Toniolo and Linder, Equation (2)

Percentage Accurate: 83.7% → 97.6%

Time: 14.8s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+113}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{\mathsf{fma}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \end{array} \]

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) 4e+113)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (/ (* t_m (/ t_m l_m)) l_m))))))
   (asin (* (/ l_m t_m) (sqrt (fma (* (/ Om Omc) (/ Om Omc)) -0.5 0.5))))))

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) <= 4e+113) {
		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 / t_m) * sqrt(fma(((Om / Omc) * (Om / Omc)), -0.5, 0.5))));
	}
	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) <= 4e+113)
		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 / t_m) * sqrt(fma(Float64(Float64(Om / Omc) * Float64(Om / Omc)), -0.5, 0.5))));
	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], 4e+113], 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 / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4e113

   1. Initial program 88.0%

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

      1. unpow2 N/A

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

      2. associate-*r/ N/A

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

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

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

      4. *-commutative N/A

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

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

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

      6. /-lowering-/.f64 87.5

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

   4. Applied egg-rr 87.5%

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

   if 4e113 < (/.f64 t l)

   1. Initial program 50.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 40.7%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 86.4

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

   8. Simplified 86.4%

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

      1. times-frac N/A

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

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

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

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

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

      4. /-lowering-/.f64 99.5

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

   10. Applied egg-rr 99.5%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+113}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{\mathsf{fma}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \end{array} \]

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) 4e+113)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0))))
   (asin (* (/ l_m t_m) (sqrt (fma (* (/ Om Omc) (/ Om Omc)) -0.5 0.5))))))

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) <= 4e+113) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(fma(((Om / Omc) * (Om / Omc)), -0.5, 0.5))));
	}
	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) <= 4e+113)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(fma(Float64(Float64(Om / Omc) * Float64(Om / Omc)), -0.5, 0.5))));
	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], 4e+113], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4e113

   1. Initial program 88.0%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. /-lowering-/.f64 88.0

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

   4. Applied egg-rr 88.0%

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

   if 4e113 < (/.f64 t l)

   1. Initial program 50.9%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 40.7%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 86.4

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

   8. Simplified 86.4%

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

      1. times-frac N/A

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

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

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

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

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

      4. /-lowering-/.f64 99.5

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 4 \cdot 10^{+113}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\mathsf{fma}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{\mathsf{fma}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \end{array} \]

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) 0.01)
   (asin (sqrt (- 1.0 (pow (/ Om Omc) 2.0))))
   (asin (* (/ l_m t_m) (sqrt (fma (* (/ Om Omc) (/ Om Omc)) -0.5 0.5))))))

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) <= 0.01) {
		tmp = asin(sqrt((1.0 - pow((Om / Omc), 2.0))));
	} else {
		tmp = asin(((l_m / t_m) * sqrt(fma(((Om / Omc) * (Om / Omc)), -0.5, 0.5))));
	}
	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) <= 0.01)
		tmp = asin(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))));
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(fma(Float64(Float64(Om / Omc) * Float64(Om / Omc)), -0.5, 0.5))));
	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], 0.01], N[ArcSin[N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.1%

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

   3. Taylor expanded in t around 0

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

   5. Simplified 63.5%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 64.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 40.2%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 89.3

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

   8. Simplified 89.3%

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

      1. times-frac N/A

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

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

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

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

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

      4. /-lowering-/.f64 98.8

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

   10. Applied egg-rr 98.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.01:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t} \cdot \sqrt{\mathsf{fma}\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}, -0.5, 0.5\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.1% accurate, 2.0× speedup

Math

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

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) 0.01)
   (asin 1.0)
   (asin (* (/ l_m t_m) (sqrt (fma (* (/ Om Omc) (/ Om Omc)) -0.5 0.5))))))

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.1%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 55.1

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

   5. Simplified 55.1%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 62.9%

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

   if 0.0100000000000000002 < (/.f64 t l)

   1. Initial program 64.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 40.2%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 89.3

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

   8. Simplified 89.3%

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

      1. times-frac N/A

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

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

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

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

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

      4. /-lowering-/.f64 98.8

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

   10. Applied egg-rr 98.8%

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

Alternative 5: 79.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{l\_m \cdot 0.5} \cdot \sqrt{l\_m}}{t\_m}\right)\\ \mathbf{elif}\;l\_m \leq 1.36 \cdot 10^{+129}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m \cdot t\_m, \frac{2}{l\_m \cdot l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 4.2e-88)
   (asin (/ (* (sqrt (* l_m 0.5)) (sqrt l_m)) t_m))
   (if (<= l_m 1.36e+129)
     (asin (sqrt (/ 1.0 (fma (* t_m t_m) (/ 2.0 (* l_m l_m)) 1.0))))
     (asin 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 <= 4.2e-88) {
		tmp = asin(((sqrt((l_m * 0.5)) * sqrt(l_m)) / t_m));
	} else if (l_m <= 1.36e+129) {
		tmp = asin(sqrt((1.0 / fma((t_m * t_m), (2.0 / (l_m * l_m)), 1.0))));
	} else {
		tmp = asin(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 <= 4.2e-88)
		tmp = asin(Float64(Float64(sqrt(Float64(l_m * 0.5)) * sqrt(l_m)) / t_m));
	elseif (l_m <= 1.36e+129)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m * t_m), Float64(2.0 / Float64(l_m * l_m)), 1.0))));
	else
		tmp = asin(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, 4.2e-88], N[ArcSin[N[(N[(N[Sqrt[N[(l$95$m * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], If[LessEqual[l$95$m, 1.36e+129], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4.1999999999999999e-88

   1. Initial program 80.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 24.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]
   6. Step-by-step derivation

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

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

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

   7. Applied egg-rr 29.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{\left(\mathsf{fma}\left(Om, \frac{Om}{\mathsf{fma}\left(Omc, 0 - Omc, 0\right)} \cdot 0.5, 0.5\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}}{t}\right)} \]

   8. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 32.6

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

   10. Simplified 32.6%

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

       1. pow1/2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

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

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

       6. pow1/2 N/A

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

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

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

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

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

       9. pow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 15.9

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

   12. Applied egg-rr 15.9%

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

   if 4.1999999999999999e-88 < l < 1.3599999999999999e129

   1. Initial program 81.4%

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

   3. Taylor expanded in Om around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

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

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

      9. unpow2 N/A

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

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

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 77.6

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

   5. Simplified 77.6%

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

   if 1.3599999999999999e129 < l

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.5

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

   5. Simplified 85.5%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 89.2%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.2 \cdot 10^{-72}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{l\_m \cdot 0.5} \cdot \sqrt{l\_m}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.2e-72)
   (asin (/ (* (sqrt (* l_m 0.5)) (sqrt l_m)) t_m))
   (asin 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 <= 1.2e-72) {
		tmp = asin(((sqrt((l_m * 0.5)) * sqrt(l_m)) / t_m));
	} else {
		tmp = asin(1.0);
	}
	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 (l_m <= 1.2d-72) then
        tmp = asin(((sqrt((l_m * 0.5d0)) * sqrt(l_m)) / t_m))
    else
        tmp = asin(1.0d0)
    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 (l_m <= 1.2e-72) {
		tmp = Math.asin(((Math.sqrt((l_m * 0.5)) * Math.sqrt(l_m)) / t_m));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 1.2e-72:
		tmp = math.asin(((math.sqrt((l_m * 0.5)) * math.sqrt(l_m)) / t_m))
	else:
		tmp = math.asin(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 <= 1.2e-72)
		tmp = asin(Float64(Float64(sqrt(Float64(l_m * 0.5)) * sqrt(l_m)) / t_m));
	else
		tmp = asin(1.0);
	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 (l_m <= 1.2e-72)
		tmp = asin(((sqrt((l_m * 0.5)) * sqrt(l_m)) / t_m));
	else
		tmp = asin(1.0);
	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[l$95$m, 1.2e-72], N[ArcSin[N[(N[(N[Sqrt[N[(l$95$m * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[l$95$m], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.2e-72

   1. Initial program 79.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 24.6%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]
   6. Step-by-step derivation

      1. div-inv N/A

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

      2. associate-*r/ N/A

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

      3. sqrt-div N/A

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

      4. sqrt-prod N/A

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

      5. rem-square-sqrt N/A

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

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

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

   7. Applied egg-rr 28.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{\left(\mathsf{fma}\left(Om, \frac{Om}{\mathsf{fma}\left(Omc, 0 - Omc, 0\right)} \cdot 0.5, 0.5\right) \cdot \left(\ell \cdot \ell\right)\right) \cdot 1}}{t}\right)} \]

   8. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 32.2

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

   10. Simplified 32.2%

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

       1. pow1/2 N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. unpow-prod-down N/A

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

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

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

       6. pow1/2 N/A

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

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

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

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

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

       9. pow1/2 N/A

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

       10. sqrt-lowering-sqrt.f64 15.7

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

   12. Applied egg-rr 15.7%

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

   if 1.2e-72 < l

   1. Initial program 90.8%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 76.3%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 72.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 2.7 \cdot 10^{-75}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{\frac{t\_m}{\sqrt{0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 2.7e-75) (asin (/ l_m (/ t_m (sqrt 0.5)))) (asin 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 <= 2.7e-75) {
		tmp = asin((l_m / (t_m / sqrt(0.5))));
	} else {
		tmp = asin(1.0);
	}
	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 (l_m <= 2.7d-75) then
        tmp = asin((l_m / (t_m / sqrt(0.5d0))))
    else
        tmp = asin(1.0d0)
    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 (l_m <= 2.7e-75) {
		tmp = Math.asin((l_m / (t_m / Math.sqrt(0.5))));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 2.7e-75:
		tmp = math.asin((l_m / (t_m / math.sqrt(0.5))))
	else:
		tmp = math.asin(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 <= 2.7e-75)
		tmp = asin(Float64(l_m / Float64(t_m / sqrt(0.5))));
	else
		tmp = asin(1.0);
	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 (l_m <= 2.7e-75)
		tmp = asin((l_m / (t_m / sqrt(0.5))));
	else
		tmp = asin(1.0);
	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[l$95$m, 2.7e-75], N[ArcSin[N[(l$95$m / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.6999999999999998e-75

   1. Initial program 79.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 24.6%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 33.2

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

   8. Simplified 33.2%

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

   9. Taylor expanded in Om around 0

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

       1. associate-/l* N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 37.0

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

   11. Simplified 37.0%

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

       1. clear-num N/A

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

       2. un-div-inv N/A

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

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

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

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

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

       5. sqrt-lowering-sqrt.f64 37.0

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

   13. Applied egg-rr 37.0%

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

   if 2.6999999999999998e-75 < l

   1. Initial program 90.8%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 76.3%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 73.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 7 \cdot 10^{-75}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 7e-75) (asin (* l_m (/ (sqrt 0.5) t_m))) (asin 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 <= 7e-75) {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	} else {
		tmp = asin(1.0);
	}
	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 (l_m <= 7d-75) then
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    else
        tmp = asin(1.0d0)
    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 (l_m <= 7e-75) {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	} else {
		tmp = Math.asin(1.0);
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if l_m <= 7e-75:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	else:
		tmp = math.asin(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 <= 7e-75)
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	else
		tmp = asin(1.0);
	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 (l_m <= 7e-75)
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	else
		tmp = asin(1.0);
	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[l$95$m, 7e-75], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 6.9999999999999997e-75

   1. Initial program 79.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 24.6%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(0.5, \frac{Om \cdot Om}{0 - Omc \cdot Omc}, 0.5\right)}{t \cdot t}}}\right) \]

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

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

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

      11. unpow2 N/A

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

      12. *-lowering-*.f64 33.2

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

   8. Simplified 33.2%

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

   9. Taylor expanded in Om around 0

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

       1. associate-/l* N/A

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

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

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

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

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

       4. sqrt-lowering-sqrt.f64 37.0

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

   11. Simplified 37.0%

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

   if 6.9999999999999997e-75 < l

   1. Initial program 90.8%

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

   3. Taylor expanded in t around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 70.1

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

   5. Simplified 70.1%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 76.3%

      \[\leadsto \sin^{-1} \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 50.4% 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 82.5%

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

3. Taylor expanded in t around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 44.6

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

5. Simplified 44.6%

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

6. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 50.9%

   \[\leadsto \sin^{-1} \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(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)))))))