Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.5%

Time: 14.7s

Alternatives: 9

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: 83.6% 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: 98.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 2 \cdot \frac{-1}{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\frac{0.5}{t\_m}}}{\sqrt{t\_m}}\right)\\ \end{array} \end{array} \]

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+31)
   (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 (/ 0.5 t_m))) (sqrt 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+31) {
		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((0.5 / t_m))) / sqrt(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) <= 5d+31) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 - (2.0d0 * ((-1.0d0) / ((l_m / t_m) * (l_m / t_m))))))))
    else
        tmp = asin(((l_m * sqrt((0.5d0 / t_m))) / sqrt(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) <= 5e+31) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 - (2.0 * (-1.0 / ((l_m / t_m) * (l_m / t_m))))))));
	} else {
		tmp = Math.asin(((l_m * Math.sqrt((0.5 / t_m))) / Math.sqrt(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) <= 5e+31:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 - (2.0 * (-1.0 / ((l_m / t_m) * (l_m / t_m))))))))
	else:
		tmp = math.asin(((l_m * math.sqrt((0.5 / t_m))) / math.sqrt(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+31)
		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(0.5 / t_m))) / sqrt(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) <= 5e+31)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 - (2.0 * (-1.0 / ((l_m / t_m) * (l_m / t_m))))))));
	else
		tmp = asin(((l_m * sqrt((0.5 / t_m))) / sqrt(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], 5e+31], 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[(0.5 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000027e31

   1. Initial program 86.8%

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

      1. lift-/.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 86.7

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

   4. Applied rewrites 86.7%

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

   if 5.00000000000000027e31 < (/.f64 t l)

   1. Initial program 62.5%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. sqrt-prod N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. pow1/2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-sqrt.f64 56.3

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

   10. Applied rewrites 56.3%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 - 2 \cdot \frac{-1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{0.5}{t}}}{\sqrt{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\frac{0.5}{t\_m}}}{\sqrt{t\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-\frac{Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
      0.4)
   (asin (/ (* l_m (sqrt (/ 0.5 t_m))) (sqrt t_m)))
   (asin (sqrt (fma (- (/ 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 (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 0.4) {
		tmp = asin(((l_m * sqrt((0.5 / t_m))) / sqrt(t_m)));
	} else {
		tmp = asin(sqrt(fma(-(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(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 0.4)
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(0.5 / t_m))) / sqrt(t_m)));
	else
		tmp = asin(sqrt(fma(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[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(0.5 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[((-N[(Om / Omc), $MachinePrecision]) * N[(Om / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $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))))) < 0.40000000000000002

   1. Initial program 67.4%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 44.1

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

   8. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. sqrt-prod N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. pow1/2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-sqrt.f64 32.8

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

   10. Applied rewrites 32.8%

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

   if 0.40000000000000002 < (/.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 97.0%

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

   3. Taylor expanded in 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 83.7

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

   5. Applied rewrites 83.7%

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

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

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

      11. lift-/.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lower-fma.f64 95.6

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. distribute-frac-neg2 N/A

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

      19. distribute-frac-neg N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 95.6

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

   7. Applied rewrites 95.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{0.5}{t}}}{\sqrt{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-\frac{Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-\frac{Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
      0.4)
   (asin (* l_m (/ (sqrt 0.5) t_m)))
   (asin (sqrt (fma (- (/ 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 (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 0.4) {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	} else {
		tmp = asin(sqrt(fma(-(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(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 0.4)
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	else
		tmp = asin(sqrt(fma(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[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.4], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[((-N[(Om / Omc), $MachinePrecision]) * N[(Om / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $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))))) < 0.40000000000000002

   1. Initial program 67.4%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 44.1

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

   8. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. lift-/.f64 N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-/.f64 N/A

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

      22. pow1/2 N/A

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

      23. lower-sqrt.f64 56.5

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

   10. Applied rewrites 56.5%

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

   if 0.40000000000000002 < (/.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 97.0%

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

   3. Taylor expanded in 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 83.7

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

   5. Applied rewrites 83.7%

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

      1. times-frac N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

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

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

      11. lift-/.f64 N/A

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

      12. distribute-frac-neg2 N/A

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lower-fma.f64 95.6

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. distribute-frac-neg2 N/A

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

      19. distribute-frac-neg N/A

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

      20. lower-/.f64 N/A

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

      21. lower-neg.f64 95.6

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

   7. Applied rewrites 95.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-\frac{Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{1}{l\_m}, t\_m \cdot \left(\frac{t\_m}{l\_m} \cdot 2\right), 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\frac{0.5}{t\_m}}}{\sqrt{t\_m}}\right)\\ \end{array} \end{array} \]

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+31)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ 1.0 l_m) (* t_m (* (/ t_m l_m) 2.0)) 1.0))))
   (asin (/ (* l_m (sqrt (/ 0.5 t_m))) (sqrt 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+31) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((1.0 / l_m), (t_m * ((t_m / l_m) * 2.0)), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt((0.5 / t_m))) / sqrt(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+31)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(1.0 / l_m), Float64(t_m * Float64(Float64(t_m / l_m) * 2.0)), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(0.5 / t_m))) / sqrt(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+31], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(0.5 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000027e31

   1. Initial program 86.8%

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

      1. lift-/.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. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*l* N/A

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

      12. lift-/.f64 N/A

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

      13. clear-num N/A

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

      14. associate-/r/ N/A

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

      15. associate-*l* N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 85.4

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

   4. Applied rewrites 85.4%

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

   if 5.00000000000000027e31 < (/.f64 t l)

   1. Initial program 62.5%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. sqrt-prod N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. pow1/2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-sqrt.f64 56.3

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

   10. Applied rewrites 56.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{1}{\ell}, t \cdot \left(\frac{t}{\ell} \cdot 2\right), 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{0.5}{t}}}{\sqrt{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(t\_m \cdot 2, \frac{\frac{t\_m}{l\_m}}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\frac{0.5}{t\_m}}}{\sqrt{t\_m}}\right)\\ \end{array} \end{array} \]

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+31)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (* t_m 2.0) (/ (/ t_m l_m) l_m) 1.0))))
   (asin (/ (* l_m (sqrt (/ 0.5 t_m))) (sqrt 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+31) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((t_m * 2.0), ((t_m / l_m) / l_m), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt((0.5 / t_m))) / sqrt(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+31)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(t_m * 2.0), Float64(Float64(t_m / l_m) / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(0.5 / t_m))) / sqrt(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+31], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m * 2.0), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(0.5 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000027e31

   1. Initial program 86.8%

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

      1. lift-/.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. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-/.f64 N/A

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

      11. div-inv N/A

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

      12. associate-*l* N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

      18. associate-/r/ N/A

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

      19. clear-num N/A

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

      20. lift-/.f64 N/A

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

      21. lower-/.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if 5.00000000000000027e31 < (/.f64 t l)

   1. Initial program 62.5%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. sqrt-prod N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. pow1/2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-sqrt.f64 56.3

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

   10. Applied rewrites 56.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(t \cdot 2, \frac{\frac{t}{\ell}}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{0.5}{t}}}{\sqrt{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+31}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{l\_m}, t\_m \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\frac{0.5}{t\_m}}}{\sqrt{t\_m}}\right)\\ \end{array} \end{array} \]

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+31)
   (asin (sqrt (/ 1.0 (fma (/ 2.0 l_m) (* t_m (/ t_m l_m)) 1.0))))
   (asin (/ (* l_m (sqrt (/ 0.5 t_m))) (sqrt 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+31) {
		tmp = asin(sqrt((1.0 / fma((2.0 / l_m), (t_m * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt((0.5 / t_m))) / sqrt(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+31)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 / l_m), Float64(t_m * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(0.5 / t_m))) / sqrt(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+31], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(0.5 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.00000000000000027e31

   1. Initial program 86.8%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 68.7

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

   5. Applied rewrites 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-/.f64 N/A

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

      14. lower-*.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if 5.00000000000000027e31 < (/.f64 t l)

   1. Initial program 62.5%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 44.3

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 45.8

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

   8. Applied rewrites 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. sqrt-div N/A

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

      5. lower-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. sqrt-prod N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. sqrt-pow1 N/A

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

      12. metadata-eval N/A

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

      13. unpow1 N/A

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

      14. pow1/2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

      19. lower-sqrt.f64 56.3

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

   10. Applied rewrites 56.3%

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

Alternative 7: 96.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0004:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{-l\_m}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.0004)
   (asin (fma (/ t_m l_m) (/ t_m (- l_m)) 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) <= 0.0004) {
		tmp = asin(fma((t_m / l_m), (t_m / -l_m), 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) <= 0.0004)
		tmp = asin(fma(Float64(t_m / l_m), Float64(t_m / Float64(-l_m)), 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], 0.0004], N[ArcSin[N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / (-l$95$m)), $MachinePrecision] + 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) < 4.00000000000000019e-4

   1. Initial program 86.5%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 53.5

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

   8. Applied rewrites 53.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 59.6

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

   10. Applied rewrites 59.6%

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

   if 4.00000000000000019e-4 < (/.f64 t l)

   1. Initial program 65.7%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. lift-/.f64 N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-/.f64 N/A

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

      22. pow1/2 N/A

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

      23. lower-sqrt.f64 95.1

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

   10. Applied rewrites 95.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.0004:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{-\ell}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0004:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.0004) (asin 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) <= 0.0004) {
		tmp = asin(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) <= 0.0004d0) then
        tmp = asin(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) <= 0.0004) {
		tmp = Math.asin(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) <= 0.0004:
		tmp = math.asin(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) <= 0.0004)
		tmp = asin(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) <= 0.0004)
		tmp = asin(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], 0.0004], N[ArcSin[1.0], $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) < 4.00000000000000019e-4

   1. Initial program 86.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}\right) \]
   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 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.8%

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

   if 4.00000000000000019e-4 < (/.f64 t l)

   1. Initial program 65.7%

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

   3. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   4. 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. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. sqrt-prod N/A

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

      7. pow1/2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. sqrt-pow1 N/A

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

      11. metadata-eval N/A

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

      12. unpow1 N/A

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

      13. lower-*.f64 N/A

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

      14. pow1/2 N/A

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

      15. lift-/.f64 N/A

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

      16. sqrt-div N/A

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

      17. pow1/2 N/A

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

      18. lift-*.f64 N/A

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

      19. sqrt-prod N/A

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

      20. rem-square-sqrt N/A

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

      21. lower-/.f64 N/A

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

      22. pow1/2 N/A

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

      23. lower-sqrt.f64 95.1

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

   10. Applied rewrites 95.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.0004:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.7% accurate, 3.5× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(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[1.0], $MachinePrecision]

Derivation

1. Initial program 81.7%

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

3. Taylor expanded in t around 0

   \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}\right) \]
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 42.9

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

5. Applied rewrites 42.9%

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

6. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \color{blue}{1} \]
7. Step-by-step derivation

8. Applied rewrites 48.2%

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

Reproduce

herbie shell --seed 2024214 
(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)))))))