Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.9%

Time: 15.1s

Alternatives: 8

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}} \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+144)
   (asin
    (sqrt
     (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
   (asin
    (* (/ l_m (* t_m (sqrt 2.0))) (sqrt (- 1.0 (/ Om (* Omc (/ Omc Om)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+144) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0))))));
	} else {
		tmp = asin(((l_m / (t_m * sqrt(2.0))) * sqrt((1.0 - (Om / (Omc * (Omc / Om)))))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+144) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0))))))
    else
        tmp = asin(((l_m / (t_m * sqrt(2.0d0))) * sqrt((1.0d0 - (om / (omc * (omc / om)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+144) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0))))));
	} else {
		tmp = Math.asin(((l_m / (t_m * Math.sqrt(2.0))) * Math.sqrt((1.0 - (Om / (Omc * (Omc / Om)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+144:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0))))))
	else:
		tmp = math.asin(((l_m / (t_m * math.sqrt(2.0))) * math.sqrt((1.0 - (Om / (Omc * (Omc / Om)))))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+144)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))))));
	else
		tmp = asin(Float64(Float64(l_m / Float64(t_m * sqrt(2.0))) * sqrt(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om)))))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+144)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0))))));
	else
		tmp = asin(((l_m / (t_m * sqrt(2.0))) * sqrt((1.0 - (Om / (Omc * (Omc / Om)))))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+144], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.9999999999999999e144

   1. Initial program 91.1%

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

   if 4.9999999999999999e144 < (/.f64 t l)

   1. Initial program 38.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 51.7%

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

      1. lift-*.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 54.6

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

   9. Applied rewrites 54.6%

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

   10. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 99.6

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

   12. Applied rewrites 99.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}} \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{1}{l\_m}, t\_m \cdot \left(\frac{t\_m}{l\_m} \cdot 2\right), 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}} \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+26)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ 1.0 l_m) (* t_m (* (/ t_m l_m) 2.0)) 1.0))))
   (asin
    (* (/ l_m (* t_m (sqrt 2.0))) (sqrt (- 1.0 (/ Om (* Omc (/ Omc Om)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+26) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((1.0 / l_m), (t_m * ((t_m / l_m) * 2.0)), 1.0))));
	} else {
		tmp = asin(((l_m / (t_m * sqrt(2.0))) * sqrt((1.0 - (Om / (Omc * (Omc / Om)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+26)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(1.0 / l_m), Float64(t_m * Float64(Float64(t_m / l_m) * 2.0)), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m / Float64(t_m * sqrt(2.0))) * sqrt(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om)))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+26], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e26

   1. Initial program 90.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. clear-num N/A

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

      14. associate-/r/ N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 88.0

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

   4. Applied rewrites 88.0%

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

   if 2.0000000000000001e26 < (/.f64 t l)

   1. Initial program 62.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.4

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

   5. Applied rewrites 36.4%

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

   7. Applied rewrites 42.5%

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

      1. lift-*.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 44.6

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

   9. Applied rewrites 44.6%

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

   10. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 99.4

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

   12. Applied rewrites 99.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{1}{\ell}, t \cdot \left(\frac{t}{\ell} \cdot 2\right), 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}} \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t\_m \cdot 2}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}} \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+43)
   (asin (sqrt (/ 1.0 (fma (/ (* t_m 2.0) l_m) (/ t_m l_m) 1.0))))
   (asin
    (* (/ l_m (* t_m (sqrt 2.0))) (sqrt (- 1.0 (/ Om (* Omc (/ Omc Om)))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+43) {
		tmp = asin(sqrt((1.0 / fma(((t_m * 2.0) / l_m), (t_m / l_m), 1.0))));
	} else {
		tmp = asin(((l_m / (t_m * sqrt(2.0))) * sqrt((1.0 - (Om / (Omc * (Omc / Om)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+43)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(Float64(t_m * 2.0) / l_m), Float64(t_m / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m / Float64(t_m * sqrt(2.0))) * sqrt(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om)))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+43], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(N[(t$95$m * 2.0), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.00000000000000003e43

   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

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-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. lower-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. lower-*.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. lower-/.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. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 89.7

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

   7. Applied rewrites 89.7%

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

   if 2.00000000000000003e43 < (/.f64 t l)

   1. Initial program 62.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 43.1%

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

      1. lift-*.f64 N/A

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

      2. associate-*r/ N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. *-lft-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 45.2

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

   9. Applied rewrites 45.2%

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

   10. Taylor expanded in l around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 99.4

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

   12. Applied rewrites 99.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t \cdot 2}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}} \cdot \sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+41) {
		tmp = asin(sqrt((1.0 / fma(((t_m * 2.0) / l_m), (t_m / l_m), 1.0))));
	} else {
		tmp = asin((sqrt(fma((Om / -Omc), (Om / Omc), 1.0)) * ((l_m * sqrt(0.5)) / t_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+41)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(Float64(t_m * 2.0) / l_m), Float64(t_m / l_m), 1.0))));
	else
		tmp = asin(Float64(sqrt(fma(Float64(Om / Float64(-Omc)), Float64(Om / Omc), 1.0)) * Float64(Float64(l_m * sqrt(0.5)) / t_m)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 90.0%

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

   3. 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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-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. lower-*.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. lower-/.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. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 89.7

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

   7. Applied rewrites 89.7%

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

   if 5.00000000000000022e41 < (/.f64 t l)

   1. Initial program 62.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 85.0

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

   5. Applied rewrites 85.0%

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-frac-neg2 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 99.5

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

      11. lift-/.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. distribute-frac-neg2 N/A

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

      14. distribute-frac-neg N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 92.1%

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

Alternative 5: 97.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.1%

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

   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. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-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. lower-*.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. lower-/.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. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 90.6

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

   7. Applied rewrites 90.6%

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

   if 2e145 < (/.f64 t l)

   1. Initial program 36.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 36.6

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

   5. Applied rewrites 36.6%

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

   7. Applied rewrites 50.3%

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

   8. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 98.5

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

   10. Applied rewrites 98.5%

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

       1. lift-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. div-inv N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 98.6

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

   12. Applied rewrites 98.6%

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

Alternative 6: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{-Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.005)
   (asin (sqrt (fma (/ Om (- Omc)) (/ Om Omc) 1.0)))
   (asin (/ l_m (* t_m (sqrt 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin(sqrt(fma((Om / -Omc), (Om / Omc), 1.0)));
	} else {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.005)
		tmp = asin(sqrt(fma(Float64(Om / Float64(-Omc)), Float64(Om / Omc), 1.0)));
	else
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(N[(Om / (-Omc)), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

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

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

      11. lift-/.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lower-fma.f64 66.6

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

   7. Applied rewrites 66.6%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 39.5

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

   5. Applied rewrites 39.5%

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 96.7

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

   10. Applied rewrites 96.7%

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

Alternative 7: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.005) (asin 1.0) (asin (/ l_m (* t_m (sqrt 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.005d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.005:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.005)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.005)
		tmp = asin(1.0);
	else
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 59.5

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 66.3%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 39.5

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

   5. Applied rewrites 39.5%

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

   7. Applied rewrites 40.8%

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

   8. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 96.7

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

   10. Applied rewrites 96.7%

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

Alternative 8: 49.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(1.0d0)
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	return Math.asin(1.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[1.0], $MachinePrecision]

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 44.5

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

5. Applied rewrites 44.5%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 49.6%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))