Toniolo and Linder, Equation (2)

Percentage Accurate: 83.1% → 98.8%

Time: 13.7s

Alternatives: 11

Speedup: 2.4×

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: 83.1% 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, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - \frac{-1}{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}} \cdot 2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+95)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (- 1.0 (* (/ -1.0 (* (/ l_m t_m) (/ l_m t_m))) 2.0)))))
   (asin
    (* (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (sqrt 0.5) t_m)) l_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+95) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 - ((-1.0 / ((l_m / t_m) * (l_m / t_m))) * 2.0)))));
	} else {
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 2d+95) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 - (((-1.0d0) / ((l_m / t_m) * (l_m / t_m))) * 2.0d0)))))
    else
        tmp = asin(((sqrt((1.0d0 - ((om / omc) * (om / omc)))) * (sqrt(0.5d0) / t_m)) * l_m))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+95) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 - ((-1.0 / ((l_m / t_m) * (l_m / t_m))) * 2.0)))));
	} else {
		tmp = Math.asin(((Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (Math.sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 2e+95:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 - ((-1.0 / ((l_m / t_m) * (l_m / t_m))) * 2.0)))))
	else:
		tmp = math.asin(((math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (math.sqrt(0.5) / t_m)) * l_m))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+95)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 - Float64(Float64(-1.0 / Float64(Float64(l_m / t_m) * Float64(l_m / t_m))) * 2.0)))));
	else
		tmp = asin(Float64(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(sqrt(0.5) / t_m)) * l_m));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+95)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 - ((-1.0 / ((l_m / t_m) * (l_m / t_m))) * 2.0)))));
	else
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (sqrt(0.5) / t_m)) * l_m));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+95], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[(-1.0 / N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.00000000000000004e95

   1. Initial program 91.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 91.4

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

   4. Applied rewrites 91.4%

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

   if 2.00000000000000004e95 < (/.f64 t l)

   1. Initial program 72.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

   5. Applied rewrites 91.3%

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \mathsf{fma}\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\sqrt{0.5}}, -0.125, \frac{\sqrt{0.5}}{t}\right)\right) \cdot \ell\right)} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.6%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+95}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - \frac{-1}{\frac{\ell}{t} \cdot \frac{\ell}{t}} \cdot 2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{{\left(\frac{t\_m}{l\_m}\right)}^{2} \cdot 2 + 1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ (* (pow (/ t_m l_m) 2.0) 2.0) 1.0))
      2e-6)
   (asin (/ (* (sqrt 0.5) l_m) t_m))
   (asin (sqrt 1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / ((pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 2e-6) {
		tmp = asin(((sqrt(0.5) * l_m) / t_m));
	} else {
		tmp = asin(sqrt(1.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (((1.0d0 - ((om / omc) ** 2.0d0)) / ((((t_m / l_m) ** 2.0d0) * 2.0d0) + 1.0d0)) <= 2d-6) then
        tmp = asin(((sqrt(0.5d0) * l_m) / t_m))
    else
        tmp = asin(sqrt(1.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - Math.pow((Om / Omc), 2.0)) / ((Math.pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 2e-6) {
		tmp = Math.asin(((Math.sqrt(0.5) * l_m) / t_m));
	} else {
		tmp = Math.asin(Math.sqrt(1.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if ((1.0 - math.pow((Om / Omc), 2.0)) / ((math.pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 2e-6:
		tmp = math.asin(((math.sqrt(0.5) * l_m) / t_m))
	else:
		tmp = math.asin(math.sqrt(1.0))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(Float64((Float64(t_m / l_m) ^ 2.0) * 2.0) + 1.0)) <= 2e-6)
		tmp = asin(Float64(Float64(sqrt(0.5) * l_m) / t_m));
	else
		tmp = asin(sqrt(1.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (((1.0 - ((Om / Omc) ^ 2.0)) / ((((t_m / l_m) ^ 2.0) * 2.0) + 1.0)) <= 2e-6)
		tmp = asin(((sqrt(0.5) * l_m) / t_m));
	else
		tmp = asin(sqrt(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 2e-6], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 1.99999999999999991e-6

   1. Initial program 74.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 66.1%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in Omc around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 92.6

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

   8. Applied rewrites 92.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

       \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2 + 1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{{\left(\frac{t\_m}{l\_m}\right)}^{2} \cdot 2 + 1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t\_m} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ (* (pow (/ t_m l_m) 2.0) 2.0) 1.0))
      2e-6)
   (asin (* (/ (sqrt 0.5) t_m) l_m))
   (asin (sqrt 1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / ((pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 2e-6) {
		tmp = asin(((sqrt(0.5) / t_m) * l_m));
	} else {
		tmp = asin(sqrt(1.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (((1.0d0 - ((om / omc) ** 2.0d0)) / ((((t_m / l_m) ** 2.0d0) * 2.0d0) + 1.0d0)) <= 2d-6) then
        tmp = asin(((sqrt(0.5d0) / t_m) * l_m))
    else
        tmp = asin(sqrt(1.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - Math.pow((Om / Omc), 2.0)) / ((Math.pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 2e-6) {
		tmp = Math.asin(((Math.sqrt(0.5) / t_m) * l_m));
	} else {
		tmp = Math.asin(Math.sqrt(1.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if ((1.0 - math.pow((Om / Omc), 2.0)) / ((math.pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 2e-6:
		tmp = math.asin(((math.sqrt(0.5) / t_m) * l_m))
	else:
		tmp = math.asin(math.sqrt(1.0))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(Float64((Float64(t_m / l_m) ^ 2.0) * 2.0) + 1.0)) <= 2e-6)
		tmp = asin(Float64(Float64(sqrt(0.5) / t_m) * l_m));
	else
		tmp = asin(sqrt(1.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (((1.0 - ((Om / Omc) ^ 2.0)) / ((((t_m / l_m) ^ 2.0) * 2.0) + 1.0)) <= 2e-6)
		tmp = asin(((sqrt(0.5) / t_m) * l_m));
	else
		tmp = asin(sqrt(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 2e-6], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 1.99999999999999991e-6

   1. Initial program 74.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 66.1%

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

   10. Applied rewrites 66.2%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in Omc around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 92.6

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

   8. Applied rewrites 92.6%

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

   9. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 97.6%

       \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{{\left(\frac{t}{\ell}\right)}^{2} \cdot 2 + 1} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\left(\frac{\frac{t\_m}{l\_m}}{l\_m} \cdot t\_m\right) \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+57)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ (* (* (/ (/ t_m l_m) l_m) t_m) 2.0) 1.0))))
   (asin
    (* (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (sqrt 0.5) t_m)) l_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+57) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (((((t_m / l_m) / l_m) * t_m) * 2.0) + 1.0))));
	} else {
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 5d+57) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (((((t_m / l_m) / l_m) * t_m) * 2.0d0) + 1.0d0))))
    else
        tmp = asin(((sqrt((1.0d0 - ((om / omc) * (om / omc)))) * (sqrt(0.5d0) / t_m)) * l_m))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+57) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (((((t_m / l_m) / l_m) * t_m) * 2.0) + 1.0))));
	} else {
		tmp = Math.asin(((Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (Math.sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 5e+57:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (((((t_m / l_m) / l_m) * t_m) * 2.0) + 1.0))))
	else:
		tmp = math.asin(((math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (math.sqrt(0.5) / t_m)) * l_m))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+57)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(Float64(Float64(Float64(Float64(t_m / l_m) / l_m) * t_m) * 2.0) + 1.0))));
	else
		tmp = asin(Float64(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(sqrt(0.5) / t_m)) * l_m));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 5e+57)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (((((t_m / l_m) / l_m) * t_m) * 2.0) + 1.0))));
	else
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (sqrt(0.5) / t_m)) * l_m));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+57], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999972e57

   1. Initial program 91.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 90.2

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

   4. Applied rewrites 90.2%

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

   if 4.99999999999999972e57 < (/.f64 t l)

   1. Initial program 76.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

   5. Applied rewrites 90.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \mathsf{fma}\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\sqrt{0.5}}, -0.125, \frac{\sqrt{0.5}}{t}\right)\right) \cdot \ell\right)} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\left(\frac{\frac{t}{\ell}}{\ell} \cdot t\right) \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\frac{t\_m}{\frac{l\_m}{t\_m} \cdot l\_m} \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+51)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ (* (/ t_m (* (/ l_m t_m) l_m)) 2.0) 1.0))))
   (asin
    (* (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (sqrt 0.5) t_m)) l_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+51) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))));
	} else {
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 2d+51) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0d0) + 1.0d0))))
    else
        tmp = asin(((sqrt((1.0d0 - ((om / omc) * (om / omc)))) * (sqrt(0.5d0) / t_m)) * l_m))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+51) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))));
	} else {
		tmp = Math.asin(((Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (Math.sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 2e+51:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))))
	else:
		tmp = math.asin(((math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (math.sqrt(0.5) / t_m)) * l_m))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+51)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(Float64(Float64(t_m / Float64(Float64(l_m / t_m) * l_m)) * 2.0) + 1.0))));
	else
		tmp = asin(Float64(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(sqrt(0.5) / t_m)) * l_m));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 2e+51)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))));
	else
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (sqrt(0.5) / t_m)) * l_m));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+51], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / N[(N[(l$95$m / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2e51

   1. Initial program 91.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 90.1

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

   4. Applied rewrites 90.1%

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

   if 2e51 < (/.f64 t l)

   1. Initial program 76.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

   5. Applied rewrites 90.8%

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \mathsf{fma}\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\sqrt{0.5}}, -0.125, \frac{\sqrt{0.5}}{t}\right)\right) \cdot \ell\right)} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+51}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\frac{t}{\frac{\ell}{t} \cdot \ell} \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+57)
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 1.0))))
   (asin
    (* (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (sqrt 0.5) t_m)) l_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+57) {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin(((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * (sqrt(0.5) / t_m)) * l_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+57)
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(sqrt(0.5) / t_m)) * l_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+57], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999972e57

   1. Initial program 91.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in Omc around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.1

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

   8. Applied rewrites 82.1%

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

   10. Applied rewrites 89.8%

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

   if 4.99999999999999972e57 < (/.f64 t l)

   1. Initial program 76.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \ell\right)} \]

   5. Applied rewrites 90.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \mathsf{fma}\left(\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\sqrt{0.5}}, -0.125, \frac{\sqrt{0.5}}{t}\right)\right) \cdot \ell\right)} \]

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.5%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \mathsf{fma}\left(-0.5, \frac{Om}{Omc} \cdot \frac{Om}{Omc}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+57)
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 1.0))))
   (asin
    (* (/ (* (sqrt 0.5) l_m) t_m) (fma -0.5 (* (/ Om Omc) (/ Om Omc)) 1.0)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+57) {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * fma(-0.5, ((Om / Omc) * (Om / Omc)), 1.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+57)
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * fma(-0.5, Float64(Float64(Om / Omc) * Float64(Om / Omc)), 1.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+57], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999972e57

   1. Initial program 91.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in Omc around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.1

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

   8. Applied rewrites 82.1%

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

   10. Applied rewrites 89.8%

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

   if 4.99999999999999972e57 < (/.f64 t l)

   1. Initial program 76.3%

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

   3. Taylor expanded in Omc around inf

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \mathsf{fma}\left(-0.5, \frac{Om}{Omc} \cdot \frac{Om}{Omc}, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+57)
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 1.0))))
   (asin (/ (* (sqrt 0.5) l_m) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+57) {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin(((sqrt(0.5) * l_m) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+57)
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) * l_m) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+57], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999972e57

   1. Initial program 91.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in Omc around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 82.1

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

   8. Applied rewrites 82.1%

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

   10. Applied rewrites 89.8%

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

   if 4.99999999999999972e57 < (/.f64 t l)

   1. Initial program 76.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 96.0

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

   5. Applied rewrites 96.0%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 97.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{\frac{Om}{Omc}}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.2)
   (asin (sqrt (fma (- Om) (/ (/ Om Omc) Omc) 1.0)))
   (asin (/ (* (sqrt 0.5) l_m) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.2) {
		tmp = asin(sqrt(fma(-Om, ((Om / Omc) / Omc), 1.0)));
	} else {
		tmp = asin(((sqrt(0.5) * l_m) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.2)
		tmp = asin(sqrt(fma(Float64(-Om), Float64(Float64(Om / Omc) / Omc), 1.0)));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) * l_m) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.2], N[ArcSin[N[Sqrt[N[((-Om) * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.20000000000000001

   1. Initial program 90.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

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

   7. Applied rewrites 71.5%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 80.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 96.8%

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

Alternative 10: 97.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{Om}{Omc} \cdot Om, \frac{-0.5}{Omc}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.2)
   (asin (fma (* (/ Om Omc) Om) (/ -0.5 Omc) 1.0))
   (asin (/ (* (sqrt 0.5) l_m) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.2) {
		tmp = asin(fma(((Om / Omc) * Om), (-0.5 / Omc), 1.0));
	} else {
		tmp = asin(((sqrt(0.5) * l_m) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.2)
		tmp = asin(fma(Float64(Float64(Om / Omc) * Om), Float64(-0.5 / Omc), 1.0));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) * l_m) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.2], N[ArcSin[N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] * N[(-0.5 / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.20000000000000001

   1. Initial program 90.6%

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

   3. Taylor expanded in Omc around inf

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.6%

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

   10. Applied rewrites 71.2%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 80.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 95.8

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 96.8%

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

Alternative 11: 50.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

3. Taylor expanded in t around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in Omc around inf

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 77.2

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

8. Applied rewrites 77.2%

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

9. Taylor expanded in t around 0

   \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
10. Step-by-step derivation

11. Applied rewrites 53.9%

    \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(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)))))))