Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 97.2%

Time: 12.6s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.8% 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: 97.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{2}{l\_m} \cdot t\_m, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+155)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ t_m l_m) (* (/ 2.0 l_m) t_m) 1.0))))
   (asin
    (*
     (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
     (/ (* (sqrt 0.5) l_m) t_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+155) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((t_m / l_m), ((2.0 / l_m) * t_m), 1.0))));
	} else {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((sqrt(0.5) * l_m) / t_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+155)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(t_m / l_m), Float64(Float64(2.0 / l_m) * t_m), 1.0))));
	else
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(sqrt(0.5) * l_m) / t_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+155], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(2.0 / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.00000000000000001e155

   1. Initial program 91.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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 49.2

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

   4. Applied rewrites 49.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 89.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. div-inv N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. div-inv N/A

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

      12. lower-fma.f64 N/A

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

      13. div-inv N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. div-inv N/A

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

      18. clear-num N/A

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

      19. lift-/.f64 N/A

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

      20. lower-*.f64 92.0

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

   8. Applied rewrites 92.0%

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

   if 1.00000000000000001e155 < (/.f64 t l)

   1. Initial program 44.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{2}{\ell} \cdot t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.0% accurate, 0.9× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ (* (pow (/ t_m l_m) 2.0) 2.0) 1.0))
      5e-10)
   (asin (sqrt (/ (* (* l_m l_m) 0.5) (* t_m t_m))))
   (asin (sqrt 1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / ((pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 5e-10) {
		tmp = asin(sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))));
	} else {
		tmp = asin(sqrt(1.0));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if (((1.0d0 - ((om / omc) ** 2.0d0)) / ((((t_m / l_m) ** 2.0d0) * 2.0d0) + 1.0d0)) <= 5d-10) then
        tmp = asin(sqrt((((l_m * l_m) * 0.5d0) / (t_m * t_m))))
    else
        tmp = asin(sqrt(1.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - Math.pow((Om / Omc), 2.0)) / ((Math.pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 5e-10) {
		tmp = Math.asin(Math.sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))));
	} else {
		tmp = Math.asin(Math.sqrt(1.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if ((1.0 - math.pow((Om / Omc), 2.0)) / ((math.pow((t_m / l_m), 2.0) * 2.0) + 1.0)) <= 5e-10:
		tmp = math.asin(math.sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))))
	else:
		tmp = math.asin(math.sqrt(1.0))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(Float64((Float64(t_m / l_m) ^ 2.0) * 2.0) + 1.0)) <= 5e-10)
		tmp = asin(sqrt(Float64(Float64(Float64(l_m * l_m) * 0.5) / Float64(t_m * t_m))));
	else
		tmp = asin(sqrt(1.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (((1.0 - ((Om / Omc) ^ 2.0)) / ((((t_m / l_m) ^ 2.0) * 2.0) + 1.0)) <= 5e-10)
		tmp = asin(sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))));
	else
		tmp = asin(sqrt(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 5e-10], N[ArcSin[N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 5.00000000000000031e-10

   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. 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

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 47.6%

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

   if 5.00000000000000031e-10 < (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))

   1. Initial program 99.3%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 95.6%

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

4. Final simplification 72.0%

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

Alternative 3: 95.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2 \cdot t\_m, \frac{\frac{t\_m}{l\_m}}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+74)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (* 2.0 t_m) (/ (/ t_m l_m) l_m) 1.0))))
   (asin
    (*
     (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
     (/ (* (sqrt 0.5) l_m) t_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+74) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((2.0 * t_m), ((t_m / l_m) / l_m), 1.0))));
	} else {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((sqrt(0.5) * l_m) / t_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+74)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(2.0 * t_m), Float64(Float64(t_m / l_m) / l_m), 1.0))));
	else
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(sqrt(0.5) * l_m) / t_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+74], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 * t$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.99999999999999963e74

   1. Initial program 91.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. associate-/r/ N/A

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

      15. clear-num N/A

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

      16. lift-/.f64 N/A

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

      17. lower-/.f64 89.7

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

   4. Applied rewrites 89.7%

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

   if 4.99999999999999963e74 < (/.f64 t l)

   1. Initial program 58.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 94.8

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

   5. Applied rewrites 94.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+74}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2 \cdot t, \frac{\frac{t}{\ell}}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 10^{+19}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\frac{l\_m}{t\_m} \cdot l\_m}, t\_m, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 \cdot l\_m}{t\_m} \cdot \frac{0.5 \cdot l\_m}{t\_m}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (pow (/ t_m l_m) 2.0) 1e+19)
   (asin (sqrt (/ 1.0 (fma (/ 2.0 (* (/ l_m t_m) l_m)) t_m 1.0))))
   (asin (sqrt (* (/ (* 1.0 l_m) t_m) (/ (* 0.5 l_m) t_m))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (pow((t_m / l_m), 2.0) <= 1e+19) {
		tmp = asin(sqrt((1.0 / fma((2.0 / ((l_m / t_m) * l_m)), t_m, 1.0))));
	} else {
		tmp = asin(sqrt((((1.0 * l_m) / t_m) * ((0.5 * l_m) / t_m))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if ((Float64(t_m / l_m) ^ 2.0) <= 1e+19)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 / Float64(Float64(l_m / t_m) * l_m)), t_m, 1.0))));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(1.0 * l_m) / t_m) * Float64(Float64(0.5 * l_m) / t_m))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision], 1e+19], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 / N[(N[(l$95$m / t$95$m), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(1.0 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(0.5 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) #s(literal 2 binary64)) < 1e19

   1. Initial program 99.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 62.3

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

   4. Applied rewrites 62.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 98.7%

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

   7. Taylor expanded in Omc around inf

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   8. 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 89.4

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

   9. Applied rewrites 89.4%

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

   11. Applied rewrites 99.4%

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

   if 1e19 < (pow.f64 (/.f64 t l) #s(literal 2 binary64))

   1. Initial program 71.5%

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

   3. Taylor expanded in t around 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

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 48.3%

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

   10. Applied rewrites 73.0%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 10^{+19}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\frac{\ell}{t} \cdot \ell}, t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 \cdot \ell}{t} \cdot \frac{0.5 \cdot \ell}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 50000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{l\_m}, \frac{t\_m}{l\_m} \cdot t\_m, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 \cdot l\_m}{t\_m} \cdot \frac{0.5 \cdot l\_m}{t\_m}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (pow (/ t_m l_m) 2.0) 50000000000000.0)
   (asin (sqrt (/ 1.0 (fma (/ 2.0 l_m) (* (/ t_m l_m) t_m) 1.0))))
   (asin (sqrt (* (/ (* 1.0 l_m) t_m) (/ (* 0.5 l_m) t_m))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (pow((t_m / l_m), 2.0) <= 50000000000000.0) {
		tmp = asin(sqrt((1.0 / fma((2.0 / l_m), ((t_m / l_m) * t_m), 1.0))));
	} else {
		tmp = asin(sqrt((((1.0 * l_m) / t_m) * ((0.5 * l_m) / t_m))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if ((Float64(t_m / l_m) ^ 2.0) <= 50000000000000.0)
		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(Float64(Float64(1.0 * l_m) / t_m) * Float64(Float64(0.5 * l_m) / t_m))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision], 50000000000000.0], 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[(N[(N[(1.0 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(0.5 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) #s(literal 2 binary64)) < 5e13

   1. Initial program 99.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 63.0

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

   4. Applied rewrites 63.0%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in Omc around inf

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   8. 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 90.6

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

   9. Applied rewrites 90.6%

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

   11. Applied rewrites 99.4%

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

   if 5e13 < (pow.f64 (/.f64 t l) #s(literal 2 binary64))

   1. Initial program 72.2%

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

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 45.1

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 48.0%

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

   10. Applied rewrites 73.6%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 50000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2}{\ell}, \frac{t}{\ell} \cdot t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 \cdot \ell}{t} \cdot \frac{0.5 \cdot \ell}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{2}{l\_m} \cdot t\_m, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1e+155)
   (asin (sqrt (/ 1.0 (fma (/ t_m l_m) (* (/ 2.0 l_m) t_m) 1.0))))
   (asin
    (*
     (sqrt (fma (- Om) (/ Om (* Omc Omc)) 1.0))
     (/ (* (sqrt 0.5) l_m) t_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1e+155) {
		tmp = asin(sqrt((1.0 / fma((t_m / l_m), ((2.0 / l_m) * t_m), 1.0))));
	} else {
		tmp = asin((sqrt(fma(-Om, (Om / (Omc * Omc)), 1.0)) * ((sqrt(0.5) * l_m) / t_m)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+155)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m / l_m), Float64(Float64(2.0 / l_m) * t_m), 1.0))));
	else
		tmp = asin(Float64(sqrt(fma(Float64(-Om), Float64(Om / Float64(Omc * Omc)), 1.0)) * Float64(Float64(sqrt(0.5) * l_m) / t_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+155], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(2.0 / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[((-Om) * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.00000000000000001e155

   1. Initial program 91.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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 49.2

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

   4. Applied rewrites 49.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. un-div-inv N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. times-frac N/A

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

      17. lower-fma.f64 N/A

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

   6. Applied rewrites 89.0%

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

   7. Taylor expanded in Omc around inf

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   8. 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 77.4

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

   9. Applied rewrites 77.4%

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

   11. Applied rewrites 91.8%

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

   if 1.00000000000000001e155 < (/.f64 t l)

   1. Initial program 44.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-/l* N/A

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

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

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+155}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{2}{\ell} \cdot t, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{Om}{Omc \cdot Omc}, 1\right)} \cdot \frac{\sqrt{0.5} \cdot \ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 \cdot l\_m}{t\_m} \cdot \frac{0.5 \cdot l\_m}{t\_m}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.1)
   (asin (sqrt 1.0))
   (asin (sqrt (* (/ (* 1.0 l_m) t_m) (/ (* 0.5 l_m) t_m))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.1) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(sqrt((((1.0 * l_m) / t_m) * ((0.5 * l_m) / t_m))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.1d0) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(sqrt((((1.0d0 * l_m) / t_m) * ((0.5d0 * l_m) / t_m))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.1) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(Math.sqrt((((1.0 * l_m) / t_m) * ((0.5 * l_m) / t_m))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.1:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(math.sqrt((((1.0 * l_m) / t_m) * ((0.5 * l_m) / t_m))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.1)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(1.0 * l_m) / t_m) * Float64(Float64(0.5 * l_m) / t_m))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.1)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt((((1.0 * l_m) / t_m) * ((0.5 * l_m) / t_m))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.1], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(1.0 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(N[(0.5 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.10000000000000001

   1. Initial program 91.1%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 66.0%

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

   if 0.10000000000000001 < (/.f64 t l)

   1. Initial program 71.8%

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

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 42.1

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 46.1%

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

   10. Applied rewrites 70.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.1:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 \cdot \ell}{t} \cdot \frac{0.5 \cdot \ell}{t}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.1% accurate, 2.3× speedup

Math

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

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 (/ t_m l_m) (* (/ 2.0 l_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((t_m / l_m), ((2.0 / l_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(t_m / l_m), Float64(Float64(2.0 / l_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[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(2.0 / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 86.4%

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

   1. lift-+.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 48.7

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

4. Applied rewrites 48.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. associate-*l* N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. clear-num N/A

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

   12. lift-/.f64 N/A

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

   13. un-div-inv N/A

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

   14. times-frac N/A

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

   15. *-commutative N/A

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

   16. times-frac N/A

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

   17. lower-fma.f64 N/A

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

6. Applied rewrites 83.8%

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

7. Taylor expanded in Omc around inf

   \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
8. 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 73.6

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

9. Applied rewrites 73.6%

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

11. Applied rewrites 86.3%

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

Alternative 9: 73.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 30000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{l\_m}{t\_m \cdot t\_m} \cdot 0.5\right) \cdot l\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 30000.0)
   (asin (sqrt 1.0))
   (asin (sqrt (* (* (/ l_m (* t_m t_m)) 0.5) 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) <= 30000.0) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5) * 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) <= 30000.0d0) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5d0) * 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) <= 30000.0) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(Math.sqrt((((l_m / (t_m * t_m)) * 0.5) * 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) <= 30000.0:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(math.sqrt((((l_m / (t_m * t_m)) * 0.5) * 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) <= 30000.0)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(l_m / Float64(t_m * t_m)) * 0.5) * 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) <= 30000.0)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5) * 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], 30000.0], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(l$95$m / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 3e4

   1. Initial program 91.2%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 65.5%

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

   if 3e4 < (/.f64 t l)

   1. Initial program 70.9%

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

   3. Taylor expanded in t around inf

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

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 N/A

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

      20. unpow2 N/A

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

      21. lower-*.f64 42.9

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

   5. Applied rewrites 42.9%

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

   7. Applied rewrites 51.9%

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

   8. Taylor expanded in Omc around inf

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

   10. Applied rewrites 56.2%

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

Alternative 10: 57.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.55 \cdot 10^{+198}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om}{Omc \cdot Omc} \cdot \left(-Om\right)}\right)\\ \end{array} \end{array} \]

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) 1.55e+198)
   (asin (sqrt 1.0))
   (asin (sqrt (* (/ Om (* Omc Omc)) (- Om))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.55e+198) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(sqrt(((Om / (Omc * Omc)) * -Om)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 1.55d+198) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(sqrt(((om / (omc * omc)) * -om)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.55e+198) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(Math.sqrt(((Om / (Omc * Omc)) * -Om)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1.55e+198:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(math.sqrt(((Om / (Omc * Omc)) * -Om)))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.55e+198)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(Om / Float64(Omc * Omc)) * Float64(-Om))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1.55e+198)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(((Om / (Omc * Omc)) * -Om)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.55e+198], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * (-Om)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.54999999999999987e198

   1. Initial program 90.0%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 53.3

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

   5. Applied rewrites 53.3%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 56.4%

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

   if 1.54999999999999987e198 < (/.f64 t l)

   1. Initial program 52.6%

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

   3. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 3.3

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

   5. Applied rewrites 3.3%

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

   6. Taylor expanded in Omc around inf

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

   8. Applied rewrites 3.4%

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

   9. Taylor expanded in Omc around 0

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

   11. Applied rewrites 37.3%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 1.55 \cdot 10^{+198}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om}{Omc \cdot Omc} \cdot \left(-Om\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.6% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in t around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-/l* N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 48.4

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

5. Applied rewrites 48.4%

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

6. Taylor expanded in Omc around inf

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

8. Applied rewrites 51.2%

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

Reproduce

herbie shell --seed 2024240 
(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)))))))