Toniolo and Linder, Equation (2)

Percentage Accurate: 82.9% → 98.9%

Time: 15.3s

Alternatives: 10

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 2 \cdot \frac{-1}{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+148)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (- 1.0 (* 2.0 (/ -1.0 (* (/ l_m t_m) (/ l_m t_m))))))))
   (asin (/ (* l_m (sqrt (* (fma (/ (/ Om Omc) Omc) (- Om) 1.0) 0.5))) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+148) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 - (2.0 * (-1.0 / ((l_m / t_m) * (l_m / t_m))))))));
	} else {
		tmp = asin(((l_m * sqrt((fma(((Om / Omc) / Omc), -Om, 1.0) * 0.5))) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+148)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 - Float64(2.0 * Float64(-1.0 / Float64(Float64(l_m / t_m) * Float64(l_m / t_m))))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0) * 0.5))) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+148], 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[(-1.0 / N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e148

   1. Initial program 92.3%

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

      1. 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-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 2.0000000000000001e148 < (/.f64 t l)

   1. Initial program 45.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 93.8%

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

   9. Applied rewrites 99.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 2 \cdot \frac{-1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+148)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin (/ (* l_m (sqrt (* (fma (/ (/ Om Omc) Omc) (- Om) 1.0) 0.5))) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+148) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin(((l_m * sqrt((fma(((Om / Omc) / Omc), -Om, 1.0) * 0.5))) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+148)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0) * 0.5))) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+148], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e148

   1. Initial program 92.3%

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

      1. 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. lower-/.f64 N/A

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

      7. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 2.0000000000000001e148 < (/.f64 t l)

   1. Initial program 45.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 93.8%

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

   9. Applied rewrites 99.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+148)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0))))
   (asin (/ (* l_m (sqrt (* (fma (/ (/ Om Omc) Omc) (- Om) 1.0) 0.5))) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+148) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt((fma(((Om / Omc) / Omc), -Om, 1.0) * 0.5))) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+148)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0) * 0.5))) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+148], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e148

   1. Initial program 92.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if 2.0000000000000001e148 < (/.f64 t l)

   1. Initial program 45.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 79.3

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

   5. Applied rewrites 79.3%

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

   7. Applied rewrites 93.8%

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

   9. Applied rewrites 99.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{\mathsf{fma}\left(\left(\frac{t\_m}{l\_m} \cdot 2\right) \cdot \frac{1}{l\_m}, t\_m, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+26)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (fma (* (* (/ t_m l_m) 2.0) (/ 1.0 l_m)) t_m 1.0))))
   (asin (/ (* l_m (sqrt (* (fma (/ (/ Om Omc) Omc) (- Om) 1.0) 0.5))) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+26) {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / fma((((t_m / l_m) * 2.0) * (1.0 / l_m)), t_m, 1.0))));
	} else {
		tmp = asin(((l_m * sqrt((fma(((Om / Omc) / Omc), -Om, 1.0) * 0.5))) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+26)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / fma(Float64(Float64(Float64(t_m / l_m) * 2.0) * Float64(1.0 / l_m)), t_m, 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0) * 0.5))) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+26], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * N[(1.0 / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.00000000000000005e26

   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-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. associate-/r/ N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 88.3

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

   4. Applied rewrites 88.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 88.3

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

   6. Applied rewrites 88.3%

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

   if 1.00000000000000005e26 < (/.f64 t l)

   1. Initial program 69.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 81.5

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

   5. Applied rewrites 81.5%

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

   7. Applied rewrites 92.2%

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

   9. Applied rewrites 99.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot 2\right) \cdot \frac{1}{\ell}, t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + 2 \cdot \frac{\frac{t\_m}{l\_m}}{\frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 4e+30)
   (asin
    (sqrt
     (/
      (- 1.0 (* Om (/ Om (* Omc Omc))))
      (+ 1.0 (* 2.0 (/ (/ t_m l_m) (/ l_m t_m)))))))
   (asin (/ (* l_m (sqrt (* (fma (/ (/ Om Omc) Omc) (- Om) 1.0) 0.5))) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 4e+30) {
		tmp = asin(sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / (1.0 + (2.0 * ((t_m / l_m) / (l_m / t_m)))))));
	} else {
		tmp = asin(((l_m * sqrt((fma(((Om / Omc) / Omc), -Om, 1.0) * 0.5))) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 4e+30)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc)))) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) / Float64(l_m / t_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0) * 0.5))) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 4e+30], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.0000000000000001e30

   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

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. lift-/.f64 N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 85.4

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

   6. Applied rewrites 85.4%

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

   if 4.0000000000000001e30 < (/.f64 t l)

   1. Initial program 67.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 99.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{\mathsf{fma}\left(\frac{t\_m \cdot 2}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 4e+30)
   (asin
    (sqrt
     (/
      (- 1.0 (* Om (/ Om (* Omc Omc))))
      (fma (/ (* t_m 2.0) l_m) (/ t_m l_m) 1.0))))
   (asin (/ (* l_m (sqrt (* (fma (/ (/ Om Omc) Omc) (- Om) 1.0) 0.5))) t_m))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 4e+30) {
		tmp = asin(sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / fma(((t_m * 2.0) / l_m), (t_m / l_m), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt((fma(((Om / Omc) / Omc), -Om, 1.0) * 0.5))) / t_m));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 4e+30)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc)))) / fma(Float64(Float64(t_m * 2.0) / l_m), Float64(t_m / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0) * 0.5))) / t_m));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 4e+30], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m * 2.0), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.0000000000000001e30

   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

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 85.4

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 85.4

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

   6. Applied rewrites 85.4%

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

   if 4.0000000000000001e30 < (/.f64 t l)

   1. Initial program 67.9%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 91.9%

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

   9. Applied rewrites 99.3%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 4 \cdot 10^{+30}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{\mathsf{fma}\left(\frac{t \cdot 2}{\ell}, \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right) \cdot 0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1

   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. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 64.1%

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

   if 1 < (/.f64 t l)

   1. Initial program 71.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 92.0%

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

   9. Applied rewrites 98.6%

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

4. Final simplification 75.4%

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

Alternative 8: 97.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1

   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. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   7. Applied rewrites 64.1%

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

   if 1 < (/.f64 t l)

   1. Initial program 71.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 97.5%

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

Alternative 9: 96.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.0) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_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) <= 1.0d0) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_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) <= 1.0) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_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) <= 1.0:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin((l_m * (math.sqrt(0.5) / t_m)))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.0)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_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) <= 1.0)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin((l_m * (sqrt(0.5) / t_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], 1.0], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1

   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. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 63.2%

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]

   if 1 < (/.f64 t l)

   1. Initial program 71.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 97.5%

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

Alternative 10: 49.7% 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 84.3%

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

3. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 40.6

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

5. Applied rewrites 40.6%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 44.2%

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

Reproduce

herbie shell --seed 2024235 
(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)))))))