Toniolo and Linder, Equation (2)

Percentage Accurate: 83.4% → 98.7%

Time: 15.2s

Alternatives: 10

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: 83.4% 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: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot {0.5}^{-0.5}}\right)\\ \end{array} \end{array} \]

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) 5e+137)
   (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 (pow 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) <= 5e+137) {
		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 * pow(0.5, -0.5))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+137) 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 * (0.5d0 ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+137) {
		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.pow(0.5, -0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+137:
		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.pow(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) <= 5e+137)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))))));
	else
		tmp = asin(Float64(l_m / Float64(t_m * (0.5 ^ -0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+137)
		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 * (0.5 ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+137], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.0000000000000002e137

   1. Initial program 91.5%

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

   if 5.0000000000000002e137 < (/.f64 t l)

   1. Initial program 55.2%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 55.2%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 99.7%

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

   8. Simplified 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. pow1/2 N/A

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

       5. pow-flip N/A

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

       6. metadata-eval N/A

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

       7. pow-lowering-pow.f64 99.7%

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

   12. Applied egg-rr 99.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+137}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot {0.5}^{-0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t\_m}{l\_m \cdot \frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot {0.5}^{-0.5}}\right)\\ \end{array} \end{array} \]

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) 5e+37)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (/ t_m (* l_m (/ l_m t_m))))))))
   (asin (/ l_m (* t_m (pow 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) <= 5e+37) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (t_m / (l_m * (l_m / t_m))))))));
	} else {
		tmp = asin((l_m / (t_m * pow(0.5, -0.5))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+37) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * (t_m / (l_m * (l_m / t_m))))))))
    else
        tmp = asin((l_m / (t_m * (0.5d0 ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+37) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (t_m / (l_m * (l_m / t_m))))))));
	} else {
		tmp = Math.asin((l_m / (t_m * Math.pow(0.5, -0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+37:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * (t_m / (l_m * (l_m / t_m))))))))
	else:
		tmp = math.asin((l_m / (t_m * math.pow(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) <= 5e+37)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(t_m / Float64(l_m * Float64(l_m / t_m))))))));
	else
		tmp = asin(Float64(l_m / Float64(t_m * (0.5 ^ -0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+37)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * (t_m / (l_m * (l_m / t_m))))))));
	else
		tmp = asin((l_m / (t_m * (0.5 ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+37], 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[(t$95$m / N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999989e37

   1. Initial program 90.5%

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

      1. unpow2 N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 88.4%

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

   4. Applied egg-rr 88.4%

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

   if 4.99999999999999989e37 < (/.f64 t l)

   1. Initial program 73.3%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 72.6%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 98.8%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. pow1/2 N/A

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

       5. pow-flip N/A

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

       6. metadata-eval N/A

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

       7. pow-lowering-pow.f64 99.0%

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

   12. Applied egg-rr 99.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t}{\ell \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot {0.5}^{-0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot {0.5}^{-0.5}}\right)\\ \end{array} \end{array} \]

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) 5e+37)
   (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 (pow 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) <= 5e+37) {
		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 * pow(0.5, -0.5))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+37) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m * (t_m / l_m)) / l_m))))))
    else
        tmp = asin((l_m / (t_m * (0.5d0 ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+37) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	} else {
		tmp = Math.asin((l_m / (t_m * Math.pow(0.5, -0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+37:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))))
	else:
		tmp = math.asin((l_m / (t_m * math.pow(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) <= 5e+37)
		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(l_m / Float64(t_m * (0.5 ^ -0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+37)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m * (t_m / l_m)) / l_m))))));
	else
		tmp = asin((l_m / (t_m * (0.5 ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+37], 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[(l$95$m / N[(t$95$m * N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999989e37

   1. Initial program 90.5%

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

      1. unpow2 N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. /-lowering-/.f64 87.6%

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

   4. Applied egg-rr 87.6%

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

   if 4.99999999999999989e37 < (/.f64 t l)

   1. Initial program 73.3%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 72.6%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 98.8%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. pow1/2 N/A

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

       5. pow-flip N/A

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

       6. metadata-eval N/A

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

       7. pow-lowering-pow.f64 99.0%

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

   12. Applied egg-rr 99.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot {0.5}^{-0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{t\_m \cdot 2}{\frac{l\_m}{\frac{t\_m}{l\_m}}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot {0.5}^{-0.5}}\right)\\ \end{array} \end{array} \]

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) 5e+37)
   (asin (pow (+ 1.0 (/ (* t_m 2.0) (/ l_m (/ t_m l_m)))) -0.5))
   (asin (/ l_m (* t_m (pow 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) <= 5e+37) {
		tmp = asin(pow((1.0 + ((t_m * 2.0) / (l_m / (t_m / l_m)))), -0.5));
	} else {
		tmp = asin((l_m / (t_m * pow(0.5, -0.5))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+37) then
        tmp = asin(((1.0d0 + ((t_m * 2.0d0) / (l_m / (t_m / l_m)))) ** (-0.5d0)))
    else
        tmp = asin((l_m / (t_m * (0.5d0 ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+37) {
		tmp = Math.asin(Math.pow((1.0 + ((t_m * 2.0) / (l_m / (t_m / l_m)))), -0.5));
	} else {
		tmp = Math.asin((l_m / (t_m * Math.pow(0.5, -0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+37:
		tmp = math.asin(math.pow((1.0 + ((t_m * 2.0) / (l_m / (t_m / l_m)))), -0.5))
	else:
		tmp = math.asin((l_m / (t_m * math.pow(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) <= 5e+37)
		tmp = asin((Float64(1.0 + Float64(Float64(t_m * 2.0) / Float64(l_m / Float64(t_m / l_m)))) ^ -0.5));
	else
		tmp = asin(Float64(l_m / Float64(t_m * (0.5 ^ -0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+37)
		tmp = asin(((1.0 + ((t_m * 2.0) / (l_m / (t_m / l_m)))) ^ -0.5));
	else
		tmp = asin((l_m / (t_m * (0.5 ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+37], N[ArcSin[N[Power[N[(1.0 + N[(N[(t$95$m * 2.0), $MachinePrecision] / N[(l$95$m / N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999989e37

   1. Initial program 90.5%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 90.0%

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

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

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

      2. pow1/2 N/A

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

      3. inv-pow N/A

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

      4. unpow2 N/A

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

      5. clear-num N/A

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

      6. times-frac N/A

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

      7. pow-pow N/A

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

      8. +-commutative N/A

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

      9. fma-define N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. frac-times N/A

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

      13. fma-define N/A

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

      14. associate-/l* N/A

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

      15. +-commutative N/A

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

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

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

   7. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\sin^{-1} \left({\left(1 + \frac{2}{\frac{\frac{\ell}{\frac{t}{\ell}}}{t}}\right)}^{-0.5}\right)} \]
   8. Step-by-step derivation

      1. associate-/r/ N/A

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

      2. associate-*l/ N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 87.9%

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

   9. Applied egg-rr 87.9%

      \[\leadsto \sin^{-1} \left({\left(1 + \color{blue}{\frac{2 \cdot t}{\frac{\ell}{\frac{t}{\ell}}}}\right)}^{-0.5}\right) \]

   if 4.99999999999999989e37 < (/.f64 t l)

   1. Initial program 73.3%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 72.6%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 98.8%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. pow1/2 N/A

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

       5. pow-flip N/A

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

       6. metadata-eval N/A

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

       7. pow-lowering-pow.f64 99.0%

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

   12. Applied egg-rr 99.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{t \cdot 2}{\frac{\ell}{\frac{t}{\ell}}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot {0.5}^{-0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{2}{\frac{\frac{l\_m}{\frac{t\_m}{l\_m}}}{t\_m}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot {0.5}^{-0.5}}\right)\\ \end{array} \end{array} \]

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) 5e+37)
   (asin (pow (+ 1.0 (/ 2.0 (/ (/ l_m (/ t_m l_m)) t_m))) -0.5))
   (asin (/ l_m (* t_m (pow 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) <= 5e+37) {
		tmp = asin(pow((1.0 + (2.0 / ((l_m / (t_m / l_m)) / t_m))), -0.5));
	} else {
		tmp = asin((l_m / (t_m * pow(0.5, -0.5))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+37) then
        tmp = asin(((1.0d0 + (2.0d0 / ((l_m / (t_m / l_m)) / t_m))) ** (-0.5d0)))
    else
        tmp = asin((l_m / (t_m * (0.5d0 ** (-0.5d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+37) {
		tmp = Math.asin(Math.pow((1.0 + (2.0 / ((l_m / (t_m / l_m)) / t_m))), -0.5));
	} else {
		tmp = Math.asin((l_m / (t_m * Math.pow(0.5, -0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+37:
		tmp = math.asin(math.pow((1.0 + (2.0 / ((l_m / (t_m / l_m)) / t_m))), -0.5))
	else:
		tmp = math.asin((l_m / (t_m * math.pow(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) <= 5e+37)
		tmp = asin((Float64(1.0 + Float64(2.0 / Float64(Float64(l_m / Float64(t_m / l_m)) / t_m))) ^ -0.5));
	else
		tmp = asin(Float64(l_m / Float64(t_m * (0.5 ^ -0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+37)
		tmp = asin(((1.0 + (2.0 / ((l_m / (t_m / l_m)) / t_m))) ^ -0.5));
	else
		tmp = asin((l_m / (t_m * (0.5 ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+37], N[ArcSin[N[Power[N[(1.0 + N[(2.0 / N[(N[(l$95$m / N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999989e37

   1. Initial program 90.5%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 90.0%

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

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

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

      2. pow1/2 N/A

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

      3. inv-pow N/A

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

      4. unpow2 N/A

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

      5. clear-num N/A

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

      6. times-frac N/A

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

      7. pow-pow N/A

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

      8. +-commutative N/A

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

      9. fma-define N/A

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

      10. times-frac N/A

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

      11. clear-num N/A

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

      12. frac-times N/A

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

      13. fma-define N/A

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

      14. associate-/l* N/A

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

      15. +-commutative N/A

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

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

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

   7. Applied egg-rr 87.9%

      \[\leadsto \color{blue}{\sin^{-1} \left({\left(1 + \frac{2}{\frac{\frac{\ell}{\frac{t}{\ell}}}{t}}\right)}^{-0.5}\right)} \]

   if 4.99999999999999989e37 < (/.f64 t l)

   1. Initial program 73.3%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 72.6%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 98.8%

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

   8. Simplified 98.8%

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

   10. Applied egg-rr 98.8%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. pow1/2 N/A

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

       5. pow-flip N/A

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

       6. metadata-eval N/A

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

       7. pow-lowering-pow.f64 99.0%

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

   12. Applied egg-rr 99.0%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\sin^{-1} \left({\left(1 + \frac{2}{\frac{\frac{\ell}{\frac{t}{\ell}}}{t}}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot {0.5}^{-0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0002:\\ \;\;\;\;\frac{\pi}{2} - \cos^{-1} \left(1 - \frac{t\_m}{\frac{l\_m}{\frac{t\_m}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot {0.5}^{-0.5}}\right)\\ \end{array} \end{array} \]

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) 0.0002)
   (- (/ PI 2.0) (acos (- 1.0 (/ t_m (/ l_m (/ t_m l_m))))))
   (asin (/ l_m (* t_m (pow 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) <= 0.0002) {
		tmp = (((double) M_PI) / 2.0) - acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	} else {
		tmp = asin((l_m / (t_m * pow(0.5, -0.5))));
	}
	return tmp;
}

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = (Math.PI / 2.0) - Math.acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	} else {
		tmp = Math.asin((l_m / (t_m * Math.pow(0.5, -0.5))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.0002:
		tmp = (math.pi / 2.0) - math.acos((1.0 - (t_m / (l_m / (t_m / l_m)))))
	else:
		tmp = math.asin((l_m / (t_m * math.pow(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) <= 0.0002)
		tmp = Float64(Float64(pi / 2.0) - acos(Float64(1.0 - Float64(t_m / Float64(l_m / Float64(t_m / l_m))))));
	else
		tmp = asin(Float64(l_m / Float64(t_m * (0.5 ^ -0.5))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.0002)
		tmp = (pi / 2.0) - acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	else
		tmp = asin((l_m / (t_m * (0.5 ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0002], N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcCos[N[(1.0 - N[(t$95$m / N[(l$95$m / N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Power[0.5, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 90.1%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 89.6%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 54.7%

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

   8. Simplified 54.7%

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

      1. asin-acos N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \color{blue}{\cos^{-1} \left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \cos^{-1} \color{blue}{\left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \cos^{-1} \left(\color{blue}{1} - \frac{t \cdot t}{\ell \cdot \ell}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

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

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

      7. times-frac N/A

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

      8. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{t}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \left(\frac{\ell}{\frac{t}{\ell}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 64.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 64.9%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 76.0%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 98.1%

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

   8. Simplified 98.1%

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

   10. Applied egg-rr 98.1%

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

       1. clear-num N/A

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

       2. associate-/r/ N/A

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

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

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

       4. pow1/2 N/A

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

       5. pow-flip N/A

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

       6. metadata-eval N/A

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

       7. pow-lowering-pow.f64 98.3%

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

   12. Applied egg-rr 98.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\frac{\pi}{2} - \cos^{-1} \left(1 - \frac{t}{\frac{\ell}{\frac{t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot {0.5}^{-0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0002:\\ \;\;\;\;\frac{\pi}{2} - \cos^{-1} \left(1 - \frac{t\_m}{\frac{l\_m}{\frac{t\_m}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.0002)
   (- (/ PI 2.0) (acos (- 1.0 (/ t_m (/ l_m (/ t_m l_m))))))
   (asin (/ (* l_m (sqrt 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = (((double) M_PI) / 2.0) - acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	} else {
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	}
	return tmp;
}

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = (Math.PI / 2.0) - Math.acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt(0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.0002:
		tmp = (math.pi / 2.0) - math.acos((1.0 - (t_m / (l_m / (t_m / l_m)))))
	else:
		tmp = math.asin(((l_m * math.sqrt(0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.0002)
		tmp = Float64(Float64(pi / 2.0) - acos(Float64(1.0 - Float64(t_m / Float64(l_m / Float64(t_m / l_m))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.0002)
		tmp = (pi / 2.0) - acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	else
		tmp = asin(((l_m * sqrt(0.5)) / t_m));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0002], N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcCos[N[(1.0 - N[(t$95$m / N[(l$95$m / N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 90.1%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 89.6%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 54.7%

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

   8. Simplified 54.7%

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

      1. asin-acos N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \color{blue}{\cos^{-1} \left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \cos^{-1} \color{blue}{\left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \cos^{-1} \left(\color{blue}{1} - \frac{t \cdot t}{\ell \cdot \ell}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

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

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

      7. times-frac N/A

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

      8. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{t}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \left(\frac{\ell}{\frac{t}{\ell}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 64.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 64.9%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 76.0%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 98.1%

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

   8. Simplified 98.1%

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

Alternative 8: 96.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0002:\\ \;\;\;\;\frac{\pi}{2} - \cos^{-1} \left(1 - \frac{t\_m}{\frac{l\_m}{\frac{t\_m}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.0002)
   (- (/ PI 2.0) (acos (- 1.0 (/ t_m (/ l_m (/ t_m l_m))))))
   (asin (* l_m (/ (sqrt 0.5) t_m)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = (((double) M_PI) / 2.0) - acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = (Math.PI / 2.0) - Math.acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.0002:
		tmp = (math.pi / 2.0) - math.acos((1.0 - (t_m / (l_m / (t_m / l_m)))))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.0002)
		tmp = Float64(Float64(pi / 2.0) - acos(Float64(1.0 - Float64(t_m / Float64(l_m / Float64(t_m / l_m))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.0002)
		tmp = (pi / 2.0) - acos((1.0 - (t_m / (l_m / (t_m / l_m)))));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0002], N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcCos[N[(1.0 - N[(t$95$m / N[(l$95$m / N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 90.1%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 89.6%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 54.7%

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

   8. Simplified 54.7%

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

      1. asin-acos N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \color{blue}{\cos^{-1} \left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \cos^{-1} \color{blue}{\left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \cos^{-1} \left(\color{blue}{1} - \frac{t \cdot t}{\ell \cdot \ell}\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\left(1 - \frac{t \cdot t}{\ell \cdot \ell}\right)\right)\right) \]

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

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

      7. times-frac N/A

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

      8. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \left(\frac{t}{\frac{\ell}{\frac{t}{\ell}}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \left(\frac{\ell}{\frac{t}{\ell}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, \left(\frac{t}{\ell}\right)\right)\right)\right)\right)\right) \]

      11. /-lowering-/.f64 64.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{acos.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \ell\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 64.9%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 76.0%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 98.1%

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

   8. Simplified 98.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sqrt-lowering-sqrt.f64 98.0%

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

   10. Applied egg-rr 98.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.0002:\\ \;\;\;\;\frac{\pi}{2} - \cos^{-1} \left(1 - \frac{t}{\frac{\ell}{\frac{t}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 2.7 \cdot 10^{-65}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 2.7e-65) (asin 1.0) (asin (* (/ l_m t_m) (sqrt 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 <= 2.7e-65) {
		tmp = asin(1.0);
	} else {
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (t_m <= 2.7d-65) then
        tmp = asin(1.0d0)
    else
        tmp = asin(((l_m / t_m) * sqrt(0.5d0)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 2.7e-65) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin(((l_m / t_m) * Math.sqrt(0.5)));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if t_m <= 2.7e-65:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin(((l_m / t_m) * math.sqrt(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 (t_m <= 2.7e-65)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(Float64(l_m / t_m) * sqrt(0.5)));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (t_m <= 2.7e-65)
		tmp = asin(1.0);
	else
		tmp = asin(((l_m / t_m) * sqrt(0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 2.7e-65], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.6999999999999999e-65

   1. Initial program 90.3%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 89.6%

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

   6. Taylor expanded in t around 0

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

   8. Simplified 58.3%

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

   if 2.6999999999999999e-65 < t

   1. Initial program 77.4%

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

   3. Taylor expanded in Om around 0

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

   5. Simplified 77.2%

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

   6. Taylor expanded in t around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 52.0%

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

   8. Simplified 52.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*r* N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. sqrt-lowering-sqrt.f64 52.0%

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

   10. Applied egg-rr 52.0%

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

Alternative 10: 50.5% accurate, 4.1× 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 86.5%

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

3. Taylor expanded in Om around 0

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

5. Simplified 86.0%

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

6. Taylor expanded in t around 0

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

8. Simplified 50.1%

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

Reproduce

herbie shell --seed 2024158 
(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)))))))