m}}{l\_m}}{\frac{1}{t\_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_1 (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0)))) 2e-204)
     (asin
      (*
       (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
       (/ (* l_m (sqrt 0.5)) t_m)))
     (asin
      (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ (/ t_m l_m) l_m) (/ 1.0 t_m))))))))))

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);
	double tmp;
	if ((t_1 / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 2e-204) {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * sqrt(0.5)) / t_m)));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (((t_m / l_m) / l_m) / (1.0 / t_m)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_1 / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 2e-204)
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(Float64(t_m / l_m) / l_m) / Float64(1.0 / t_m)))))));
	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[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-204], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] / N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $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))))) < 2e-204

   1. Initial program 53.9%

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

   3. Taylor expanded in t around 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. 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 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 24.5%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 59.4

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

   11. Applied rewrites 59.4%

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

   if 2e-204 < (/.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 98.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. div-inv N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. associate-*r/ N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*r/ N/A

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

      16. associate-/r/ N/A

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

      17. clear-num N/A

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

      18. lift-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

Alternative 3: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t\_1}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 2 \cdot 10^{-193}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot \frac{t\_m}{l\_m \cdot \frac{l\_m}{t\_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t_1 (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0)))) 2e-193)
     (asin
      (*
       (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
       (/ (* l_m (sqrt 0.5)) t_m)))
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ t_m (* l_m (/ l_m t_m)))))))))))

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);
	double tmp;
	if ((t_1 / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 2e-193) {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * sqrt(0.5)) / t_m)));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (t_m / (l_m * (l_m / t_m))))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_1 / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 2e-193)
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(t_m / Float64(l_m * Float64(l_m / t_m))))))));
	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[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-193], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(t$95$m / N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < 2.0000000000000001e-193

   1. Initial program 55.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 24.0%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if 2.0000000000000001e-193 < (/.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 98.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. unpow1 N/A

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

      7. sqr-pow N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 58.0

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

   4. Applied rewrites 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. un-div-inv N/A

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

      11. lift-/.f64 N/A

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

      12. clear-num N/A

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

      13. clear-num N/A

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

      14. frac-times N/A

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

      15. metadata-eval N/A

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

      16. associate-/r/ N/A

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

      17. remove-double-div N/A

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

      18. lower-/.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 97.0

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

   6. Applied rewrites 97.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 2 \cdot 10^{-193}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t\_1}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 2 \cdot 10^{-193}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{fma}\left(t\_m \cdot 2, \frac{\frac{t\_m}{l\_m}}{l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

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))))
   (if (<= (/ t_1 (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0)))) 2e-193)
     (asin
      (*
       (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
       (/ (* l_m (sqrt 0.5)) t_m)))
     (asin (sqrt (/ t_1 (fma (* t_m 2.0) (/ (/ t_m l_m) l_m) 1.0)))))))

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);
	double tmp;
	if ((t_1 / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 2e-193) {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * sqrt(0.5)) / t_m)));
	} else {
		tmp = asin(sqrt((t_1 / fma((t_m * 2.0), ((t_m / l_m) / l_m), 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t_1 / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 2e-193)
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(t_1 / fma(Float64(t_m * 2.0), Float64(Float64(t_m / l_m) / l_m), 1.0))));
	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[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-193], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $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))))) < 2.0000000000000001e-193

   1. Initial program 55.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 24.0%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 59.2

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

   11. Applied rewrites 59.2%

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

   if 2.0000000000000001e-193 < (/.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 98.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. associate-/r/ N/A

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

      15. clear-num N/A

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

      16. lift-/.f64 N/A

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

      17. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 2 \cdot 10^{-193}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(t \cdot 2, \frac{\frac{t}{\ell}}{\ell}, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{\frac{t\_m}{l\_m}}\\ \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + 2 \cdot \left(\frac{t\_1}{l\_m} \cdot \frac{t\_1}{\frac{1}{t\_m}}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (sqrt (/ t_m l_m))))
   (if (<=
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
        4e-21)
     (asin
      (*
       (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
       (/ (* l_m (sqrt 0.5)) t_m)))
     (asin
      (sqrt
       (/
        (- 1.0 (/ (* Om (/ Om Omc)) Omc))
        (+ 1.0 (* 2.0 (* (/ t_1 l_m) (/ t_1 (/ 1.0 t_m)))))))))))

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((t_m / l_m));
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 4e-21) {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * sqrt(0.5)) / t_m)));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 + (2.0 * ((t_1 / l_m) * (t_1 / (1.0 / t_m))))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = sqrt(Float64(t_m / l_m))
	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)))) <= 4e-21)
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_1 / l_m) * Float64(t_1 / Float64(1.0 / t_m))))))));
	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[Sqrt[N[(t$95$m / l$95$m), $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], 4e-21], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$1 / l$95$m), $MachinePrecision] * N[(t$95$1 / N[(1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < 3.99999999999999963e-21

   1. Initial program 69.3%

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

   3. Taylor expanded in 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. 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 4.5

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 17.0%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 59.5

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

   11. Applied rewrites 59.5%

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

   if 3.99999999999999963e-21 < (/.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 98.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. unpow1 N/A

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

      7. sqr-pow N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 58.2

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

   4. Applied rewrites 58.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 58.2

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

   6. Applied rewrites 58.2%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 + 2 \cdot \left(\frac{\sqrt{\frac{t}{\ell}}}{\ell} \cdot \frac{\sqrt{\frac{t}{\ell}}}{\frac{1}{t}}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{\frac{t\_m}{l\_m}}\\ \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{\mathsf{fma}\left(\left(t\_m \cdot 2\right) \cdot t\_1, \frac{t\_1}{l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (sqrt (/ t_m l_m))))
   (if (<=
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
        5e-20)
     (asin
      (*
       (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
       (/ (* l_m (sqrt 0.5)) t_m)))
     (asin
      (sqrt
       (/
        (- 1.0 (/ (* Om (/ Om Omc)) Omc))
        (fma (* (* t_m 2.0) t_1) (/ t_1 l_m) 1.0)))))))

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((t_m / l_m));
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 5e-20) {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * sqrt(0.5)) / t_m)));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / fma(((t_m * 2.0) * t_1), (t_1 / l_m), 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = sqrt(Float64(t_m / l_m))
	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)))) <= 5e-20)
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / fma(Float64(Float64(t_m * 2.0) * t_1), Float64(t_1 / l_m), 1.0))));
	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[Sqrt[N[(t$95$m / l$95$m), $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], 5e-20], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$1 / l$95$m), $MachinePrecision] + 1.0), $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))))) < 4.9999999999999999e-20

   1. Initial program 69.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. 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. 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 4.5

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

   5. Applied rewrites 4.5%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 16.9%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 59.0

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

   11. Applied rewrites 59.0%

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

   if 4.9999999999999999e-20 < (/.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 98.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. unpow1 N/A

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

      7. sqr-pow N/A

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

      8. div-inv N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lower-/.f64 58.6

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

   4. Applied rewrites 58.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 58.6

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

   6. Applied rewrites 58.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. /-rgt-identity N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. lift-sqrt.f64 N/A

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

      16. sqrt-div N/A

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

      17. lift-/.f64 N/A

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

      18. clear-num N/A

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

      19. lower-*.f64 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. lower-/.f64 48.7

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

   8. Applied rewrites 48.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

   10. Applied rewrites 58.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{\mathsf{fma}\left(\left(t \cdot 2\right) \cdot \sqrt{\frac{t}{\ell}}, \frac{\sqrt{\frac{t}{\ell}}}{\ell}, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 - 2 \cdot \frac{\frac{t\_m}{l\_m}}{l\_m \cdot \frac{-1}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 2e-18)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (if (<= (/ t_m l_m) 2e+94)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ (* Om Om) Omc) Omc))
        (- 1.0 (* 2.0 (/ (/ t_m l_m) (* l_m (/ -1.0 t_m))))))))
     (asin
      (*
       (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
       (/ (* 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) <= 2e-18) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else if ((t_m / l_m) <= 2e+94) {
		tmp = asin(sqrt(((1.0 - (((Om * Om) / Omc) / Omc)) / (1.0 - (2.0 * ((t_m / l_m) / (l_m * (-1.0 / t_m))))))));
	} else {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * 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) <= 2e-18)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	elseif (Float64(t_m / l_m) <= 2e+94)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om * Om) / Omc) / Omc)) / Float64(1.0 - Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m * Float64(-1.0 / t_m))))))));
	else
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e-18], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+94], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om * Om), $MachinePrecision] / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m * N[(-1.0 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < 2.0000000000000001e-18

   1. Initial program 88.5%

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

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 65.9%

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

   if 2.0000000000000001e-18 < (/.f64 t l) < 2e94

   1. Initial program 98.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 52.3

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

   4. Applied rewrites 52.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 49.1

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

   6. Applied rewrites 49.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. div-inv N/A

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

      9. div-inv N/A

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

      10. lift-/.f64 N/A

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

      11. rem-square-sqrt N/A

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

      12. lift-sqrt.f64 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. *-commutative N/A

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

      15. frac-times N/A

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

      16. frac-2neg N/A

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

      17. frac-2neg N/A

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

      18. frac-times N/A

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

      19. sqr-neg N/A

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

      20. lift-sqrt.f64 N/A

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

      21. lift-sqrt.f64 N/A

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

      22. rem-square-sqrt N/A

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

   8. Applied rewrites 95.7%

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

   if 2e94 < (/.f64 t l)

   1. Initial program 49.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 92.9

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

   11. Applied rewrites 92.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{1 - 2 \cdot \frac{\frac{t}{\ell}}{\ell \cdot \frac{-1}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 2e-18)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (if (<= (/ t_m l_m) 2e+94)
     (asin
      (sqrt
       (/
        (- 1.0 (/ (/ (* Om Om) Omc) Omc))
        (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0))))
     (asin
      (*
       (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
       (/ (* 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) <= 2e-18) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else if ((t_m / l_m) <= 2e+94) {
		tmp = asin(sqrt(((1.0 - (((Om * Om) / Omc) / Omc)) / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	} else {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * 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) <= 2e-18)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	elseif (Float64(t_m / l_m) <= 2e+94)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om * Om) / Omc) / Omc)) / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	else
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e-18], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+94], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om * Om), $MachinePrecision] / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < 2.0000000000000001e-18

   1. Initial program 88.5%

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

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 65.9%

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

   if 2.0000000000000001e-18 < (/.f64 t l) < 2e94

   1. Initial program 98.9%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 52.3

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

   4. Applied rewrites 52.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 49.1

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

   6. Applied rewrites 49.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lift-*.f64 N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 95.8

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

   8. Applied rewrites 95.8%

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

   if 2e94 < (/.f64 t l)

   1. Initial program 49.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.2

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

   5. Applied rewrites 3.2%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 22.0%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 92.9

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

   11. Applied rewrites 92.9%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{-18}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.01)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (asin
    (*
     (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
     (/ (* 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) <= 0.01) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((l_m * 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) <= 0.01)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	else
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.01], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0100000000000000002

   1. Initial program 88.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 65.3%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 71.9%

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

   3. Taylor expanded in t around 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. 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 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 13.1%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. associate-/l* N/A

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

       8. distribute-lft-neg-in N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. mul-1-neg N/A

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

       12. lower-neg.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-sqrt.f64 90.3

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

   11. Applied rewrites 90.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5}}{t\_m}\right)\right)\\ \end{array} \end{array} \]

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) 0.01)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (asin
    (*
     l_m
     (* (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0)) (/ (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) <= 0.01) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else {
		tmp = asin((l_m * (sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * (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) <= 0.01)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(sqrt(0.5) / t_m))));
	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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.01], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0100000000000000002

   1. Initial program 88.5%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 65.3%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 71.9%

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

   3. Taylor expanded in t around 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. 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 5.0

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

   5. Applied rewrites 5.0%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 5.8%

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]

   9. Taylor expanded in t around inf

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

       1. associate-/l* N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

   11. Applied rewrites 90.3%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{\frac{Omc \cdot Omc}{Om \cdot Om}}}\right)\\ \end{array} \end{array} \]

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) 2.65e+222)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (asin (sqrt (/ -1.0 (/ (* Omc Omc) (* Om Om)))))))

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) <= 2.65e+222) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else {
		tmp = asin(sqrt((-1.0 / ((Omc * Omc) / (Om * Om)))));
	}
	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) <= 2.65e+222)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	else
		tmp = asin(sqrt(Float64(-1.0 / Float64(Float64(Omc * Omc) / Float64(Om * Om)))));
	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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2.65e+222], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(-1.0 / N[(N[(Omc * Omc), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.64999999999999996e222

   1. Initial program 86.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 47.9

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 54.0%

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

   if 2.64999999999999996e222 < (/.f64 t l)

   1. Initial program 55.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 38.8%

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

   10. Applied rewrites 44.1%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{\frac{Omc \cdot Omc}{Om \cdot Om}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{Omc \cdot \left(-Omc\right)}}\right)\\ \end{array} \end{array} \]

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) 2.65e+222)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (asin (sqrt (/ (* Om Om) (* Omc (- 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 ((t_m / l_m) <= 2.65e+222) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else {
		tmp = asin(sqrt(((Om * Om) / (Omc * -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(t_m / l_m) <= 2.65e+222)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	else
		tmp = asin(sqrt(Float64(Float64(Om * Om) / Float64(Omc * Float64(-Omc)))));
	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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2.65e+222], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Om * Om), $MachinePrecision] / N[(Omc * (-Omc)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.64999999999999996e222

   1. Initial program 86.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 47.9

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

   5. Applied rewrites 47.9%

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

   7. Applied rewrites 54.0%

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

   if 2.64999999999999996e222 < (/.f64 t l)

   1. Initial program 55.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 38.8%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{Omc \cdot \left(-Omc\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-145}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, t\_m \cdot \frac{2}{l\_m \cdot l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 1.1e-145)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (asin (sqrt (/ 1.0 (fma t_m (* t_m (/ 2.0 (* l_m l_m))) 1.0))))))

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 <= 1.1e-145) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else {
		tmp = asin(sqrt((1.0 / fma(t_m, (t_m * (2.0 / (l_m * l_m))), 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 1.1e-145)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(t_m * Float64(2.0 / Float64(l_m * l_m))), 1.0))));
	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_] := If[LessEqual[t$95$m, 1.1e-145], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(t$95$m * N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.1e-145

   1. Initial program 90.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 51.2

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

   5. Applied rewrites 51.2%

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

   7. Applied rewrites 58.3%

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

   if 1.1e-145 < t

   1. Initial program 75.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 34.5

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

   5. Applied rewrites 34.5%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 64.0

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

   8. Applied rewrites 64.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-145}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, t \cdot \frac{2}{\ell \cdot \ell}, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{Omc \cdot \left(-Omc\right)}}\right)\\ \end{array} \end{array} \]

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) 2.65e+222)
   (asin (sqrt 1.0))
   (asin (sqrt (/ (* Om Om) (* Omc (- 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 ((t_m / l_m) <= 2.65e+222) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(sqrt(((Om * Om) / (Omc * -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 ((t_m / l_m) <= 2.65d+222) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(sqrt(((om * om) / (omc * -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 ((t_m / l_m) <= 2.65e+222) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(Math.sqrt(((Om * Om) / (Omc * -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 (t_m / l_m) <= 2.65e+222:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(math.sqrt(((Om * Om) / (Omc * -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(t_m / l_m) <= 2.65e+222)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(Om * Om) / Float64(Omc * Float64(-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 ((t_m / l_m) <= 2.65e+222)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(((Om * Om) / (Omc * -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[(t$95$m / l$95$m), $MachinePrecision], 2.65e+222], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Om * Om), $MachinePrecision] / N[(Omc * (-Omc)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.64999999999999996e222

   1. Initial program 86.7%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 53.4%

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]

   if 2.64999999999999996e222 < (/.f64 t l)

   1. Initial program 55.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 38.8%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2.65 \cdot 10^{+222}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{Omc \cdot \left(-Omc\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.5 \cdot 10^{+200}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(-Om\right) \cdot \frac{Om}{Omc \cdot Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.5e+200)
   (asin (sqrt 1.0))
   (asin (sqrt (* (- Om) (/ Om (* Omc 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 ((t_m / l_m) <= 1.5e+200) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(sqrt((-Om * (Om / (Omc * 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 ((t_m / l_m) <= 1.5d+200) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(sqrt((-om * (om / (omc * 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 ((t_m / l_m) <= 1.5e+200) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(Math.sqrt((-Om * (Om / (Omc * 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 (t_m / l_m) <= 1.5e+200:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(math.sqrt((-Om * (Om / (Omc * 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(t_m / l_m) <= 1.5e+200)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(-Om) * Float64(Om / Float64(Omc * 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 ((t_m / l_m) <= 1.5e+200)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt((-Om * (Om / (Omc * 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[(t$95$m / l$95$m), $MachinePrecision], 1.5e+200], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.49999999999999995e200

   1. Initial program 87.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. 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. 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 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 53.6%

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]

   if 1.49999999999999995e200 < (/.f64 t l)

   1. Initial program 52.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. 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 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in Om around inf

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

   8. Applied rewrites 36.8%

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

   10. Applied rewrites 34.4%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 1.5 \cdot 10^{+200}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(-Om\right) \cdot \frac{Om}{Omc \cdot Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.8% accurate, 3.2× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc) :precision binary64 (asin (sqrt 1.0)))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin(sqrt(1.0));
}

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(sqrt(1.0d0))
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(Math.sqrt(1.0));
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	return math.asin(math.sqrt(1.0))

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(1.0))
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(sqrt(1.0));
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[Sqrt[1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.3%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. 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. 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 44.6

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

5. Applied rewrites 44.6%

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

6. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
7. Step-by-step derivation

8. Applied rewrites 49.6%

   \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024232 
(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)))))))