Toniolo and Linder, Equation (2)

Percentage Accurate: 83.7% → 98.8%

Time: 17.7s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+150}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{1}{\frac{t\_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+150)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (/ Om Omc) (/ Omc Om)))
      (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m)))))))
   (asin (* l_m (/ 1.0 (/ 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 / l_m) <= 5e+150) {
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = asin((l_m * (1.0 / (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 / l_m) <= 5d+150) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) / (omc / om))) / (1.0d0 + (2.0d0 * ((t_m / l_m) * (t_m / l_m)))))))
    else
        tmp = asin((l_m * (1.0d0 / (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 / l_m) <= 5e+150) {
		tmp = Math.asin(Math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = Math.asin((l_m * (1.0 / (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 / l_m) <= 5e+150:
		tmp = math.asin(math.sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))))
	else:
		tmp = math.asin((l_m * (1.0 / (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 (Float64(t_m / l_m) <= 5e+150)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Omc) / Float64(Omc / Om))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(l_m * Float64(1.0 / Float64(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 / l_m) <= 5e+150)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) / (Omc / Om))) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	else
		tmp = asin((l_m * (1.0 / (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[N[(t$95$m / l$95$m), $MachinePrecision], 5e+150], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $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], N[ArcSin[N[(l$95$m * N[(1.0 / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000009e150

   1. Initial program 87.0%

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

      1. unpow2 87.0%

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

      2. clear-num 87.0%

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

      3. un-div-inv 87.0%

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

   4. Applied egg-rr 87.0%

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

      1. unpow2 87.0%

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

   6. Applied egg-rr 87.0%

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

   if 5.00000000000000009e150 < (/.f64 t l)

   1. Initial program 50.4%

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

   3. Taylor expanded in t around inf 86.2%

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

      1. *-commutative 86.2%

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

      2. unpow2 86.2%

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

      3. unpow2 86.2%

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

      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%

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

   6. Taylor expanded in Om around 0 99.7%

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

      1. associate-*r/ 99.7%

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

   8. Simplified 99.7%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. unpow-1 99.8%

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

   12. Simplified 99.8%

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

Alternative 2: 98.3% accurate, 0.8× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin
  (/
   (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
   (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 - pow((Om / Omc), 2.0))) / 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 - Math.pow((Om / Omc), 2.0))) / 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 - math.pow((Om / Omc), 2.0))) / 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 / Omc) ^ 2.0))) / 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) ^ 2.0))) / 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[Power[N[(Om / Omc), $MachinePrecision], 2.0], $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 81.7%

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

   1. sqrt-div 81.7%

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

   2. div-inv 81.7%

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

   3. add-sqr-sqrt 81.7%

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

   4. hypot-1-def 81.7%

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

   5. *-commutative 81.7%

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

   6. sqrt-prod 81.6%

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

   7. sqrt-pow1 98.0%

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

   8. metadata-eval 98.0%

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

   9. pow1 98.0%

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

4. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

6. Simplified 98.0%

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

Alternative 3: 97.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{t\_m \cdot \sqrt{2}}{l\_m}\right)}\right)\right)\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (expm1 (log1p (asin (/ 1.0 (hypot 1.0 (/ (* t_m (sqrt 2.0)) l_m)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return expm1(log1p(asin((1.0 / hypot(1.0, ((t_m * sqrt(2.0)) / l_m))))));
}

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.expm1(Math.log1p(Math.asin((1.0 / Math.hypot(1.0, ((t_m * Math.sqrt(2.0)) / l_m))))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.expm1(math.log1p(math.asin((1.0 / math.hypot(1.0, ((t_m * math.sqrt(2.0)) / l_m))))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return expm1(log1p(asin(Float64(1.0 / hypot(1.0, Float64(Float64(t_m * sqrt(2.0)) / 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[(Exp[N[Log[1 + N[ArcSin[N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. sqrt-div 81.7%

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

   2. div-inv 81.7%

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

   3. add-sqr-sqrt 81.7%

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

   4. hypot-1-def 81.7%

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

   5. *-commutative 81.7%

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

   6. sqrt-prod 81.6%

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

   7. sqrt-pow1 98.0%

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

   8. metadata-eval 98.0%

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

   9. pow1 98.0%

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

4. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in Om around 0 97.6%

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

   1. expm1-log1p-u 97.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)\right)} \]

   2. expm1-undefine 65.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{t}{\ell} \cdot \sqrt{2}\right)}\right)\right)} - 1} \]

   3. associate-*l/ 65.1%

      \[\leadsto e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}\right)}\right)\right)} - 1 \]

9. Applied egg-rr 65.1%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-define 97.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)\right)\right)} \]

11. Simplified 97.6%

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{1}{\mathsf{hypot}\left(1, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right)\right)\right)} \]
12. Add Preprocessing

Alternative 4: 97.4% accurate, 1.3× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (/ 1.0 (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((1.0 / 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((1.0 / 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((1.0 / 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(1.0 / 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((1.0 / 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[(1.0 / 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 81.7%

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

   1. sqrt-div 81.7%

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

   2. div-inv 81.7%

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

   3. add-sqr-sqrt 81.7%

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

   4. hypot-1-def 81.7%

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

   5. *-commutative 81.7%

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

   6. sqrt-prod 81.6%

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

   7. sqrt-pow1 98.0%

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

   8. metadata-eval 98.0%

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

   9. pow1 98.0%

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

4. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in Om around 0 97.6%

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

Alternative 5: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t\_m}{l\_m \cdot \frac{l\_m}{t\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}} \cdot -0.5\right) \cdot \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\right)\\ \end{array} \end{array} \]

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) 0.02)
   (asin (/ 1.0 (+ 1.0 (/ t_m (* l_m (/ l_m t_m))))))
   (asin
    (*
     (+ 1.0 (* (/ (/ Om Omc) (/ Omc Om)) -0.5))
     (* 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) <= 0.02) {
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	} else {
		tmp = asin(((1.0 + (((Om / Omc) / (Omc / Om)) * -0.5)) * (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) <= 0.02d0) then
        tmp = asin((1.0d0 / (1.0d0 + (t_m / (l_m * (l_m / t_m))))))
    else
        tmp = asin(((1.0d0 + (((om / omc) / (omc / om)) * (-0.5d0))) * (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) <= 0.02) {
		tmp = Math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	} else {
		tmp = Math.asin(((1.0 + (((Om / Omc) / (Omc / Om)) * -0.5)) * (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) <= 0.02:
		tmp = math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))))
	else:
		tmp = math.asin(((1.0 + (((Om / Omc) / (Omc / Om)) * -0.5)) * (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) <= 0.02)
		tmp = asin(Float64(1.0 / Float64(1.0 + Float64(t_m / Float64(l_m * Float64(l_m / t_m))))));
	else
		tmp = asin(Float64(Float64(1.0 + Float64(Float64(Float64(Om / Omc) / Float64(Omc / Om)) * -0.5)) * 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) <= 0.02)
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	else
		tmp = asin(((1.0 + (((Om / Omc) / (Omc / Om)) * -0.5)) * (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], 0.02], N[ArcSin[N[(1.0 / N[(1.0 + N[(t$95$m / N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 + N[(N[(N[(Om / Omc), $MachinePrecision] / N[(Omc / Om), $MachinePrecision]), $MachinePrecision] * -0.5), $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) < 0.0200000000000000004

   1. Initial program 85.3%

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

      1. sqrt-div 85.4%

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

      2. div-inv 85.4%

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

      3. add-sqr-sqrt 85.4%

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

      4. hypot-1-def 85.4%

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

      5. *-commutative 85.4%

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

      6. sqrt-prod 85.3%

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

      7. sqrt-pow1 97.8%

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

      8. metadata-eval 97.8%

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

      9. pow1 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. associate-*r/ 97.8%

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

      2. *-rgt-identity 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in Om around 0 97.3%

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

   8. Taylor expanded in t around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. unpow2 64.3%

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

      3. rem-square-sqrt 64.3%

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

      4. associate-*r/ 64.3%

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

      5. unpow2 64.3%

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

      6. unpow2 64.3%

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

      7. times-frac 72.3%

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

      8. unpow2 72.3%

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

   10. Simplified 72.3%

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

       1. associate-*r* 72.3%

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

       2. metadata-eval 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. unpow2 72.3%

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

       5. clear-num 72.3%

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

       6. frac-times 72.2%

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

       7. *-un-lft-identity 72.2%

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

   12. Applied egg-rr 72.2%

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

   if 0.0200000000000000004 < (/.f64 t l)

   1. Initial program 70.3%

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

   3. Taylor expanded in t around inf 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. times-frac 97.2%

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

      5. unpow2 97.2%

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

      6. associate-/l* 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in Om around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. times-frac 97.2%

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

      5. unpow2 97.2%

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

   8. Simplified 97.2%

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

      1. unpow2 70.3%

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

      2. clear-num 70.3%

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

      3. un-div-inv 70.3%

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

   10. Applied egg-rr 97.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t}{\ell \cdot \frac{\ell}{t}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(1 + \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}} \cdot -0.5\right) \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t\_m}{l\_m \cdot \frac{l\_m}{t\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{1}{\frac{t\_m}{\sqrt{0.5}}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.02)
   (asin (/ 1.0 (+ 1.0 (/ t_m (* l_m (/ l_m t_m))))))
   (asin (* l_m (/ 1.0 (/ 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 / l_m) <= 0.02) {
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	} else {
		tmp = asin((l_m * (1.0 / (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 / l_m) <= 0.02d0) then
        tmp = asin((1.0d0 / (1.0d0 + (t_m / (l_m * (l_m / t_m))))))
    else
        tmp = asin((l_m * (1.0d0 / (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 / l_m) <= 0.02) {
		tmp = Math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	} else {
		tmp = Math.asin((l_m * (1.0 / (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 / l_m) <= 0.02:
		tmp = math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))))
	else:
		tmp = math.asin((l_m * (1.0 / (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 (Float64(t_m / l_m) <= 0.02)
		tmp = asin(Float64(1.0 / Float64(1.0 + Float64(t_m / Float64(l_m * Float64(l_m / t_m))))));
	else
		tmp = asin(Float64(l_m * Float64(1.0 / Float64(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 / l_m) <= 0.02)
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	else
		tmp = asin((l_m * (1.0 / (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[N[(t$95$m / l$95$m), $MachinePrecision], 0.02], N[ArcSin[N[(1.0 / N[(1.0 + N[(t$95$m / N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(1.0 / N[(t$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0200000000000000004

   1. Initial program 85.3%

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

      1. sqrt-div 85.4%

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

      2. div-inv 85.4%

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

      3. add-sqr-sqrt 85.4%

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

      4. hypot-1-def 85.4%

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

      5. *-commutative 85.4%

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

      6. sqrt-prod 85.3%

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

      7. sqrt-pow1 97.8%

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

      8. metadata-eval 97.8%

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

      9. pow1 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. associate-*r/ 97.8%

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

      2. *-rgt-identity 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in Om around 0 97.3%

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

   8. Taylor expanded in t around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. unpow2 64.3%

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

      3. rem-square-sqrt 64.3%

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

      4. associate-*r/ 64.3%

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

      5. unpow2 64.3%

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

      6. unpow2 64.3%

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

      7. times-frac 72.3%

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

      8. unpow2 72.3%

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

   10. Simplified 72.3%

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

       1. associate-*r* 72.3%

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

       2. metadata-eval 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. unpow2 72.3%

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

       5. clear-num 72.3%

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

       6. frac-times 72.2%

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

       7. *-un-lft-identity 72.2%

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

   12. Applied egg-rr 72.2%

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

   if 0.0200000000000000004 < (/.f64 t l)

   1. Initial program 70.3%

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

   3. Taylor expanded in t around inf 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. times-frac 97.2%

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

      5. unpow2 97.2%

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

      6. associate-/l* 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in Om around 0 97.2%

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

      1. associate-*r/ 97.2%

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

   8. Simplified 97.2%

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

      1. clear-num 97.2%

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

      2. inv-pow 97.2%

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

   10. Applied egg-rr 97.2%

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

       1. unpow-1 97.2%

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

   12. Simplified 97.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t}{\ell \cdot \frac{\ell}{t}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{1}{\frac{t}{\sqrt{0.5}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t\_m}{l\_m \cdot \frac{l\_m}{t\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

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) 0.02)
   (asin (/ 1.0 (+ 1.0 (/ t_m (* l_m (/ l_m t_m))))))
   (asin (/ l_m (* 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) <= 0.02) {
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	} else {
		tmp = asin((l_m / (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) <= 0.02d0) then
        tmp = asin((1.0d0 / (1.0d0 + (t_m / (l_m * (l_m / t_m))))))
    else
        tmp = asin((l_m / (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) <= 0.02) {
		tmp = Math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	} else {
		tmp = Math.asin((l_m / (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) <= 0.02:
		tmp = math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))))
	else:
		tmp = math.asin((l_m / (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) <= 0.02)
		tmp = asin(Float64(1.0 / Float64(1.0 + Float64(t_m / Float64(l_m * Float64(l_m / t_m))))));
	else
		tmp = asin(Float64(l_m / 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) <= 0.02)
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	else
		tmp = asin((l_m / (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], 0.02], N[ArcSin[N[(1.0 / N[(1.0 + N[(t$95$m / N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0200000000000000004

   1. Initial program 85.3%

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

      1. sqrt-div 85.4%

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

      2. div-inv 85.4%

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

      3. add-sqr-sqrt 85.4%

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

      4. hypot-1-def 85.4%

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

      5. *-commutative 85.4%

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

      6. sqrt-prod 85.3%

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

      7. sqrt-pow1 97.8%

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

      8. metadata-eval 97.8%

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

      9. pow1 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. associate-*r/ 97.8%

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

      2. *-rgt-identity 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in Om around 0 97.3%

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

   8. Taylor expanded in t around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. unpow2 64.3%

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

      3. rem-square-sqrt 64.3%

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

      4. associate-*r/ 64.3%

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

      5. unpow2 64.3%

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

      6. unpow2 64.3%

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

      7. times-frac 72.3%

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

      8. unpow2 72.3%

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

   10. Simplified 72.3%

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

       1. associate-*r* 72.3%

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

       2. metadata-eval 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. unpow2 72.3%

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

       5. clear-num 72.3%

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

       6. frac-times 72.2%

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

       7. *-un-lft-identity 72.2%

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

   12. Applied egg-rr 72.2%

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

   if 0.0200000000000000004 < (/.f64 t l)

   1. Initial program 70.3%

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

      1. sqrt-div 70.2%

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

      2. div-inv 70.2%

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

      3. add-sqr-sqrt 70.1%

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

      4. hypot-1-def 70.1%

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

      5. *-commutative 70.1%

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

      6. sqrt-prod 70.1%

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

      7. sqrt-pow1 98.5%

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

      8. metadata-eval 98.5%

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

      9. pow1 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. associate-*r/ 98.5%

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

      2. *-rgt-identity 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in Om around 0 98.5%

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

   8. Taylor expanded in t around inf 97.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t}{\ell \cdot \frac{\ell}{t}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t\_m}{l\_m \cdot \frac{l\_m}{t\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.02)
   (asin (/ 1.0 (+ 1.0 (/ t_m (* l_m (/ l_m t_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) <= 0.02) {
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_m))))));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

t_m = abs(t)
l_m = abs(l)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.02d0) then
        tmp = asin((1.0d0 / (1.0d0 + (t_m / (l_m * (l_m / t_m))))))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.02) {
		tmp = Math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_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) <= 0.02:
		tmp = math.asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_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) <= 0.02)
		tmp = asin(Float64(1.0 / Float64(1.0 + Float64(t_m / Float64(l_m * Float64(l_m / t_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) <= 0.02)
		tmp = asin((1.0 / (1.0 + (t_m / (l_m * (l_m / t_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], 0.02], N[ArcSin[N[(1.0 / N[(1.0 + N[(t$95$m / N[(l$95$m * N[(l$95$m / t$95$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) < 0.0200000000000000004

   1. Initial program 85.3%

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

      1. sqrt-div 85.4%

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

      2. div-inv 85.4%

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

      3. add-sqr-sqrt 85.4%

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

      4. hypot-1-def 85.4%

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

      5. *-commutative 85.4%

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

      6. sqrt-prod 85.3%

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

      7. sqrt-pow1 97.8%

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

      8. metadata-eval 97.8%

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

      9. pow1 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. associate-*r/ 97.8%

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

      2. *-rgt-identity 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in Om around 0 97.3%

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

   8. Taylor expanded in t around 0 64.3%

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

      1. *-commutative 64.3%

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

      2. unpow2 64.3%

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

      3. rem-square-sqrt 64.3%

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

      4. associate-*r/ 64.3%

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

      5. unpow2 64.3%

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

      6. unpow2 64.3%

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

      7. times-frac 72.3%

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

      8. unpow2 72.3%

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

   10. Simplified 72.3%

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

       1. associate-*r* 72.3%

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

       2. metadata-eval 72.3%

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

       3. *-un-lft-identity 72.3%

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

       4. unpow2 72.3%

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

       5. clear-num 72.3%

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

       6. frac-times 72.2%

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

       7. *-un-lft-identity 72.2%

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

   12. Applied egg-rr 72.2%

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

   if 0.0200000000000000004 < (/.f64 t l)

   1. Initial program 70.3%

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

   3. Taylor expanded in t around inf 82.8%

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

      1. *-commutative 82.8%

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

      2. unpow2 82.8%

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

      3. unpow2 82.8%

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

      4. times-frac 97.2%

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

      5. unpow2 97.2%

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

      6. associate-/l* 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in Om around 0 97.2%

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

      1. associate-*r/ 97.2%

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

   8. Simplified 97.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.02:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{1 + \frac{t}{\ell \cdot \frac{\ell}{t}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 3.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (/ 1.0 (+ 1.0 (/ 1.0 (* (/ l_m t_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((1.0 / (1.0 + (1.0 / ((l_m / t_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((1.0d0 / (1.0d0 + (1.0d0 / ((l_m / t_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((1.0 / (1.0 + (1.0 / ((l_m / t_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((1.0 / (1.0 + (1.0 / ((l_m / t_m) * (l_m / t_m))))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(Float64(1.0 / Float64(1.0 + Float64(1.0 / Float64(Float64(l_m / t_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((1.0 / (1.0 + (1.0 / ((l_m / t_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[(1.0 / N[(1.0 + N[(1.0 / N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. sqrt-div 81.7%

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

   2. div-inv 81.7%

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

   3. add-sqr-sqrt 81.7%

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

   4. hypot-1-def 81.7%

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

   5. *-commutative 81.7%

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

   6. sqrt-prod 81.6%

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

   7. sqrt-pow1 98.0%

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

   8. metadata-eval 98.0%

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

   9. pow1 98.0%

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

4. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in Om around 0 97.6%

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

8. Taylor expanded in t around 0 56.5%

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

   1. *-commutative 56.5%

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

   2. unpow2 56.5%

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

   3. rem-square-sqrt 56.5%

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

   4. associate-*r/ 56.5%

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

   5. unpow2 56.5%

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

   6. unpow2 56.5%

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

   7. times-frac 62.8%

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

   8. unpow2 62.8%

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

10. Simplified 62.8%

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

    1. associate-*r* 62.8%

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

    2. metadata-eval 62.8%

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

    3. *-un-lft-identity 62.8%

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

    4. unpow2 62.8%

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

    5. clear-num 62.8%

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

    6. clear-num 62.8%

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

    7. frac-times 62.8%

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

    8. metadata-eval 62.8%

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

12. Applied egg-rr 62.8%

    \[\leadsto \sin^{-1} \left(\frac{1}{1 + \color{blue}{\frac{1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right) \]
13. Add Preprocessing

Alternative 10: 62.1% accurate, 3.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (/ 1.0 (+ 1.0 (/ 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((1.0 / (1.0 + (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((1.0d0 / (1.0d0 + (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((1.0 / (1.0 + (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((1.0 / (1.0 + (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(Float64(1.0 / Float64(1.0 + Float64(t_m / Float64(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((1.0 / (1.0 + (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[(1.0 / N[(1.0 + N[(t$95$m / N[(l$95$m * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.7%

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

   1. sqrt-div 81.7%

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

   2. div-inv 81.7%

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

   3. add-sqr-sqrt 81.7%

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

   4. hypot-1-def 81.7%

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

   5. *-commutative 81.7%

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

   6. sqrt-prod 81.6%

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

   7. sqrt-pow1 98.0%

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

   8. metadata-eval 98.0%

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

   9. pow1 98.0%

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

4. Applied egg-rr 98.0%

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

   1. associate-*r/ 98.0%

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

   2. *-rgt-identity 98.0%

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

6. Simplified 98.0%

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

7. Taylor expanded in Om around 0 97.6%

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

8. Taylor expanded in t around 0 56.5%

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

   1. *-commutative 56.5%

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

   2. unpow2 56.5%

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

   3. rem-square-sqrt 56.5%

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

   4. associate-*r/ 56.5%

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

   5. unpow2 56.5%

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

   6. unpow2 56.5%

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

   7. times-frac 62.8%

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

   8. unpow2 62.8%

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

10. Simplified 62.8%

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

    1. associate-*r* 62.8%

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

    2. metadata-eval 62.8%

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

    3. *-un-lft-identity 62.8%

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

    4. unpow2 62.8%

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

    5. clear-num 62.8%

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

    6. frac-times 62.8%

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

    7. *-un-lft-identity 62.8%

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

12. Applied egg-rr 62.8%

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

13. Final simplification 62.8%

    \[\leadsto \sin^{-1} \left(\frac{1}{1 + \frac{t}{\ell \cdot \frac{\ell}{t}}}\right) \]
14. Add Preprocessing

Alternative 11: 49.5% 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 81.7%

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

3. Taylor expanded in t around 0 46.3%

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

   1. unpow2 46.3%

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

   2. unpow2 46.3%

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

   3. times-frac 51.2%

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

   4. unpow2 51.2%

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

5. Simplified 51.2%

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

6. Taylor expanded in Om around 0 50.9%

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

Reproduce

herbie shell --seed 2024135 
(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)))))))