Toniolo and Linder, Equation (2)

Percentage Accurate: 84.2% → 99.0%

Time: 15.5s

Alternatives: 6

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+134)
   (asin
    (sqrt
     (/
      (- 1.0 (/ Om (* Omc (/ Omc Om))))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (* 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) <= 5e+134) {
		tmp = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * (l_m * (sqrt(0.5) / t_m))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+134) then
        tmp = asin(sqrt(((1.0d0 - (om / (omc * (omc / om)))) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_m)))))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) ** 2.0d0))) * (l_m * (sqrt(0.5d0) / t_m))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+134) {
		tmp = Math.asin(Math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) * (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) <= 5e+134:
		tmp = math.asin(math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))))
	else:
		tmp = math.asin((math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) * (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) <= 5e+134)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om)))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * 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) <= 5e+134)
		tmp = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) ^ 2.0))) * (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], 5e+134], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999981e134

   1. Initial program 92.2%

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

      1. unpow2 92.2%

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

      2. clear-num 92.3%

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

      3. un-div-inv 92.3%

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

   4. Applied egg-rr 92.3%

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

      1. unpow2 98.7%

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

      2. clear-num 98.7%

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

      3. frac-times 98.7%

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

      4. *-un-lft-identity 98.7%

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

   6. Applied egg-rr 92.3%

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

   if 4.99999999999999981e134 < (/.f64 t l)

   1. Initial program 54.5%

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

   3. Taylor expanded in t around inf 97.1%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 97.1%

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

      2. unpow2 97.1%

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

      3. unpow2 97.1%

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

      4. times-frac 99.7%

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

      5. unpow2 99.7%

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

      6. associate-/l* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

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+142)
   (asin
    (sqrt
     (/
      (- 1.0 (/ Om (* Omc (/ Omc Om))))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin (* l_m (/ (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (* t_m (sqrt 2.0)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+142) {
		tmp = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin((l_m * (sqrt((1.0 - pow((Om / Omc), 2.0))) / (t_m * sqrt(2.0)))));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+142) then
        tmp = asin(sqrt(((1.0d0 - (om / (omc * (omc / om)))) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_m)))))))
    else
        tmp = asin((l_m * (sqrt((1.0d0 - ((om / omc) ** 2.0d0))) / (t_m * sqrt(2.0d0)))))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+142) {
		tmp = Math.asin(Math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt((1.0 - Math.pow((Om / Omc), 2.0))) / (t_m * Math.sqrt(2.0)))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+142:
		tmp = math.asin(math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))))
	else:
		tmp = math.asin((l_m * (math.sqrt((1.0 - math.pow((Om / Omc), 2.0))) / (t_m * math.sqrt(2.0)))))
	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+142)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om)))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) / Float64(t_m * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+142)
		tmp = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	else
		tmp = asin((l_m * (sqrt((1.0 - ((Om / Omc) ^ 2.0))) / (t_m * sqrt(2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+142], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.0000000000000001e142

   1. Initial program 92.3%

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

      1. unpow2 92.3%

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

      2. clear-num 92.4%

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

      3. un-div-inv 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. unpow2 98.7%

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

      2. clear-num 98.7%

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

      3. frac-times 98.7%

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

      4. *-un-lft-identity 98.7%

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

   6. Applied egg-rr 92.4%

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

   if 5.0000000000000001e142 < (/.f64 t l)

   1. Initial program 50.7%

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

      1. sqrt-div 50.7%

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

      2. add-sqr-sqrt 50.7%

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

      3. hypot-1-def 50.7%

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

      4. *-commutative 50.7%

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

      5. sqrt-prod 50.8%

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

      6. sqrt-pow1 99.6%

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

      7. metadata-eval 99.6%

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

      8. pow1 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in t around inf 96.9%

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

      1. associate-*l/ 96.9%

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

      2. associate-*r/ 96.8%

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

      3. unpow2 96.8%

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

      4. unpow2 96.8%

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

      5. times-frac 99.6%

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

      6. unpow2 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{t \cdot \sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc \cdot \frac{Omc}{Om}}}}{\mathsf{hypot}\left(1, \frac{t\_m}{l\_m} \cdot \sqrt{2}\right)}\right) \end{array} \]

FPCore

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

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin((sqrt((1.0 - (Om / (Omc * (Omc / Om))))) / hypot(1.0, ((t_m / l_m) * sqrt(2.0)))));
}

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((Math.sqrt((1.0 - (Om / (Omc * (Omc / Om))))) / Math.hypot(1.0, ((t_m / l_m) * Math.sqrt(2.0)))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin((math.sqrt((1.0 - (Om / (Omc * (Omc / Om))))) / math.hypot(1.0, ((t_m / l_m) * math.sqrt(2.0)))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(sqrt(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om))))) / hypot(1.0, Float64(Float64(t_m / l_m) * sqrt(2.0)))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((sqrt((1.0 - (Om / (Omc * (Omc / Om))))) / hypot(1.0, ((t_m / l_m) * sqrt(2.0)))));
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[N[(N[Sqrt[N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. sqrt-div 86.5%

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

   2. add-sqr-sqrt 86.5%

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

   3. hypot-1-def 86.5%

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

   4. *-commutative 86.5%

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

   5. sqrt-prod 86.5%

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

   6. sqrt-pow1 98.8%

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

   7. metadata-eval 98.8%

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

   8. pow1 98.8%

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

4. Applied egg-rr 98.8%

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

   1. unpow2 98.8%

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

   2. clear-num 98.8%

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

   3. frac-times 98.8%

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

   4. *-un-lft-identity 98.8%

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

6. Applied egg-rr 98.8%

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

7. Final simplification 98.8%

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

Alternative 4: 84.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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(sqrt(((1.0d0 - (om / (omc * (omc / om)))) / (1.0d0 + (2.0d0 * ((t_m / l_m) / (l_m / t_m)))))))
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(Math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
}

Python

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

Julia

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

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
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[N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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. Step-by-step derivation

   1. unpow2 86.5%

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

   2. clear-num 86.5%

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

   3. un-div-inv 86.5%

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

4. Applied egg-rr 86.5%

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

   1. unpow2 98.8%

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

   2. clear-num 98.8%

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

   3. frac-times 98.8%

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

   4. *-un-lft-identity 98.8%

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

6. Applied egg-rr 86.5%

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

7. Final simplification 86.5%

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

Alternative 5: 84.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

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(sqrt(((1.0d0 - (om / (omc * (omc / om)))) / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_m)))))))
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(Math.sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
}

Python

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

Julia

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

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(sqrt(((1.0 - (Om / (Omc * (Omc / Om)))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
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[N[Sqrt[N[(N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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. Step-by-step derivation

   1. unpow2 86.5%

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

   2. clear-num 86.5%

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

   3. un-div-inv 86.5%

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

4. Applied egg-rr 86.5%

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

   1. unpow2 98.8%

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

   2. clear-num 98.8%

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

   3. frac-times 98.8%

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

   4. *-un-lft-identity 98.8%

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

6. Applied egg-rr 86.5%

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

   1. div-inv 86.5%

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

   2. clear-num 86.5%

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

8. Applied egg-rr 86.5%

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

9. Final simplification 86.5%

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

Alternative 6: 51.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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(sqrt((1.0d0 - (om / (omc * (omc / om))))))
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(Math.sqrt((1.0 - (Om / (Omc * (Omc / Om))))));
}

Python

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

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(Float64(1.0 - Float64(Om / Float64(Omc * Float64(Omc / Om))))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(sqrt((1.0 - (Om / (Omc * (Omc / Om))))));
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[N[Sqrt[N[(1.0 - N[(Om / N[(Omc * N[(Omc / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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 t around 0 45.2%

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

   1. unpow2 45.2%

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

   2. unpow2 45.2%

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

   3. times-frac 49.9%

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

   4. unpow2 49.9%

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

5. Simplified 49.9%

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

   1. unpow2 98.8%

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

   2. clear-num 98.8%

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

   3. frac-times 98.8%

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

   4. *-un-lft-identity 98.8%

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

7. Applied egg-rr 49.9%

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

8. Final simplification 49.9%

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

Reproduce

herbie shell --seed 2024102 
(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)))))))