Toniolo and Linder, Equation (2)

Percentage Accurate: 82.9% → 98.6%

Time: 14.0s

Alternatives: 12

Speedup: 1.5×

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: 82.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, 2 \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\mathsf{fma}\left(-0.125, \frac{l\_m \cdot l\_m}{\sqrt{0.5} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}, \frac{\sqrt{0.5}}{t\_m}\right) \cdot \sqrt{t\_1}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (/ (* (/ Om Omc) Om) Omc))))
   (if (<= (/ t_m l_m) 2e+148)
     (asin (sqrt (/ t_1 (fma (/ t_m l_m) (* 2.0 (/ t_m l_m)) 1.0))))
     (asin
      (*
       (*
        (fma
         -0.125
         (/ (* l_m l_m) (* (sqrt 0.5) (* (* t_m t_m) t_m)))
         (/ (sqrt 0.5) t_m))
        (sqrt t_1))
       l_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 - (((Om / Omc) * Om) / Omc);
	double tmp;
	if ((t_m / l_m) <= 2e+148) {
		tmp = asin(sqrt((t_1 / fma((t_m / l_m), (2.0 * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin(((fma(-0.125, ((l_m * l_m) / (sqrt(0.5) * ((t_m * t_m) * t_m))), (sqrt(0.5) / t_m)) * sqrt(t_1)) * l_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(Float64(Float64(Om / Omc) * Om) / Omc))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+148)
		tmp = asin(sqrt(Float64(t_1 / fma(Float64(t_m / l_m), Float64(2.0 * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(fma(-0.125, Float64(Float64(l_m * l_m) / Float64(sqrt(0.5) * Float64(Float64(t_m * t_m) * t_m))), Float64(sqrt(0.5) / t_m)) * sqrt(t_1)) * l_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[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+148], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(-0.125 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e148

   1. Initial program 92.3%

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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 92.3

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

   4. Applied rewrites 92.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 92.3

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

   6. Applied rewrites 92.3%

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

   if 2.0000000000000001e148 < (/.f64 t l)

   1. Initial program 45.9%

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

      1. lift-+.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. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 45.9

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

   4. Applied rewrites 45.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 79.2%

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

   9. Applied rewrites 99.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell}, 2 \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\sqrt{0.5} \cdot \left(\left(t \cdot t\right) \cdot t\right)}, \frac{\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.4% accurate, 0.9× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ (* (pow (/ t_m l_m) 2.0) 2.0) 1.0))
      2e-5)
   (asin (sqrt (/ (* 0.5 (* l_m l_m)) (* t_m t_m))))
   (asin (sqrt (fma (/ (- Om) Omc) (/ Om Omc) 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 (((1.0 - pow((Om / Omc), 2.0)) / ((pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 2e-5) {
		tmp = asin(sqrt(((0.5 * (l_m * l_m)) / (t_m * t_m))));
	} else {
		tmp = asin(sqrt(fma((-Om / Omc), (Om / Omc), 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 (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(Float64((Float64(t_m / l_m) ^ 2.0) * 2.0) + 1.0)) <= 2e-5)
		tmp = asin(sqrt(Float64(Float64(0.5 * Float64(l_m * l_m)) / Float64(t_m * t_m))));
	else
		tmp = asin(sqrt(fma(Float64(Float64(-Om) / Omc), Float64(Om / Omc), 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[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 2e-5], N[ArcSin[N[Sqrt[N[(N[(0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[((-Om) / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + 1.0), $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))))) < 2.00000000000000016e-5

   1. Initial program 73.6%

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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 36.5

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

   7. Applied rewrites 36.5%

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

   8. Taylor expanded in Omc around inf

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

   10. Applied rewrites 44.6%

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

   if 2.00000000000000016e-5 < (/.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 97.8%

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

   7. Applied rewrites 94.6%

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

4. Final simplification 66.7%

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

Alternative 3: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \cdot 2 + 1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, 2 \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{l\_m \cdot l\_m}{\sqrt{0.5} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}, -0.125, \frac{\sqrt{0.5}}{t\_m}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ (* (pow (/ t_m l_m) 2.0) 2.0) 1.0) 4e+304)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* (/ Om Omc) Om) Omc))
      (fma (/ t_m l_m) (* 2.0 (/ t_m l_m)) 1.0))))
   (asin
    (*
     (fma
      (/ (* l_m l_m) (* (sqrt 0.5) (* (* t_m t_m) t_m)))
      -0.125
      (/ (sqrt 0.5) t_m))
     l_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((pow((t_m / l_m), 2.0) * 2.0) + 1.0) <= 4e+304) {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / fma((t_m / l_m), (2.0 * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((fma(((l_m * l_m) / (sqrt(0.5) * ((t_m * t_m) * t_m))), -0.125, (sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(Float64((Float64(t_m / l_m) ^ 2.0) * 2.0) + 1.0) <= 4e+304)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / fma(Float64(t_m / l_m), Float64(2.0 * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(fma(Float64(Float64(l_m * l_m) / Float64(sqrt(0.5) * Float64(Float64(t_m * t_m) * t_m))), -0.125, Float64(sqrt(0.5) / t_m)) * l_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[(N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision], 4e+304], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 3.9999999999999998e304

   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-+.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. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

   if 3.9999999999999998e304 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 48.9%

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

      1. lift-+.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. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 48.9

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in Omc around inf

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

   10. Applied rewrites 79.6%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \cdot 2 + 1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell}, 2 \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{\sqrt{0.5} \cdot \left(\left(t \cdot t\right) \cdot t\right)}, -0.125, \frac{\sqrt{0.5}}{t}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \cdot 2 + 1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{t\_m}{l\_m}, 2 \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{l\_m \cdot l\_m}{\sqrt{0.5} \cdot \left(\left(t\_m \cdot t\_m\right) \cdot t\_m\right)}, -0.125, \frac{\sqrt{0.5}}{t\_m}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ (* (pow (/ t_m l_m) 2.0) 2.0) 1.0) 4e+304)
   (asin (/ 1.0 (sqrt (fma (/ t_m l_m) (* 2.0 (/ t_m l_m)) 1.0))))
   (asin
    (*
     (fma
      (/ (* l_m l_m) (* (sqrt 0.5) (* (* t_m t_m) t_m)))
      -0.125
      (/ (sqrt 0.5) t_m))
     l_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((pow((t_m / l_m), 2.0) * 2.0) + 1.0) <= 4e+304) {
		tmp = asin((1.0 / sqrt(fma((t_m / l_m), (2.0 * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((fma(((l_m * l_m) / (sqrt(0.5) * ((t_m * t_m) * t_m))), -0.125, (sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(Float64((Float64(t_m / l_m) ^ 2.0) * 2.0) + 1.0) <= 4e+304)
		tmp = asin(Float64(1.0 / sqrt(fma(Float64(t_m / l_m), Float64(2.0 * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(fma(Float64(Float64(l_m * l_m) / Float64(sqrt(0.5) * Float64(Float64(t_m * t_m) * t_m))), -0.125, Float64(sqrt(0.5) / t_m)) * l_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[(N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision], 4e+304], N[ArcSin[N[(1.0 / N[Sqrt[N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.125 + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 3.9999999999999998e304

   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-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in Omc around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 97.3%

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

   if 3.9999999999999998e304 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 48.9%

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

      1. lift-+.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. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 48.9

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 63.1%

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

   8. Taylor expanded in Omc around inf

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

   10. Applied rewrites 79.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \cdot 2 + 1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{t}{\ell}, 2 \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{\ell \cdot \ell}{\sqrt{0.5} \cdot \left(\left(t \cdot t\right) \cdot t\right)}, -0.125, \frac{\sqrt{0.5}}{t}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \cdot 2 + 1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{t\_m}{l\_m}, 2 \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ (* (pow (/ t_m l_m) 2.0) 2.0) 1.0) 4e+304)
   (asin (/ 1.0 (sqrt (fma (/ t_m l_m) (* 2.0 (/ t_m l_m)) 1.0))))
   (asin
    (*
     (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
     (/ (* (sqrt 0.5) 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 tmp;
	if (((pow((t_m / l_m), 2.0) * 2.0) + 1.0) <= 4e+304) {
		tmp = asin((1.0 / sqrt(fma((t_m / l_m), (2.0 * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((sqrt(0.5) * l_m) / 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(Float64((Float64(t_m / l_m) ^ 2.0) * 2.0) + 1.0) <= 4e+304)
		tmp = asin(Float64(1.0 / sqrt(fma(Float64(t_m / l_m), Float64(2.0 * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(sqrt(0.5) * 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_] := If[LessEqual[N[(N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision], 4e+304], N[ArcSin[N[(1.0 / N[Sqrt[N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $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[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 3.9999999999999998e304

   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-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in Omc around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.8

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

   7. Applied rewrites 65.8%

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

   9. Applied rewrites 97.3%

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

   if 3.9999999999999998e304 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 48.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \cdot 2 + 1 \leq 4 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{t}{\ell}, 2 \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-95}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{t\_m}{l\_m \cdot l\_m}, 2 \cdot t\_m, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 5.3e-95)
   (asin (sqrt (fma (/ (- Om) Omc) (/ Om Omc) 1.0)))
   (asin (/ 1.0 (sqrt (fma (/ t_m (* l_m l_m)) (* 2.0 t_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 <= 5.3e-95) {
		tmp = asin(sqrt(fma((-Om / Omc), (Om / Omc), 1.0)));
	} else {
		tmp = asin((1.0 / sqrt(fma((t_m / (l_m * l_m)), (2.0 * t_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 <= 5.3e-95)
		tmp = asin(sqrt(fma(Float64(Float64(-Om) / Omc), Float64(Om / Omc), 1.0)));
	else
		tmp = asin(Float64(1.0 / sqrt(fma(Float64(t_m / Float64(l_m * l_m)), Float64(2.0 * t_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, 5.3e-95], N[ArcSin[N[Sqrt[N[(N[((-Om) / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 / N[Sqrt[N[(N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * N[(2.0 * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.2999999999999998e-95

   1. Initial program 86.5%

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 52.1%

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

   if 5.2999999999999998e-95 < t

   1. Initial program 79.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in Omc around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 57.9

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

   7. Applied rewrites 57.9%

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

   9. Applied rewrites 64.4%

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

Alternative 7: 73.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-95}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{-Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{l\_m \cdot l\_m} \cdot t\_m, t\_m, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 5.3e-95)
   (asin (sqrt (fma (/ (- Om) Omc) (/ Om Omc) 1.0)))
   (asin (sqrt (/ 1.0 (fma (* (/ 2.0 (* l_m l_m)) t_m) t_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 <= 5.3e-95) {
		tmp = asin(sqrt(fma((-Om / Omc), (Om / Omc), 1.0)));
	} else {
		tmp = asin(sqrt((1.0 / fma(((2.0 / (l_m * l_m)) * t_m), t_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 <= 5.3e-95)
		tmp = asin(sqrt(fma(Float64(Float64(-Om) / Omc), Float64(Om / Omc), 1.0)));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(Float64(2.0 / Float64(l_m * l_m)) * t_m), t_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, 5.3e-95], 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[(N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.2999999999999998e-95

   1. Initial program 86.5%

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 50.1

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 52.1%

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

   if 5.2999999999999998e-95 < t

   1. Initial program 79.4%

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

   3. Taylor expanded in Omc around inf

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

Alternative 8: 82.0% accurate, 2.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (/ 1.0 (sqrt (fma (/ t_m l_m) (* 2.0 (/ t_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) {
	return asin((1.0 / sqrt(fma((t_m / l_m), (2.0 * (t_m / l_m)), 1.0))));
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(1.0 / sqrt(fma(Float64(t_m / l_m), Float64(2.0 * Float64(t_m / l_m)), 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[(1.0 / N[Sqrt[N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. sqrt-div N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.3%

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

5. Taylor expanded in Omc around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 61.0

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

7. Applied rewrites 61.0%

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

9. Applied rewrites 83.5%

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

Alternative 9: 79.3% accurate, 2.3× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (/ 1.0 (sqrt (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) {
	return asin((1.0 / sqrt(fma(t_m, ((2.0 * (t_m / l_m)) / l_m), 1.0))));
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(1.0 / sqrt(fma(t_m, Float64(Float64(2.0 * Float64(t_m / l_m)) / l_m), 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[(1.0 / N[Sqrt[N[(t$95$m * N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $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. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. sqrt-div N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. frac-2neg N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 62.3%

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

5. Taylor expanded in Omc around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 61.0

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

7. Applied rewrites 61.0%

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

9. Applied rewrites 79.8%

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

Alternative 10: 69.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 50:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{0.5 \cdot \left(l\_m \cdot l\_m\right)}{t\_m \cdot t\_m}}\right)\\ \end{array} \end{array} \]

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) 50.0)
   (asin (sqrt 1.0))
   (asin (sqrt (/ (* 0.5 (* l_m l_m)) (* t_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 tmp;
	if ((t_m / l_m) <= 50.0) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(sqrt(((0.5 * (l_m * l_m)) / (t_m * t_m))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 50.0d0) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(sqrt(((0.5d0 * (l_m * l_m)) / (t_m * t_m))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 50.0) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(Math.sqrt(((0.5 * (l_m * l_m)) / (t_m * t_m))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 50.0:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(math.sqrt(((0.5 * (l_m * l_m)) / (t_m * 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) <= 50.0)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(0.5 * Float64(l_m * l_m)) / Float64(t_m * t_m))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 50.0)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(((0.5 * (l_m * l_m)) / (t_m * t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 50.0], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 50

   1. Initial program 90.7%

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

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 62.9%

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

   if 50 < (/.f64 t l)

   1. Initial program 71.0%

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

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 36.6

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

   7. Applied rewrites 36.6%

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

   8. Taylor expanded in Omc around inf

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

   10. Applied rewrites 42.0%

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

Alternative 11: 57.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.42 \cdot 10^{+221}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{\left(-Omc\right) \cdot Omc}}\right)\\ \end{array} \end{array} \]

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) 1.42e+221)
   (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) <= 1.42e+221) {
		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) <= 1.42d+221) 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) <= 1.42e+221) {
		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) <= 1.42e+221:
		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) <= 1.42e+221)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(Om * Om) / Float64(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) <= 1.42e+221)
		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], 1.42e+221], 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) < 1.42000000000000001e221

   1. Initial program 88.0%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 49.0

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

   5. Applied rewrites 49.0%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 50.0%

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

   if 1.42000000000000001e221 < (/.f64 t l)

   1. Initial program 58.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in Omc around 0

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

   8. Applied rewrites 34.9%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 1.42 \cdot 10^{+221}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{\left(-Omc\right) \cdot Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.7% 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. sub-neg N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 43.2

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

5. Applied rewrites 43.2%

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

6. Taylor expanded in Omc around inf

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

8. Applied rewrites 44.2%

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

Reproduce

herbie shell --seed 2024235 
(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)))))))