Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.9%

Time: 15.6s

Alternatives: 8

Speedup: 2.5×

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: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{\mathsf{fma}\left(\frac{Om \cdot \frac{Om}{Omc}}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+145)
   (asin
    (sqrt
     (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
   (asin (* (/ l_m t_m) (sqrt (fma (/ (* Om (/ Om Omc)) Omc) -0.5 0.5))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+145) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0))))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(fma(((Om * (Om / Omc)) / Omc), -0.5, 0.5))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+145)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(fma(Float64(Float64(Om * Float64(Om / Omc)) / Omc), -0.5, 0.5))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+145], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2e145

   1. Initial program 91.1%

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

   if 2e145 < (/.f64 t l)

   1. Initial program 36.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 83.5%

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

   12. Applied rewrites 99.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\mathsf{fma}\left(\frac{Om \cdot \frac{Om}{Omc}}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{1}{l\_m}, t\_m \cdot \left(\frac{t\_m}{l\_m} \cdot 2\right), 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{\mathsf{fma}\left(\frac{Om \cdot \frac{Om}{Omc}}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+22)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ 1.0 l_m) (* t_m (* (/ t_m l_m) 2.0)) 1.0))))
   (asin (* (/ l_m t_m) (sqrt (fma (/ (* Om (/ Om Omc)) Omc) -0.5 0.5))))))

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) <= 5e+22) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((1.0 / l_m), (t_m * ((t_m / l_m) * 2.0)), 1.0))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(fma(((Om * (Om / Omc)) / Omc), -0.5, 0.5))));
	}
	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) <= 5e+22)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(1.0 / l_m), Float64(t_m * Float64(Float64(t_m / l_m) * 2.0)), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(fma(Float64(Float64(Om * Float64(Om / Omc)) / Omc), -0.5, 0.5))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+22], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.9999999999999996e22

   1. Initial program 90.0%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. associate-/r/ N/A

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

      11. associate-*l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 88.0

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

   4. Applied rewrites 88.0%

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

   if 4.9999999999999996e22 < (/.f64 t l)

   1. Initial program 62.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 36.1%

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

   5. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 28.7%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 85.2%

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

   12. Applied rewrites 99.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+22}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{1}{\ell}, t \cdot \left(\frac{t}{\ell} \cdot 2\right), 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\mathsf{fma}\left(\frac{Om \cdot \frac{Om}{Omc}}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{\mathsf{fma}\left(\frac{Om \cdot \frac{Om}{Omc}}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+41)
   (asin (sqrt (/ -1.0 (fma (* (/ t_m l_m) (/ t_m l_m)) -2.0 -1.0))))
   (asin (* (/ l_m t_m) (sqrt (fma (/ (* Om (/ Om Omc)) Omc) -0.5 0.5))))))

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) <= 5e+41) {
		tmp = asin(sqrt((-1.0 / fma(((t_m / l_m) * (t_m / l_m)), -2.0, -1.0))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(fma(((Om * (Om / Omc)) / Omc), -0.5, 0.5))));
	}
	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) <= 5e+41)
		tmp = asin(sqrt(Float64(-1.0 / fma(Float64(Float64(t_m / l_m) * Float64(t_m / l_m)), -2.0, -1.0))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(fma(Float64(Float64(Om * Float64(Om / Omc)) / Omc), -0.5, 0.5))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+41], N[ArcSin[N[Sqrt[N[(-1.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000022e41

   1. Initial program 90.0%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{0 - \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{0 - \color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      6. associate--r- N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\left(0 - 1\right) + {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

   4. Applied rewrites 71.7%

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

   5. Taylor expanded in Om around 0

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

   7. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 89.7

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

   9. Applied rewrites 89.7%

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

   if 5.00000000000000022e41 < (/.f64 t l)

   1. Initial program 62.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in l around 0

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

   10. Applied rewrites 85.0%

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

   12. Applied rewrites 99.5%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t}{\ell} \cdot \frac{t}{\ell}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\mathsf{fma}\left(\frac{Om \cdot \frac{Om}{Omc}}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 6e+144)
   (asin (sqrt (/ -1.0 (fma (* (/ t_m l_m) (/ t_m l_m)) -2.0 -1.0))))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 6e+144) {
		tmp = asin(sqrt((-1.0 / fma(((t_m / l_m) * (t_m / l_m)), -2.0, -1.0))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 6e+144)
		tmp = asin(sqrt(Float64(-1.0 / fma(Float64(Float64(t_m / l_m) * Float64(t_m / l_m)), -2.0, -1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 6e+144], N[ArcSin[N[Sqrt[N[(-1.0 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.9999999999999998e144

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{0 - \left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      5. lift--.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{0 - \color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      6. associate--r- N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\left(0 - 1\right) + {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-times N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. distribute-neg-in N/A

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in Om around 0

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

   7. Applied rewrites 70.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. frac-times N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 90.6

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

   9. Applied rewrites 90.6%

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

   if 5.9999999999999998e144 < (/.f64 t l)

   1. Initial program 36.6%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 98.5%

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

Alternative 5: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.005)
   (asin (sqrt (fma (/ Om Omc) (/ Om (- Omc)) 1.0)))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin(sqrt(fma((Om / Omc), (Om / -Omc), 1.0)));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.005)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(Om / Float64(-Omc)), 1.0)));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(Om / (-Omc)), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   7. Applied rewrites 66.6%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 28.4%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 96.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{-Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.005)
   (asin (sqrt 1.0))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.005d0) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.005:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.005)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.005)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 66.3%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 28.4%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 96.7%

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

Alternative 7: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.005)
   (asin (sqrt 1.0))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.005d0) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.005:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.005)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.005)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 66.3%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 39.6%

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

   5. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

   7. Applied rewrites 28.4%

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

   8. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 36.6

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

   10. Applied rewrites 36.6%

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

   11. Taylor expanded in Om around 0

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

   13. Applied rewrites 96.7%

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

Alternative 8: 49.9% 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 83.2%

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

3. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 44.5

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

5. Applied rewrites 44.5%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 49.6%

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

Reproduce

herbie shell --seed 2024219 
(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)))))))