Toniolo and Linder, Equation (2)

Percentage Accurate: 84.9% → 98.1%

Time: 16.8s

Alternatives: 14

Speedup: 2.1×

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: 84.9% 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: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{l\_m \cdot l\_m}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot \sqrt{0.5}\right)}, \frac{\sqrt{0.5}}{t\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+118)
   (asin
    (sqrt
     (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
   (asin
    (*
     l_m
     (*
      (sqrt (fma (/ Om Omc) (- 0.0 (/ Om Omc)) 1.0))
      (fma
       -0.125
       (/ (* l_m l_m) (* (* t_m t_m) (* t_m (sqrt 0.5))))
       (/ (sqrt 0.5) t_m)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+118) {
		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 * (sqrt(fma((Om / Omc), (0.0 - (Om / Omc)), 1.0)) * fma(-0.125, ((l_m * l_m) / ((t_m * t_m) * (t_m * sqrt(0.5)))), (sqrt(0.5) / t_m)))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+118)
		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(l_m * Float64(sqrt(fma(Float64(Om / Omc), Float64(0.0 - Float64(Om / Omc)), 1.0)) * fma(-0.125, Float64(Float64(l_m * l_m) / Float64(Float64(t_m * t_m) * Float64(t_m * sqrt(0.5)))), Float64(sqrt(0.5) / t_m)))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+118], 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[(l$95$m * N[(N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(0.0 - N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.99999999999999967e117

   1. Initial program 90.1%

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

   if 9.99999999999999967e117 < (/.f64 t l)

   1. Initial program 58.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)}\right) \]

   5. Simplified 89.4%

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

      1. times-frac N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{Om}{Omc} \cdot \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\left(\frac{Om}{Omc}\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 99.7%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(Om, Omc\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 99.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sqrt{0.5}\right)}, \frac{\sqrt{0.5}}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \sqrt{\frac{t\_m}{l\_m}}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+101}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\left(\frac{t\_m}{l\_m} \cdot 2\right) \cdot t\_1, t\_1, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{l\_m \cdot l\_m}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot \sqrt{0.5}\right)}, \frac{\sqrt{0.5}}{t\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (sqrt (/ t_m l_m))))
   (if (<= (/ t_m l_m) 1e+101)
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (fma (* (* (/ t_m l_m) 2.0) t_1) t_1 1.0))))
     (asin
      (*
       l_m
       (*
        (sqrt (fma (/ Om Omc) (- 0.0 (/ Om Omc)) 1.0))
        (fma
         -0.125
         (/ (* l_m l_m) (* (* t_m t_m) (* t_m (sqrt 0.5))))
         (/ (sqrt 0.5) t_m))))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = sqrt((t_m / l_m));
	double tmp;
	if ((t_m / l_m) <= 1e+101) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((((t_m / l_m) * 2.0) * t_1), t_1, 1.0))));
	} else {
		tmp = asin((l_m * (sqrt(fma((Om / Omc), (0.0 - (Om / Omc)), 1.0)) * fma(-0.125, ((l_m * l_m) / ((t_m * t_m) * (t_m * sqrt(0.5)))), (sqrt(0.5) / t_m)))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = sqrt(Float64(t_m / l_m))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+101)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(Float64(Float64(t_m / l_m) * 2.0) * t_1), t_1, 1.0))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(fma(Float64(Om / Omc), Float64(0.0 - Float64(Om / Omc)), 1.0)) * fma(-0.125, Float64(Float64(l_m * l_m) / Float64(Float64(t_m * t_m) * Float64(t_m * sqrt(0.5)))), Float64(sqrt(0.5) / t_m)))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[Sqrt[N[(t$95$m / l$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+101], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(0.0 - N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.9999999999999998e100

   1. Initial program 90.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(2 \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell} + 1\right)\right)\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(2 \cdot \frac{t}{\ell}\right) \cdot {\left(\frac{t}{\ell}\right)}^{1} + 1\right)\right)\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(2 \cdot \frac{t}{\ell}\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) + 1\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(\left(2 \cdot \frac{t}{\ell}\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)} + 1\right)\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\left(\left(2 \cdot \frac{t}{\ell}\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\left(\left(\frac{t}{\ell} \cdot 2\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot 2\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{t}{\ell}\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{t}{\ell}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\frac{1}{2}}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      14. unpow1/2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\sqrt{\frac{t}{\ell}}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{t}{\ell}\right)\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\frac{1}{2}}\right), 1\right)\right)\right)\right) \]

      18. unpow1/2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \left(\sqrt{\frac{t}{\ell}}\right), 1\right)\right)\right)\right) \]

      19. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{t}{\ell}\right)\right), 1\right)\right)\right)\right) \]

      20. /-lowering-/.f64 43.9%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right), 1\right)\right)\right)\right) \]

   4. Applied egg-rr 43.9%

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

   if 9.9999999999999998e100 < (/.f64 t l)

   1. Initial program 59.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)}\right) \]

   5. Simplified 89.7%

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

      1. times-frac N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{Om}{Omc} \cdot \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\left(\frac{Om}{Omc}\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 99.6%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(Om, Omc\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 99.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+101}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot 2\right) \cdot \sqrt{\frac{t}{\ell}}, \sqrt{\frac{t}{\ell}}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sqrt{0.5}\right)}, \frac{\sqrt{0.5}}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2000000000000.0)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (/ (* t_m (/ t_m l_m)) l_m))))))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2000000000000.0) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
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) <= 2000000000000.0d0) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m * (t_m / l_m)) / l_m))))))
    else
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2000000000000.0) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 2000000000000.0:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))))
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2000000000000.0)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m * Float64(t_m / l_m)) / l_m))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 2000000000000.0)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2000000000000.0], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision]), $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) < 2e12

   1. Initial program 89.4%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)\right)\right)\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \left(\frac{\frac{t}{\ell} \cdot t}{\ell}\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(\frac{t}{\ell} \cdot t\right), \ell\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot \frac{t}{\ell}\right), \ell\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{t}{\ell}\right)\right), \ell\right)\right)\right)\right)\right)\right) \]

      6. /-lowering-/.f64 88.5%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{/.f64}\left(t, \ell\right)\right), \ell\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 88.5%

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

   if 2e12 < (/.f64 t l)

   1. Initial program 69.9%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 48.3%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 48.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 98.4%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 98.4%

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

Alternative 4: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(t\_m \cdot 2, \frac{\frac{t\_m}{l\_m}}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2000000000000.0)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (* t_m 2.0) (/ (/ t_m l_m) l_m) 1.0))))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2000000000000.0) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((t_m * 2.0), ((t_m / l_m) / l_m), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2000000000000.0)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(t_m * 2.0), Float64(Float64(t_m / l_m) / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2000000000000.0], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 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) < 2e12

   1. Initial program 89.4%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1\right)\right)\right)\right) \]

      3. div-inv N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot \left(\left(t \cdot \frac{1}{\ell}\right) \cdot \frac{t}{\ell}\right) + 1\right)\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot \left(t \cdot \left(\frac{1}{\ell} \cdot \frac{t}{\ell}\right)\right) + 1\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(2 \cdot t\right) \cdot \left(\frac{1}{\ell} \cdot \frac{t}{\ell}\right) + 1\right)\right)\right)\right) \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\left(2 \cdot t\right), \left(\frac{1}{\ell} \cdot \frac{t}{\ell}\right), 1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\frac{1}{\ell} \cdot \frac{t}{\ell}\right), 1\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\frac{\frac{1}{\ell} \cdot t}{\ell}\right), 1\right)\right)\right)\right) \]

      9. associate-/r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\frac{\frac{1}{\frac{\ell}{t}}}{\ell}\right), 1\right)\right)\right)\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, t\right), \left(\frac{\frac{t}{\ell}}{\ell}\right), 1\right)\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{/.f64}\left(\left(\frac{t}{\ell}\right), \ell\right), 1\right)\right)\right)\right) \]

      12. /-lowering-/.f64 86.6%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(2, t\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \ell\right), 1\right)\right)\right)\right) \]

   4. Applied egg-rr 86.6%

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

   if 2e12 < (/.f64 t l)

   1. Initial program 69.9%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 48.3%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 48.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 98.4%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 98.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(t \cdot 2, \frac{\frac{t}{\ell}}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{l\_m \cdot l\_m}{\left(t\_m \cdot t\_m\right) \cdot \left(t\_m \cdot \sqrt{0.5}\right)}, \frac{\sqrt{0.5}}{t\_m}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+118)
   (asin (sqrt (/ 1.0 (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0))))
   (asin
    (*
     l_m
     (*
      (sqrt (fma (/ Om Omc) (- 0.0 (/ Om Omc)) 1.0))
      (fma
       -0.125
       (/ (* l_m l_m) (* (* t_m t_m) (* t_m (sqrt 0.5))))
       (/ (sqrt 0.5) t_m)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+118) {
		tmp = asin(sqrt((1.0 / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	} else {
		tmp = asin((l_m * (sqrt(fma((Om / Omc), (0.0 - (Om / Omc)), 1.0)) * fma(-0.125, ((l_m * l_m) / ((t_m * t_m) * (t_m * sqrt(0.5)))), (sqrt(0.5) / t_m)))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+118)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(fma(Float64(Om / Omc), Float64(0.0 - Float64(Om / Omc)), 1.0)) * fma(-0.125, Float64(Float64(l_m * l_m) / Float64(Float64(t_m * t_m) * Float64(t_m * sqrt(0.5)))), Float64(sqrt(0.5) / t_m)))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+118], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(0.0 - N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(-0.125 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(t$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.99999999999999967e117

   1. Initial program 90.1%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 67.2%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell} + 1\right)\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t \cdot \left(t \cdot 2\right)}{\ell \cdot \ell} + 1\right)\right)\right)\right) \]

      3. frac-times N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t}{\ell} \cdot \frac{t \cdot 2}{\ell} + 1\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1\right)\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{t}{\ell} \cdot 2\right), 1\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{t}{\ell} \cdot 2\right), 1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), 2\right), 1\right)\right)\right)\right) \]

      8. /-lowering-/.f64 88.6%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right), 1\right)\right)\right)\right) \]

   7. Applied egg-rr 88.6%

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

   if 9.99999999999999967e117 < (/.f64 t l)

   1. Initial program 58.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)}\right) \]

   5. Simplified 89.4%

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

      1. times-frac N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{Om}{Omc} \cdot \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right) + 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\left(\frac{Om}{Omc}\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\left(\frac{Om}{Omc}\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 99.7%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(Om, Omc\right)\right), 1\right)\right), \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(t, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 99.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)} \cdot \mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot t\right) \cdot \left(t \cdot \sqrt{0.5}\right)}, \frac{\sqrt{0.5}}{t}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+118}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \mathsf{fma}\left(-0.125, \frac{l\_m \cdot l\_m}{\sqrt{0.5} \cdot \left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right)}, \frac{\sqrt{0.5}}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+118)
   (asin (sqrt (/ 1.0 (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0))))
   (asin
    (*
     l_m
     (fma
      -0.125
      (/ (* l_m l_m) (* (sqrt 0.5) (* t_m (* t_m t_m))))
      (/ (sqrt 0.5) t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+118) {
		tmp = asin(sqrt((1.0 / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	} else {
		tmp = asin((l_m * fma(-0.125, ((l_m * l_m) / (sqrt(0.5) * (t_m * (t_m * t_m)))), (sqrt(0.5) / t_m))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+118)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	else
		tmp = asin(Float64(l_m * fma(-0.125, Float64(Float64(l_m * l_m) / Float64(sqrt(0.5) * Float64(t_m * Float64(t_m * t_m)))), Float64(sqrt(0.5) / t_m))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+118], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(-0.125 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[Sqrt[0.5], $MachinePrecision] * N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 9.99999999999999967e117

   1. Initial program 90.1%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 67.2%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 67.2%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell} + 1\right)\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t \cdot \left(t \cdot 2\right)}{\ell \cdot \ell} + 1\right)\right)\right)\right) \]

      3. frac-times N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t}{\ell} \cdot \frac{t \cdot 2}{\ell} + 1\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1\right)\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{t}{\ell} \cdot 2\right), 1\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{t}{\ell} \cdot 2\right), 1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), 2\right), 1\right)\right)\right)\right) \]

      8. /-lowering-/.f64 88.6%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right), 1\right)\right)\right)\right) \]

   7. Applied egg-rr 88.6%

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

   if 9.99999999999999967e117 < (/.f64 t l)

   1. Initial program 58.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)}\right) \]

   5. Simplified 89.4%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{8} \cdot \frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} + \frac{\sqrt{\frac{1}{2}}}{t}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}}\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({t}^{3} \cdot \sqrt{\frac{1}{2}}\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\sqrt{\frac{1}{2}} \cdot {t}^{3}\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{2}}\right), \left({t}^{3}\right)\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left({t}^{3}\right)\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      8. cube-mult N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(t \cdot \left(t \cdot t\right)\right)\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \left(t \cdot {t}^{2}\right)\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(t, \left({t}^{2}\right)\right)\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(t, \left(t \cdot t\right)\right)\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \left(\frac{\sqrt{\frac{1}{2}}}{t}\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\left(\sqrt{\frac{1}{2}}\right), t\right)\right)\right)\right) \]

      14. sqrt-lowering-sqrt.f64 98.1%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{fma.f64}\left(\frac{-1}{8}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(t, t\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\frac{1}{2}\right), t\right)\right)\right)\right) \]

   8. Simplified 98.1%

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

Alternative 7: 98.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+59)
   (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

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+59) {
		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

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+59)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m / l_m), Float64(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

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+59], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(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) < 9.99999999999999972e58

   1. Initial program 90.0%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 68.1%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 68.1%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell} + 1\right)\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t \cdot \left(t \cdot 2\right)}{\ell \cdot \ell} + 1\right)\right)\right)\right) \]

      3. frac-times N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t}{\ell} \cdot \frac{t \cdot 2}{\ell} + 1\right)\right)\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1\right)\right)\right)\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(\frac{t}{\ell}\right), \left(\frac{t}{\ell} \cdot 2\right), 1\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \left(\frac{t}{\ell} \cdot 2\right), 1\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\left(\frac{t}{\ell}\right), 2\right), 1\right)\right)\right)\right) \]

      8. /-lowering-/.f64 88.4%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{/.f64}\left(t, \ell\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right), 1\right)\right)\right)\right) \]

   7. Applied egg-rr 88.4%

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

   if 9.99999999999999972e58 < (/.f64 t l)

   1. Initial program 61.4%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 45.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 45.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 98.1%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 98.1%

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

Alternative 8: 80.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 3.2 \cdot 10^{-129}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{elif}\;t\_m \leq 1.22 \cdot 10^{+145}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t\_m \cdot t\_m}{l\_m \cdot l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 3.2e-129)
   (asin (sqrt (fma (/ Om Omc) (- 0.0 (/ Om Omc)) 1.0)))
   (if (<= t_m 1.22e+145)
     (asin (sqrt (/ 1.0 (fma 2.0 (/ (* t_m t_m) (* l_m l_m)) 1.0))))
     (asin (/ (* l_m (sqrt 0.5)) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 3.2e-129) {
		tmp = asin(sqrt(fma((Om / Omc), (0.0 - (Om / Omc)), 1.0)));
	} else if (t_m <= 1.22e+145) {
		tmp = asin(sqrt((1.0 / fma(2.0, ((t_m * t_m) / (l_m * l_m)), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 3.2e-129)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(0.0 - Float64(Om / Omc)), 1.0)));
	elseif (t_m <= 1.22e+145)
		tmp = asin(sqrt(Float64(1.0 / fma(2.0, Float64(Float64(t_m * t_m) / Float64(l_m * l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 3.2e-129], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(0.0 - N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$m, 1.22e+145], N[ArcSin[N[Sqrt[N[(1.0 / N[(2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + 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 3 regimes

2. if t < 3.2000000000000003e-129

   1. Initial program 88.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 53.3%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

   5. Simplified 53.3%

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

      1. times-frac N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) + 1\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{Om}{Omc} \cdot \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right) + 1\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\left(\frac{Om}{Omc}\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right)\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\left(\frac{Om}{Omc}\right)\right), 1\right)\right)\right) \]

      10. /-lowering-/.f64 59.1%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(Om, Omc\right)\right), 1\right)\right)\right) \]

   7. Applied egg-rr 59.1%

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

   if 3.2000000000000003e-129 < t < 1.21999999999999994e145

   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. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) + 1\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(2 \cdot \frac{t}{\ell}\right) \cdot \frac{t}{\ell} + 1\right)\right)\right)\right) \]

      4. unpow1 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(2 \cdot \frac{t}{\ell}\right) \cdot {\left(\frac{t}{\ell}\right)}^{1} + 1\right)\right)\right)\right) \]

      5. sqr-pow N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(2 \cdot \frac{t}{\ell}\right) \cdot \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) + 1\right)\right)\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \left(\left(\left(2 \cdot \frac{t}{\ell}\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)} + 1\right)\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\left(\left(2 \cdot \frac{t}{\ell}\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\left(\left(\frac{t}{\ell} \cdot 2\right) \cdot {\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(\frac{t}{\ell} \cdot 2\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \frac{t}{\ell}\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{t}{\ell}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\frac{1}{2}}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      14. unpow1/2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \left(\sqrt{\frac{t}{\ell}}\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      15. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{t}{\ell}\right)\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right), 1\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \left({\left(\frac{t}{\ell}\right)}^{\frac{1}{2}}\right), 1\right)\right)\right)\right) \]

      18. unpow1/2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \left(\sqrt{\frac{t}{\ell}}\right), 1\right)\right)\right)\right) \]

      19. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{t}{\ell}\right)\right), 1\right)\right)\right)\right) \]

      20. /-lowering-/.f64 41.7%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), 2\right)\right), \mathsf{fma.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, \mathsf{/.f64}\left(t, \ell\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(t, \ell\right)\right), 1\right)\right)\right)\right) \]

   4. Applied egg-rr 41.7%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(2, \left(\frac{{t}^{2}}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(2, \mathsf{/.f64}\left(\left({t}^{2}\right), \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot t\right), \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      8. *-lowering-*.f64 84.4%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   7. Simplified 84.4%

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

   if 1.21999999999999994e145 < t

   1. Initial program 69.7%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 38.0%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 38.0%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 65.5%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 65.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{-129}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+145}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\ell \cdot \ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.9 \cdot 10^{-158}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{\mathsf{fma}\left(t\_m, t\_m \cdot \frac{2}{l\_m \cdot l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.9e-158)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin (/ 1.0 (sqrt (fma t_m (* t_m (/ 2.0 (* l_m l_m))) 1.0))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.9e-158) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin((1.0 / sqrt(fma(t_m, (t_m * (2.0 / (l_m * l_m))), 1.0))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.9e-158)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(Float64(1.0 / sqrt(fma(t_m, Float64(t_m * Float64(2.0 / Float64(l_m * l_m))), 1.0))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 1.9e-158], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 / N[Sqrt[N[(t$95$m * N[(t$95$m * N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8999999999999999e-158

   1. Initial program 84.0%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 60.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 60.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 33.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 33.7%

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

   if 1.8999999999999999e-158 < l

   1. Initial program 87.5%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 70.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 70.5%

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

      1. frac-2neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\left(t \cdot t\right) \cdot \frac{2}{\ell \cdot \ell} + 1\right)\right)}}\right)\right) \]

      2. distribute-neg-in N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(1\right)}{\left(\mathsf{neg}\left(\left(t \cdot t\right) \cdot \frac{2}{\ell \cdot \ell}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(1\right)}{\left(\mathsf{neg}\left(\frac{\left(t \cdot t\right) \cdot 2}{\ell \cdot \ell}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(1\right)}{\left(\mathsf{neg}\left(\frac{t \cdot \left(t \cdot 2\right)}{\ell \cdot \ell}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right)\right) \]

      5. frac-times N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(1\right)}{\left(\mathsf{neg}\left(\frac{t}{\ell} \cdot \frac{t \cdot 2}{\ell}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(1\right)}{\left(\mathsf{neg}\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right)\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1\right)\right)}}\right)\right) \]

      8. frac-2neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\sqrt{\frac{1}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1}}\right)\right) \]

      9. sqrt-div N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{\sqrt{1}}{\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\left(\frac{1}{\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(1, \left(\sqrt{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1}\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(1, \mathsf{sqrt.f64}\left(\left(\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot 2\right) + 1\right)\right)\right)\right) \]

   7. Applied egg-rr 80.7%

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

Alternative 10: 84.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.9 \cdot 10^{-158}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, t\_m \cdot \frac{2}{l\_m \cdot l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.9e-158)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin (sqrt (/ 1.0 (fma t_m (* t_m (/ 2.0 (* l_m l_m))) 1.0))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (l_m <= 1.9e-158) {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	} else {
		tmp = asin(sqrt((1.0 / fma(t_m, (t_m * (2.0 / (l_m * l_m))), 1.0))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (l_m <= 1.9e-158)
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(t_m * Float64(2.0 / Float64(l_m * l_m))), 1.0))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[l$95$m, 1.9e-158], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(t$95$m * N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8999999999999999e-158

   1. Initial program 84.0%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 60.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 60.5%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 33.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 33.7%

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

   if 1.8999999999999999e-158 < l

   1. Initial program 87.5%

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

      1. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right)\right)\right)\right)\right)\right) \]

      2. frac-times N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{Om \cdot Om}{Omc \cdot Omc}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right)\right)\right)\right)\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left(Om \cdot \frac{Om}{Omc \cdot Omc}\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(Om, \left(\frac{Om}{Omc \cdot Omc}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right)\right)\right)\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, \left(Omc \cdot Omc\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 83.9%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(Om, \mathsf{/.f64}\left(Om, \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{pow.f64}\left(\mathsf{/.f64}\left(t, \ell\right), 2\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 83.9%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\left(t \cdot t\right) \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(t \cdot \left(t \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right)\right) + 1\right)\right)\right)\right) \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(t, \left(t \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right)\right), 1\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(t, \mathsf{*.f64}\left(t, \left(2 \cdot \frac{1}{{\ell}^{2}}\right)\right), 1\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(t, \mathsf{*.f64}\left(t, \left(\frac{2 \cdot 1}{{\ell}^{2}}\right)\right), 1\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(t, \mathsf{*.f64}\left(t, \left(\frac{2}{{\ell}^{2}}\right)\right), 1\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right)\right), 1\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right)\right), 1\right)\right)\right)\right) \]

      16. *-lowering-*.f64 80.7%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(t, \mathsf{*.f64}\left(t, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), 1\right)\right)\right)\right) \]

   7. Simplified 80.7%

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

Alternative 11: 73.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 4.6 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 4.6e+59)
   (asin (sqrt (fma (/ Om Omc) (- 0.0 (/ Om Omc)) 1.0)))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 4.6e+59) {
		tmp = asin(sqrt(fma((Om / Omc), (0.0 - (Om / Omc)), 1.0)));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 4.6e+59)
		tmp = asin(sqrt(fma(Float64(Om / Omc), Float64(0.0 - Float64(Om / Omc)), 1.0)));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 4.6e+59], N[ArcSin[N[Sqrt[N[(N[(Om / Omc), $MachinePrecision] * N[(0.0 - N[(Om / Omc), $MachinePrecision]), $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 t < 4.60000000000000016e59

   1. Initial program 87.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

   5. Simplified 54.8%

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

      1. times-frac N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(1 + \left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left({\left(\frac{Om}{Omc}\right)}^{2}\right)\right) + 1\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\mathsf{neg}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) + 1\right)\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{Om}{Omc} \cdot \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right) + 1\right)\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\left(\frac{Om}{Omc}\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \left(\mathsf{neg}\left(\frac{Om}{Omc}\right)\right), 1\right)\right)\right) \]

      9. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\left(\frac{Om}{Omc}\right)\right), 1\right)\right)\right) \]

      10. /-lowering-/.f64 60.5%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(Om, Omc\right)\right), 1\right)\right)\right) \]

   7. Applied egg-rr 60.5%

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

   if 4.60000000000000016e59 < t

   1. Initial program 75.6%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 61.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 61.7%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.6 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc}, 0 - \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 5.6 \cdot 10^{+59}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc} \cdot -0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 5.6e+59)
   (asin (fma (/ Om Omc) (* (/ Om Omc) -0.5) 1.0))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 5.6e+59) {
		tmp = asin(fma((Om / Omc), ((Om / Omc) * -0.5), 1.0));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 5.6e+59)
		tmp = asin(fma(Float64(Om / Omc), Float64(Float64(Om / Omc) * -0.5), 1.0));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 5.6e+59], N[ArcSin[N[(N[(Om / Omc), $MachinePrecision] * N[(N[(Om / Omc), $MachinePrecision] * -0.5), $MachinePrecision] + 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 t < 5.5999999999999996e59

   1. Initial program 87.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

   5. Simplified 54.8%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{{Om}^{2}}{{Omc}^{2}}\right)}\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{-1}{2} \cdot \frac{{Om}^{2}}{{Omc}^{2}} + 1\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{{Om}^{2}}{{Omc}^{2}} \cdot \frac{-1}{2} + 1\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\left(\frac{{Om}^{2}}{{Omc}^{2}}\right), \frac{-1}{2}, 1\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right), \frac{-1}{2}, 1\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right), \frac{-1}{2}, 1\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right), \frac{-1}{2}, 1\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right), \frac{-1}{2}, 1\right)\right) \]

      8. *-lowering-*.f64 54.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right), \frac{-1}{2}, 1\right)\right) \]

   8. Simplified 54.7%

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

      1. times-frac N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \frac{-1}{2} + 1\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{asin.f64}\left(\left(\frac{Om}{Omc} \cdot \left(\frac{Om}{Omc} \cdot \frac{-1}{2}\right) + 1\right)\right) \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\left(\frac{Om}{Omc}\right), \left(\frac{Om}{Omc} \cdot \frac{-1}{2}\right), 1\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \left(\frac{Om}{Omc} \cdot \frac{-1}{2}\right), 1\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{*.f64}\left(\left(\frac{Om}{Omc}\right), \frac{-1}{2}\right), 1\right)\right) \]

      6. /-lowering-/.f64 60.2%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{fma.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(Om, Omc\right), \frac{-1}{2}\right), 1\right)\right) \]

   10. Applied egg-rr 60.2%

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

   if 5.5999999999999996e59 < t

   1. Initial program 75.6%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 61.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 61.7%

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

Alternative 13: 72.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 3.6 \cdot 10^{+65}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 3.6e+65) (asin 1.0) (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 3.6e+65) {
		tmp = asin(1.0);
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
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 <= 3.6d+65) then
        tmp = asin(1.0d0)
    else
        tmp = asin(((l_m * sqrt(0.5d0)) / t_m))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 3.6e+65) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 3.6e+65:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 3.6e+65)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (t_m <= 3.6e+65)
		tmp = asin(1.0);
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 3.6e+65], N[ArcSin[1.0], $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 t < 3.59999999999999978e65

   1. Initial program 87.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 54.8%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

   5. Simplified 54.8%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{1}\right) \]
   7. Step-by-step derivation

   8. Simplified 59.2%

      \[\leadsto \sin^{-1} \color{blue}{1} \]

   if 3.59999999999999978e65 < t

   1. Initial program 75.6%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot {t}^{2}}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{{t}^{2} \cdot 2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \frac{2 \cdot 1}{{\ell}^{2}} + 1\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \left({t}^{2} \cdot \left(2 \cdot \frac{1}{{\ell}^{2}}\right) + 1\right)\right)\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left({t}^{2}\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\left(t \cdot t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(2 \cdot \frac{1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2 \cdot 1}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \left(\frac{2}{{\ell}^{2}}\right), 1\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left({\ell}^{2}\right)\right), 1\right)\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \left(\ell \cdot \ell\right)\right), 1\right)\right)\right)\right) \]

      15. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, \mathsf{fma.f64}\left(\mathsf{*.f64}\left(t, t\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\ell, \ell\right)\right), 1\right)\right)\right)\right) \]

   5. Simplified 52.2%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{asin.f64}\left(\color{blue}{\left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{t}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \sqrt{\frac{1}{2}}\right), t\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\sqrt{\frac{1}{2}}\right)\right), t\right)\right) \]

      3. sqrt-lowering-sqrt.f64 61.7%

         \[\leadsto \mathsf{asin.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{sqrt.f64}\left(\frac{1}{2}\right)\right), t\right)\right) \]

   8. Simplified 61.7%

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

Alternative 14: 51.6% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

t_m = abs(t)
l_m = abs(l)
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(1.0d0)
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	return Math.asin(1.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}\right)\right) \]
4. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{{Om}^{2}}{{Omc}^{2}}\right)\right)\right)\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left({Om}^{2}\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(Om \cdot Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left({Omc}^{2}\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \left(Omc \cdot Omc\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 47.6%

      \[\leadsto \mathsf{asin.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(Om, Om\right), \mathsf{*.f64}\left(Omc, Omc\right)\right)\right)\right)\right) \]

5. Simplified 47.6%

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

6. Taylor expanded in Om around 0

   \[\leadsto \mathsf{asin.f64}\left(\color{blue}{1}\right) \]
7. Step-by-step derivation

8. Simplified 51.6%

   \[\leadsto \sin^{-1} \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024193 
(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)))))))