Toniolo and Linder, Equation (2)

Percentage Accurate: 84.3% → 97.3%

Time: 15.8s

Alternatives: 8

Speedup: 2.1×

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: 84.3% 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: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 0:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, -Om, 1\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 - 2 \cdot \frac{-1}{\frac{l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}}}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
      0.0)
   (asin (* (sqrt (* (fma (/ Om (* Omc Omc)) (- Om) 1.0) 0.5)) (/ l t)))
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (- 1.0 (* 2.0 (/ -1.0 (* (/ l_m t_m) (/ 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 (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 0.0) {
		tmp = asin((sqrt((fma((Om / (Omc * Omc)), -Om, 1.0) * 0.5)) * (l / t)));
	} else {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / (1.0 - (2.0 * (-1.0 / ((l_m / t_m) * (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(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 0.0)
		tmp = asin(Float64(sqrt(Float64(fma(Float64(Om / Float64(Omc * Omc)), Float64(-Om), 1.0) * 0.5)) * Float64(l / t)));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / Float64(1.0 - Float64(2.0 * Float64(-1.0 / Float64(Float64(l_m / t_m) * Float64(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[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[ArcSin[N[(N[Sqrt[N[(N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(2.0 * N[(-1.0 / N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 0.0

   1. Initial program 47.6%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 47.6

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

   9. Applied rewrites 67.6%

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

   if 0.0 < (/.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 98.8%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 98.8

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

   6. Applied rewrites 98.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 0:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, -Om, 1\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{1 - 2 \cdot \frac{-1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 0:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, -Om, 1\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{-\frac{\mathsf{fma}\left(Om, \frac{\frac{Om}{Omc}}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, 2 \cdot \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
      0.0)
   (asin (* (sqrt (* (fma (/ Om (* Omc Omc)) (- Om) 1.0) 0.5)) (/ l t)))
   (asin
    (sqrt
     (-
      (/
       (fma Om (/ (/ Om Omc) Omc) -1.0)
       (fma (/ t_m l_m) (* 2.0 (/ t_m l_m)) 1.0)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 0.0) {
		tmp = asin((sqrt((fma((Om / (Omc * Omc)), -Om, 1.0) * 0.5)) * (l / t)));
	} else {
		tmp = asin(sqrt(-(fma(Om, ((Om / Omc) / Omc), -1.0) / fma((t_m / l_m), (2.0 * (t_m / l_m)), 1.0))));
	}
	return tmp;
}

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[ArcSin[N[(N[Sqrt[N[(N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[(-N[(N[(Om * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 0.0

   1. Initial program 47.6%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 47.6

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

   4. Applied rewrites 47.6%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 47.6

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

   7. Applied rewrites 47.6%

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

   9. Applied rewrites 67.6%

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

   if 0.0 < (/.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 98.8%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 98.8

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

   4. Applied rewrites 98.8%

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

   6. Applied rewrites 78.4%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lower-/.f64 84.2

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

   8. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 98.8

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

   10. Applied rewrites 98.8%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}} \leq 0:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, -Om, 1\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{-\frac{\mathsf{fma}\left(Om, \frac{\frac{Om}{Omc}}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, 2 \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}} \leq 0.002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, -Om, 1\right) \cdot 0.5} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \end{array} \end{array} \]

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)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))
      0.002)
   (asin (* (sqrt (* (fma (/ Om (* Omc Omc)) (- Om) 1.0) 0.5)) (/ l t)))
   (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))) <= 0.002) {
		tmp = asin((sqrt((fma((Om / (Omc * Omc)), -Om, 1.0) * 0.5)) * (l / t)));
	} else {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))) <= 0.002)
		tmp = asin(Float64(sqrt(Float64(fma(Float64(Om / Float64(Omc * Omc)), Float64(-Om), 1.0) * 0.5)) * Float64(l / t)));
	else
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.002], N[ArcSin[N[(N[Sqrt[N[(N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))) < 2e-3

   1. Initial program 69.7%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 49.2

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

   7. Applied rewrites 49.2%

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

   9. Applied rewrites 55.6%

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

   if 2e-3 < (/.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 98.6%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 98.6

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

   4. Applied rewrites 98.6%

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

   6. Applied rewrites 84.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 88.7%

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

       1. associate-/r* N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. associate-*l/ N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 96.8

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

   11. Applied rewrites 96.8%

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

Alternative 4: 92.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 40:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right)}}{t}\right)\\ \end{array} \end{array} \]

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) 40.0)
   (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))
   (asin (* l (/ (sqrt (* 0.5 (- 1.0 (/ (* Om Om) (* Omc Omc))))) t)))))

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) <= 40.0) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	} else {
		tmp = asin((l * (sqrt((0.5 * (1.0 - ((Om * Om) / (Omc * Omc))))) / t)));
	}
	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) <= 40.0)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	else
		tmp = asin(Float64(l * Float64(sqrt(Float64(0.5 * Float64(1.0 - Float64(Float64(Om * Om) / Float64(Omc * Omc))))) / t)));
	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], 40.0], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l * N[(N[Sqrt[N[(0.5 * N[(1.0 - N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 40

   1. Initial program 89.5%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 89.5

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

   4. Applied rewrites 89.5%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 60.5%

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

       1. associate-/r* N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. associate-*l/ N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 65.9

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

   11. Applied rewrites 65.9%

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

   if 40 < (/.f64 t l)

   1. Initial program 67.4%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 4.1

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

   5. Applied rewrites 4.1%

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

      1. lift-approx 86.5

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

   7. Applied rewrites 86.7%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 40:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5 \cdot \left(1 - \frac{Om \cdot Om}{Omc \cdot Omc}\right)}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\ell \cdot \ell}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.0002)
   (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))
   (asin (sqrt (/ 1.0 (fma 2.0 (/ (* t t) (* l l)) 1.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	} else {
		tmp = asin(sqrt((1.0 / fma(2.0, ((t * t) / (l * l)), 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.0002)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(2.0, Float64(Float64(t * t) / Float64(l * l)), 1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0002], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(2.0 * N[(N[(t * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 89.5%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 89.5

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

   4. Applied rewrites 89.5%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 60.5%

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

       1. associate-/r* N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*r/ N/A

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

       4. associate-*l/ N/A

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

       5. lift-/.f64 N/A

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

       6. lower-fma.f64 65.9

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

   11. Applied rewrites 65.9%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

Alternative 6: 71.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot t}{\ell \cdot \ell}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.0002)
   (asin (sqrt (/ -1.0 -1.0)))
   (asin (sqrt (/ 1.0 (fma 2.0 (/ (* t t) (* l l)) 1.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.0002) {
		tmp = asin(sqrt((-1.0 / -1.0)));
	} else {
		tmp = asin(sqrt((1.0 / fma(2.0, ((t * t) / (l * l)), 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.0002)
		tmp = asin(sqrt(Float64(-1.0 / -1.0)));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(2.0, Float64(Float64(t * t) / Float64(l * l)), 1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0002], N[ArcSin[N[Sqrt[N[(-1.0 / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(2.0 * N[(N[(t * t), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e-4

   1. Initial program 89.5%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 89.5

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

   4. Applied rewrites 89.5%

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 60.5%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 65.5%

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

   if 2.0000000000000001e-4 < (/.f64 t l)

   1. Initial program 67.4%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 67.6

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{1 + 2 \cdot \frac{{t}^{2}}{{\ell}^{2}}}}}\right) \]
   6. 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. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 42.4

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

   7. Applied rewrites 42.4%

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

Alternative 7: 59.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.52 \cdot 10^{+240}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\frac{Om \cdot Om}{Omc \cdot Omc}}{-1}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.52e+240)
   (asin (sqrt (/ -1.0 -1.0)))
   (asin (sqrt (/ (/ (* Om Om) (* Omc Omc)) -1.0)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.52e+240) {
		tmp = asin(sqrt((-1.0 / -1.0)));
	} else {
		tmp = asin(sqrt((((Om * Om) / (Omc * Omc)) / -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 ((t_m / l_m) <= 1.52d+240) then
        tmp = asin(sqrt(((-1.0d0) / (-1.0d0))))
    else
        tmp = asin(sqrt((((om * om) / (omc * omc)) / (-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 ((t_m / l_m) <= 1.52e+240) {
		tmp = Math.asin(Math.sqrt((-1.0 / -1.0)));
	} else {
		tmp = Math.asin(Math.sqrt((((Om * Om) / (Omc * Omc)) / -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 (t_m / l_m) <= 1.52e+240:
		tmp = math.asin(math.sqrt((-1.0 / -1.0)))
	else:
		tmp = math.asin(math.sqrt((((Om * Om) / (Omc * Omc)) / -1.0)))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.52e+240)
		tmp = asin(sqrt(Float64(-1.0 / -1.0)));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(Om * Om) / Float64(Omc * Omc)) / -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 ((t_m / l_m) <= 1.52e+240)
		tmp = asin(sqrt((-1.0 / -1.0)));
	else
		tmp = asin(sqrt((((Om * Om) / (Omc * Omc)) / -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[(t$95$m / l$95$m), $MachinePrecision], 1.52e+240], N[ArcSin[N[Sqrt[N[(-1.0 / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.5200000000000001e240

   1. Initial program 87.2%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 87.2

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

   4. Applied rewrites 87.2%

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

   6. Applied rewrites 70.0%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 53.2%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 57.6%

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

   if 1.5200000000000001e240 < (/.f64 t l)

   1. Initial program 61.1%

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

      1. lift-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-2neg N/A

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

      9. metadata-eval N/A

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

      10. associate-/l/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 61.1

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

   4. Applied rewrites 61.1%

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

   6. Applied rewrites 56.7%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 3.1%

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

   10. Taylor expanded in Om around inf

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 51.3

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

   12. Applied rewrites 51.3%

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

Alternative 8: 50.6% accurate, 2.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 84.8%

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

   1. lift-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. clear-num N/A

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

   8. frac-2neg N/A

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

   9. metadata-eval N/A

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

   10. associate-/l/ N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. distribute-neg-frac2 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 84.8

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

4. Applied rewrites 84.8%

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

6. Applied rewrites 68.8%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 48.7%

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

10. Taylor expanded in Om around 0

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

12. Applied rewrites 52.7%

    \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{-1}}{-1}}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(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)))))))