Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.8%

Time: 12.1s

Alternatives: 14

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.6% 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: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left({\left(\frac{t\_m}{l\_m}\right)}^{2}, 2, 1\right)\\ t_2 := {t\_1}^{-2}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{t\_2 - {\left(\frac{Om}{Omc}\right)}^{4} \cdot t\_2}}{\sqrt{{t\_1}^{-1} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t\_m} \cdot l\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (fma (pow (/ t_m l_m) 2.0) 2.0 1.0)) (t_2 (pow t_1 -2.0)))
   (if (<= (/ t_m l_m) 2e+76)
     (asin
      (/
       (sqrt (- t_2 (* (pow (/ Om Omc) 4.0) t_2)))
       (sqrt (* (pow t_1 -1.0) (+ (pow (/ Om Omc) 2.0) 1.0)))))
     (asin
      (*
       (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))
       (* (/ (sqrt 0.5) t_m) l_m))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = fma(pow((t_m / l_m), 2.0), 2.0, 1.0);
	double t_2 = pow(t_1, -2.0);
	double tmp;
	if ((t_m / l_m) <= 2e+76) {
		tmp = asin((sqrt((t_2 - (pow((Om / Omc), 4.0) * t_2))) / sqrt((pow(t_1, -1.0) * (pow((Om / Omc), 2.0) + 1.0)))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((sqrt(0.5) / t_m) * l_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = fma((Float64(t_m / l_m) ^ 2.0), 2.0, 1.0)
	t_2 = t_1 ^ -2.0
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+76)
		tmp = asin(Float64(sqrt(Float64(t_2 - Float64((Float64(Om / Omc) ^ 4.0) * t_2))) / sqrt(Float64((t_1 ^ -1.0) * Float64((Float64(Om / Omc) ^ 2.0) + 1.0)))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(sqrt(0.5) / t_m) * l_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, -2.0], $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+76], N[ArcSin[N[(N[Sqrt[N[(t$95$2 - N[(N[Power[N[(Om / Omc), $MachinePrecision], 4.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$1, -1.0], $MachinePrecision] * N[(N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e76

   1. Initial program 94.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. flip-- N/A

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

      6. sqrt-div N/A

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

      7. lower-/.f64 N/A

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

   4. Applied rewrites 82.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{{\left(\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)\right)}^{-2} - {\left(\frac{Om}{Omc}\right)}^{4} \cdot {\left(\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)\right)}^{-2}}}{\sqrt{\left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) \cdot {\left(\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)\right)}^{-1}}}\right)} \]

   if 2.0000000000000001e76 < (/.f64 t l)

   1. Initial program 69.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 61.1

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

   4. Applied rewrites 61.1%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 61.1

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

   6. Applied rewrites 61.1%

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

   7. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower--.f64 N/A

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

   9. Applied rewrites 99.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+76}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{{\left(\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)\right)}^{-2} - {\left(\frac{Om}{Omc}\right)}^{4} \cdot {\left(\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)\right)}^{-2}}}{\sqrt{{\left(\mathsf{fma}\left({\left(\frac{t}{\ell}\right)}^{2}, 2, 1\right)\right)}^{-1} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\frac{t\_m}{\frac{l\_m}{t\_m} \cdot l\_m} \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t\_m} \cdot l\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+43)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ (* (/ t_m (* (/ l_m t_m) l_m)) 2.0) 1.0))))
   (asin
    (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (* (/ (sqrt 0.5) t_m) l_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+43) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((sqrt(0.5) / t_m) * l_m)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 1d+43) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0d0) + 1.0d0))))
    else
        tmp = asin((sqrt((1.0d0 - ((om / omc) * (om / omc)))) * ((sqrt(0.5d0) / t_m) * l_m)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+43) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))));
	} else {
		tmp = Math.asin((Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((Math.sqrt(0.5) / t_m) * l_m)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1e+43:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))))
	else:
		tmp = math.asin((math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((math.sqrt(0.5) / t_m) * l_m)))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+43)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(Float64(Float64(t_m / Float64(Float64(l_m / t_m) * l_m)) * 2.0) + 1.0))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(sqrt(0.5) / t_m) * l_m)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1e+43)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (((t_m / ((l_m / t_m) * l_m)) * 2.0) + 1.0))));
	else
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((sqrt(0.5) / t_m) * l_m)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+43], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m / N[(N[(l$95$m / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.00000000000000001e43

   1. Initial program 94.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 93.0

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

   4. Applied rewrites 93.0%

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

   if 1.00000000000000001e43 < (/.f64 t l)

   1. Initial program 73.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 65.4

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

   4. Applied rewrites 65.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 65.4

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

   6. Applied rewrites 65.4%

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

   7. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower--.f64 N/A

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

   9. Applied rewrites 99.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\frac{t}{\frac{\ell}{t} \cdot \ell} \cdot 2 + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, 2 \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t\_m} \cdot l\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2000000000.0)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ t_m l_m) (* 2.0 (/ t_m l_m)) 1.0))))
   (asin
    (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (* (/ (sqrt 0.5) t_m) l_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2000000000.0) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((t_m / l_m), (2.0 * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((sqrt(0.5) / t_m) * l_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2000000000.0)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(t_m / l_m), Float64(2.0 * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(sqrt(0.5) / t_m) * l_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2000000000.0], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2e9

   1. Initial program 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 94.1

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

   4. Applied rewrites 94.1%

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

   if 2e9 < (/.f64 t l)

   1. Initial program 77.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 66.5

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

   6. Applied rewrites 66.5%

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

   7. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower--.f64 N/A

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

   9. Applied rewrites 99.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t}{\ell}, 2 \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+67}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t\_m} \cdot l\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+67)
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 1.0))))
   (asin
    (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (* (/ (sqrt 0.5) t_m) l_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+67) {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((sqrt(0.5) / t_m) * l_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+67)
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(sqrt(0.5) / t_m) * l_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+67], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999976e67

   1. Initial program 94.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.7

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

   8. Applied rewrites 80.7%

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

   10. Applied rewrites 92.5%

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

   if 4.99999999999999976e67 < (/.f64 t l)

   1. Initial program 71.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 62.7

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

   4. Applied rewrites 62.7%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. sqrt-unprod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. unpow-prod-down N/A

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

      10. pow-prod-up N/A

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

      11. lower-pow.f64 N/A

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

      12. metadata-eval 62.7

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

   6. Applied rewrites 62.7%

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

   7. Taylor expanded in t around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-frac2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sqrt.f64 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. lower--.f64 N/A

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

   9. Applied rewrites 99.7%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+67}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+32)
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 1.0))))
   (asin
    (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+32) {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+32)
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+32], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.00000000000000005e32

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 15.9

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

   5. Applied rewrites 15.9%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.2

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

   8. Applied rewrites 81.2%

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

   10. Applied rewrites 92.2%

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

   if 1.00000000000000005e32 < (/.f64 t l)

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. unpow1 N/A

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

      8. sqr-pow N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 75.3

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \mathsf{fma}\left(-0.5, \frac{Om}{Omc} \cdot \frac{Om}{Omc}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+32)
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 1.0))))
   (asin
    (* (/ (* (sqrt 0.5) l_m) t_m) (fma -0.5 (* (/ Om Omc) (/ Om Omc)) 1.0)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+32) {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 1.0))));
	} else {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * fma(-0.5, ((Om / Omc) * (Om / Omc)), 1.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+32)
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 1.0))));
	else
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * fma(-0.5, Float64(Float64(Om / Omc) * Float64(Om / Omc)), 1.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+32], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.00000000000000005e32

   1. Initial program 94.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 15.9

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

   5. Applied rewrites 15.9%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.2

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

   8. Applied rewrites 81.2%

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

   10. Applied rewrites 92.2%

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

   if 1.00000000000000005e32 < (/.f64 t l)

   1. Initial program 75.3%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.5%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+32}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \mathsf{fma}\left(-0.5, \frac{Om}{Omc} \cdot \frac{Om}{Omc}, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{l\_m}, \frac{t\_m}{\frac{l\_m}{t\_m}}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 5e-43)
   (asin (sqrt (/ 1.0 (fma (/ 2.0 l_m) (/ t_m (/ l_m t_m)) 1.0))))
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 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 (l_m <= 5e-43) {
		tmp = asin(sqrt((1.0 / fma((2.0 / l_m), (t_m / (l_m / t_m)), 1.0))));
	} else {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 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 (l_m <= 5e-43)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 / l_m), Float64(t_m / Float64(l_m / t_m)), 1.0))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 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[l$95$m, 5e-43], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.00000000000000019e-43

   1. Initial program 89.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 78.0

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

   8. Applied rewrites 78.0%

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

   10. Applied rewrites 87.6%

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

   if 5.00000000000000019e-43 < l

   1. Initial program 90.1%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 9.0

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

   5. Applied rewrites 9.0%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

   10. Applied rewrites 90.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{t}{\frac{\ell}{t}}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 3.2 \cdot 10^{-269}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{l\_m}, \frac{t\_m}{l\_m} \cdot t\_m, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m, \frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 3.2e-269)
   (asin (sqrt (/ 1.0 (fma (/ 2.0 l_m) (* (/ t_m l_m) t_m) 1.0))))
   (asin (sqrt (/ 1.0 (fma t_m (* (/ 2.0 l_m) (/ t_m l_m)) 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 (l_m <= 3.2e-269) {
		tmp = asin(sqrt((1.0 / fma((2.0 / l_m), ((t_m / l_m) * t_m), 1.0))));
	} else {
		tmp = asin(sqrt((1.0 / fma(t_m, ((2.0 / l_m) * (t_m / l_m)), 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 (l_m <= 3.2e-269)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 / l_m), Float64(Float64(t_m / l_m) * t_m), 1.0))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(t_m, Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), 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[l$95$m, 3.2e-269], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(t$95$m * N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.2000000000000001e-269

   1. Initial program 90.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 76.3

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

   8. Applied rewrites 76.3%

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

   10. Applied rewrites 87.9%

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

   if 3.2000000000000001e-269 < l

   1. Initial program 88.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 25.6

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 79.4

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

   8. Applied rewrites 79.4%

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

   10. Applied rewrites 88.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.2 \cdot 10^{-269}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{t}{\ell} \cdot t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t, \frac{2}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;l\_m \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\frac{t\_m \cdot t\_m}{l\_m} \cdot \frac{2}{l\_m}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{l\_m \cdot l\_m} \cdot t\_m, t\_m, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= l_m 1.05e-154)
   (asin (sqrt (/ 1.0 (* (/ (* t_m t_m) l_m) (/ 2.0 l_m)))))
   (asin (sqrt (/ 1.0 (fma (* (/ 2.0 (* l_m l_m)) t_m) t_m 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 (l_m <= 1.05e-154) {
		tmp = asin(sqrt((1.0 / (((t_m * t_m) / l_m) * (2.0 / l_m)))));
	} else {
		tmp = asin(sqrt((1.0 / fma(((2.0 / (l_m * l_m)) * t_m), t_m, 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 (l_m <= 1.05e-154)
		tmp = asin(sqrt(Float64(1.0 / Float64(Float64(Float64(t_m * t_m) / l_m) * Float64(2.0 / l_m)))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(Float64(2.0 / Float64(l_m * l_m)) * t_m), t_m, 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[l$95$m, 1.05e-154], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(2.0 / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.04999999999999992e-154

   1. Initial program 91.0%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 29.0

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

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.2

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

   8. Applied rewrites 77.2%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 34.0%

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

   if 1.04999999999999992e-154 < l

   1. Initial program 88.1%

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

   3. Taylor expanded in 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. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.5

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

   5. Applied rewrites 84.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\frac{t \cdot t}{\ell} \cdot \frac{2}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\ell \cdot \ell} \cdot t, t, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 200:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{l\_m \cdot l\_m}{t\_m} \cdot \frac{0.5}{t\_m}}\right)\\ \end{array} \end{array} \]

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) 200.0)
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))
   (asin (sqrt (* (/ (* l_m l_m) t_m) (/ 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) <= 200.0) {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = asin(sqrt((((l_m * l_m) / t_m) * (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) <= 200.0d0) then
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    else
        tmp = asin(sqrt((((l_m * l_m) / t_m) * (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) <= 200.0) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	} else {
		tmp = Math.asin(Math.sqrt((((l_m * l_m) / t_m) * (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) <= 200.0:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	else:
		tmp = math.asin(math.sqrt((((l_m * l_m) / t_m) * (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) <= 200.0)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(l_m * l_m) / t_m) * Float64(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) <= 200.0)
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	else
		tmp = asin(sqrt((((l_m * l_m) / t_m) * (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], 200.0], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(0.5 / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 200

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-*r* N/A

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

      7. unpow1 N/A

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

      8. sqr-pow N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. unpow1/2 N/A

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

      16. lower-sqrt.f64 N/A

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

      17. metadata-eval N/A

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

      18. unpow1/2 N/A

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

      19. lower-sqrt.f64 42.5

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 73.5

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

   7. Applied rewrites 73.5%

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

   if 200 < (/.f64 t l)

   1. Initial program 77.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 55.6

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

   5. Applied rewrites 55.6%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 58.9%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 200:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\ell \cdot \ell}{t} \cdot \frac{0.5}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in t around inf

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

   1. associate-/l* N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-/l* N/A

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

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

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

   13. mul-1-neg N/A

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

   14. lower-fma.f64 N/A

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

   15. mul-1-neg N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. unpow2 N/A

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

   19. lower-*.f64 N/A

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

   20. unpow2 N/A

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

   21. lower-*.f64 25.0

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

5. Applied rewrites 25.0%

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

6. Taylor expanded in Om around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. associate-*r/ N/A

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

   4. unpow2 N/A

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

   5. times-frac N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 77.7

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

8. Applied rewrites 77.7%

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

10. Applied rewrites 87.2%

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

11. Final simplification 87.2%

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

Alternative 12: 71.2% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. unpow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 80.2

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

5. Applied rewrites 80.2%

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

Alternative 13: 51.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. unpow1 N/A

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

   8. sqr-pow N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. unpow1/2 N/A

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

   16. lower-sqrt.f64 N/A

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

   17. metadata-eval N/A

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

   18. unpow1/2 N/A

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

   19. lower-sqrt.f64 51.6

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 55.9

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

7. Applied rewrites 55.9%

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

Alternative 14: 50.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \sin^{-1} \left(\mathsf{fma}\left(\frac{-0.5}{Omc}, \frac{Om}{Omc} \cdot Om, 1\right)\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (fma (/ -0.5 Omc) (* (/ Om Omc) Om) 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(fma((-0.5 / Omc), ((Om / Omc) * Om), 1.0));
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	return asin(fma(Float64(-0.5 / Omc), Float64(Float64(Om / Omc) * Om), 1.0))
end

Wolfram

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

Derivation

1. Initial program 89.8%

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

3. Taylor expanded in Om around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

5. Applied rewrites 80.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 51.4%

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

10. Applied rewrites 55.6%

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

Reproduce

herbie shell --seed 2024288 
(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)))))))