Toniolo and Linder, Equation (2)

Percentage Accurate: 83.9% → 91.5%

Time: 6.8s

Alternatives: 9

Speedup: 1.3×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t, l, om, omc)
use fmin_fmax_functions
    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: 91.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, t\_m \cdot \frac{-2}{l\_m}, -1\right)}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 10^{+217}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{t\_m \cdot t\_m} \cdot -0.5} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(-0.5 \cdot \left(l\_m \cdot l\_m\right)\right) \cdot t\_1}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (fma (/ Om (* Omc Omc)) Om -1.0)))
   (if (<= (/ t_m l_m) 4e+148)
     (asin
      (sqrt
       (/
        (fma (/ Om Omc) (/ Om Omc) -1.0)
        (fma (/ t_m l_m) (* t_m (/ -2.0 l_m)) -1.0))))
     (if (<= (/ t_m l_m) 1e+217)
       (asin (* (sqrt (* (/ t_1 (* t_m t_m)) -0.5)) l_m))
       (asin (/ (sqrt (* (* -0.5 (* l_m l_m)) t_1)) t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = fma((Om / (Omc * Omc)), Om, -1.0);
	double tmp;
	if ((t_m / l_m) <= 4e+148) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / fma((t_m / l_m), (t_m * (-2.0 / l_m)), -1.0))));
	} else if ((t_m / l_m) <= 1e+217) {
		tmp = asin((sqrt(((t_1 / (t_m * t_m)) * -0.5)) * l_m));
	} else {
		tmp = asin((sqrt(((-0.5 * (l_m * l_m)) * t_1)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 4e+148)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / fma(Float64(t_m / l_m), Float64(t_m * Float64(-2.0 / l_m)), -1.0))));
	elseif (Float64(t_m / l_m) <= 1e+217)
		tmp = asin(Float64(sqrt(Float64(Float64(t_1 / Float64(t_m * t_m)) * -0.5)) * l_m));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(-0.5 * Float64(l_m * l_m)) * t_1)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 4e+148], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m * N[(-2.0 / l$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+217], N[ArcSin[N[(N[Sqrt[N[(N[(t$95$1 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(-0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < 4.0000000000000002e148

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   if 4.0000000000000002e148 < (/.f64 t l) < 9.9999999999999996e216

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 33.0%

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

   if 9.9999999999999996e216 < (/.f64 t l)

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 32.8%

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

Alternative 2: 90.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{+15}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{-2}{l\_m}, t\_m \cdot \frac{t\_m}{l\_m}, -1\right)}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, t\_m \cdot \frac{-2}{l\_m}, -1\right)}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 10^{+217}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{t\_m \cdot t\_m} \cdot -0.5} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(-0.5 \cdot \left(l\_m \cdot l\_m\right)\right) \cdot t\_1}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (fma (/ Om (* Omc Omc)) Om -1.0)))
   (if (<= (/ t_m l_m) 1e+15)
     (asin
      (sqrt
       (/
        (fma (/ Om Omc) (/ Om Omc) -1.0)
        (fma (/ -2.0 l_m) (* t_m (/ t_m l_m)) -1.0))))
     (if (<= (/ t_m l_m) 4e+148)
       (asin (sqrt (/ t_1 (fma (/ t_m l_m) (* t_m (/ -2.0 l_m)) -1.0))))
       (if (<= (/ t_m l_m) 1e+217)
         (asin (* (sqrt (* (/ t_1 (* t_m t_m)) -0.5)) l_m))
         (asin (/ (sqrt (* (* -0.5 (* l_m l_m)) t_1)) t_m)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = fma((Om / (Omc * Omc)), Om, -1.0);
	double tmp;
	if ((t_m / l_m) <= 1e+15) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / fma((-2.0 / l_m), (t_m * (t_m / l_m)), -1.0))));
	} else if ((t_m / l_m) <= 4e+148) {
		tmp = asin(sqrt((t_1 / fma((t_m / l_m), (t_m * (-2.0 / l_m)), -1.0))));
	} else if ((t_m / l_m) <= 1e+217) {
		tmp = asin((sqrt(((t_1 / (t_m * t_m)) * -0.5)) * l_m));
	} else {
		tmp = asin((sqrt(((-0.5 * (l_m * l_m)) * t_1)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e+15)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / fma(Float64(-2.0 / l_m), Float64(t_m * Float64(t_m / l_m)), -1.0))));
	elseif (Float64(t_m / l_m) <= 4e+148)
		tmp = asin(sqrt(Float64(t_1 / fma(Float64(t_m / l_m), Float64(t_m * Float64(-2.0 / l_m)), -1.0))));
	elseif (Float64(t_m / l_m) <= 1e+217)
		tmp = asin(Float64(sqrt(Float64(Float64(t_1 / Float64(t_m * t_m)) * -0.5)) * l_m));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(-0.5 * Float64(l_m * l_m)) * t_1)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+15], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(-2.0 / l$95$m), $MachinePrecision] * N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 4e+148], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m * N[(-2.0 / l$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+217], N[ArcSin[N[(N[Sqrt[N[(N[(t$95$1 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(-0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 t l) < 1e15

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-/.f64 N/A

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

      11. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if 1e15 < (/.f64 t l) < 4.0000000000000002e148

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. pow2 N/A

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

      7. unpow2 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 79.1

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

   9. Applied rewrites 79.1%

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

   if 4.0000000000000002e148 < (/.f64 t l) < 9.9999999999999996e216

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 33.0%

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

   if 9.9999999999999996e216 < (/.f64 t l)

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 32.8%

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

Alternative 3: 89.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{-50}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 4 \cdot 10^{+148}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, t\_m \cdot \frac{-2}{l\_m}, -1\right)}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 10^{+217}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{t\_m \cdot t\_m} \cdot -0.5} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(-0.5 \cdot \left(l\_m \cdot l\_m\right)\right) \cdot t\_1}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (fma (/ Om (* Omc Omc)) Om -1.0)))
   (if (<= (/ t_m l_m) 2e-50)
     (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))
     (if (<= (/ t_m l_m) 4e+148)
       (asin (sqrt (/ t_1 (fma (/ t_m l_m) (* t_m (/ -2.0 l_m)) -1.0))))
       (if (<= (/ t_m l_m) 1e+217)
         (asin (* (sqrt (* (/ t_1 (* t_m t_m)) -0.5)) l_m))
         (asin (/ (sqrt (* (* -0.5 (* l_m l_m)) t_1)) t_m)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = fma((Om / (Omc * Omc)), Om, -1.0);
	double tmp;
	if ((t_m / l_m) <= 2e-50) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	} else if ((t_m / l_m) <= 4e+148) {
		tmp = asin(sqrt((t_1 / fma((t_m / l_m), (t_m * (-2.0 / l_m)), -1.0))));
	} else if ((t_m / l_m) <= 1e+217) {
		tmp = asin((sqrt(((t_1 / (t_m * t_m)) * -0.5)) * l_m));
	} else {
		tmp = asin((sqrt(((-0.5 * (l_m * l_m)) * t_1)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e-50)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	elseif (Float64(t_m / l_m) <= 4e+148)
		tmp = asin(sqrt(Float64(t_1 / fma(Float64(t_m / l_m), Float64(t_m * Float64(-2.0 / l_m)), -1.0))));
	elseif (Float64(t_m / l_m) <= 1e+217)
		tmp = asin(Float64(sqrt(Float64(Float64(t_1 / Float64(t_m * t_m)) * -0.5)) * l_m));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(-0.5 * Float64(l_m * l_m)) * t_1)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e-50], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 4e+148], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m * N[(-2.0 / l$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+217], N[ArcSin[N[(N[Sqrt[N[(N[(t$95$1 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(-0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 t l) < 2.00000000000000002e-50

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 51.3%

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

   if 2.00000000000000002e-50 < (/.f64 t l) < 4.0000000000000002e148

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. frac-times N/A

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

      5. unpow2 N/A

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

      6. pow2 N/A

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

      7. unpow2 N/A

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

      8. associate-*l/ N/A

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

      9. pow2 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 79.1

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

   9. Applied rewrites 79.1%

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

   if 4.0000000000000002e148 < (/.f64 t l) < 9.9999999999999996e216

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 33.0%

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

   if 9.9999999999999996e216 < (/.f64 t l)

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 32.8%

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

Alternative 4: 87.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \frac{Om}{Omc \cdot Omc}\\ t_2 := \mathsf{fma}\left(t\_1, Om, -1\right)\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 10^{-120}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 10^{+133}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(Om, t\_1, -1\right)}{\mathsf{fma}\left(t\_m, \frac{-2 \cdot \frac{t\_m}{l\_m}}{l\_m}, -1\right)}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 10^{+217}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{t\_m \cdot t\_m} \cdot -0.5} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(-0.5 \cdot \left(l\_m \cdot l\_m\right)\right) \cdot t\_2}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (/ Om (* Omc Omc))) (t_2 (fma t_1 Om -1.0)))
   (if (<= (/ t_m l_m) 1e-120)
     (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))
     (if (<= (/ t_m l_m) 1e+133)
       (asin
        (sqrt
         (/ (fma Om t_1 -1.0) (fma t_m (/ (* -2.0 (/ t_m l_m)) l_m) -1.0))))
       (if (<= (/ t_m l_m) 1e+217)
         (asin (* (sqrt (* (/ t_2 (* t_m t_m)) -0.5)) l_m))
         (asin (/ (sqrt (* (* -0.5 (* l_m l_m)) t_2)) t_m)))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = Om / (Omc * Omc);
	double t_2 = fma(t_1, Om, -1.0);
	double tmp;
	if ((t_m / l_m) <= 1e-120) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	} else if ((t_m / l_m) <= 1e+133) {
		tmp = asin(sqrt((fma(Om, t_1, -1.0) / fma(t_m, ((-2.0 * (t_m / l_m)) / l_m), -1.0))));
	} else if ((t_m / l_m) <= 1e+217) {
		tmp = asin((sqrt(((t_2 / (t_m * t_m)) * -0.5)) * l_m));
	} else {
		tmp = asin((sqrt(((-0.5 * (l_m * l_m)) * t_2)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(Om / Float64(Omc * Omc))
	t_2 = fma(t_1, Om, -1.0)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1e-120)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	elseif (Float64(t_m / l_m) <= 1e+133)
		tmp = asin(sqrt(Float64(fma(Om, t_1, -1.0) / fma(t_m, Float64(Float64(-2.0 * Float64(t_m / l_m)) / l_m), -1.0))));
	elseif (Float64(t_m / l_m) <= 1e+217)
		tmp = asin(Float64(sqrt(Float64(Float64(t_2 / Float64(t_m * t_m)) * -0.5)) * l_m));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(-0.5 * Float64(l_m * l_m)) * t_2)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * Om + -1.0), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e-120], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+133], N[ArcSin[N[Sqrt[N[(N[(Om * t$95$1 + -1.0), $MachinePrecision] / N[(t$95$m * N[(N[(-2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+217], N[ArcSin[N[(N[Sqrt[N[(N[(t$95$2 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(-0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 t l) < 9.99999999999999979e-121

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 51.3%

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

   if 9.99999999999999979e-121 < (/.f64 t l) < 1e133

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-fma.f64 N/A

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

      2. pow2 N/A

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

      3. lift-pow.f64 N/A

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

      4. *-lft-identity N/A

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

      5. *-lft-identity N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. frac-times N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. mult-flip N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 76.8

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

      11. lower-fma.f64 N/A

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

      12. metadata-eval N/A

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

      13. distribute-lft-neg-out N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

   7. Applied rewrites 76.8%

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

   if 1e133 < (/.f64 t l) < 9.9999999999999996e216

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 33.0%

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

   if 9.9999999999999996e216 < (/.f64 t l)

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 32.8%

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

Alternative 5: 81.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.5:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq 10^{+217}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{t\_m \cdot t\_m} \cdot -0.5} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(-0.5 \cdot \left(l\_m \cdot l\_m\right)\right) \cdot t\_1}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (fma (/ Om (* Omc Omc)) Om -1.0)))
   (if (<= (/ t_m l_m) 0.5)
     (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))
     (if (<= (/ t_m l_m) 1e+217)
       (asin (* (sqrt (* (/ t_1 (* t_m t_m)) -0.5)) l_m))
       (asin (/ (sqrt (* (* -0.5 (* l_m l_m)) t_1)) t_m))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = fma((Om / (Omc * Omc)), Om, -1.0);
	double tmp;
	if ((t_m / l_m) <= 0.5) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	} else if ((t_m / l_m) <= 1e+217) {
		tmp = asin((sqrt(((t_1 / (t_m * t_m)) * -0.5)) * l_m));
	} else {
		tmp = asin((sqrt(((-0.5 * (l_m * l_m)) * t_1)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.5)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	elseif (Float64(t_m / l_m) <= 1e+217)
		tmp = asin(Float64(sqrt(Float64(Float64(t_1 / Float64(t_m * t_m)) * -0.5)) * l_m));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(-0.5 * Float64(l_m * l_m)) * t_1)) / t_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.5], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1e+217], N[ArcSin[N[(N[Sqrt[N[(N[(t$95$1 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(-0.5 * N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < 0.5

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. div-flip N/A

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

      7. lift-/.f64 N/A

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

      8. mult-flip N/A

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

      9. lift-/.f64 83.9

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. lift-/.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 51.3%

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

   if 0.5 < (/.f64 t l) < 9.9999999999999996e216

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

      12. +-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      15. distribute-lft-neg-out N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

   3. Applied rewrites 66.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

      6. div-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      8. mult-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      9. lift-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

      13. mult-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

      15. div-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      17. associate-*l* N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

      19. lower-*.f64 83.9

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   5. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      6. lower-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \sqrt{\frac{-1}{2} \cdot \frac{\frac{{Om}^{2}}{{Omc}^{2}} - 1}{{t}^{2}}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot \frac{\frac{{Om}^{2}}{{Omc}^{2}} - 1}{{t}^{2}}} \cdot \color{blue}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot \frac{\frac{{Om}^{2}}{{Omc}^{2}} - 1}{{t}^{2}}} \cdot \color{blue}{\ell}\right) \]

   10. Applied rewrites 33.0%

       \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}{t \cdot t} \cdot -0.5} \cdot \ell\right)} \]

   if 9.9999999999999996e216 < (/.f64 t l)

   1. Initial program 83.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right) \]

      2. frac-2neg N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      5. sub-negate-rev N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} - 1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      6. sub-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

      12. +-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      15. distribute-lft-neg-out N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

   3. Applied rewrites 66.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

      6. div-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      8. mult-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      9. lift-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

      13. mult-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

      15. div-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      17. associate-*l* N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

      19. lower-*.f64 83.9

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   5. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      6. lower-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   8. Taylor expanded in t around inf

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{\frac{-1}{2} \cdot \left({\ell}^{2} \cdot \left(\frac{{Om}^{2}}{{Omc}^{2}} - 1\right)\right)}}{t}\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{\frac{-1}{2} \cdot \left({\ell}^{2} \cdot \left(\frac{{Om}^{2}}{{Omc}^{2}} - 1\right)\right)}}{\color{blue}{t}}\right) \]

   10. Applied rewrites 32.8%

       \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{\left(-0.5 \cdot \left(\ell \cdot \ell\right)\right) \cdot \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}}{t}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}{t\_m \cdot t\_m} \cdot -0.5} \cdot l\_m\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))) 2.0)
   (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))
   (asin
    (* (sqrt (* (/ (fma (/ Om (* Omc Omc)) Om -1.0) (* t_m t_m)) -0.5)) l_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((1.0 + (2.0 * pow((t_m / l_m), 2.0))) <= 2.0) {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	} else {
		tmp = asin((sqrt(((fma((Om / (Omc * Omc)), Om, -1.0) / (t_m * t_m)) * -0.5)) * l_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))) <= 2.0)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0) / Float64(t_m * t_m)) * -0.5)) * l_m));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], 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[(N[Sqrt[N[(N[(N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 2

   1. Initial program 83.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right) \]

      2. frac-2neg N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      5. sub-negate-rev N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} - 1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      6. sub-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

      12. +-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      15. distribute-lft-neg-out N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

   3. Applied rewrites 66.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

      6. div-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      8. mult-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      9. lift-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

      13. mult-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

      15. div-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      17. associate-*l* N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

      19. lower-*.f64 83.9

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   5. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      6. lower-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   8. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-1}}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.3%

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-1}}}\right) \]

   if 2 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 83.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right) \]

      2. frac-2neg N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      5. sub-negate-rev N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} - 1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      6. sub-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

      12. +-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      15. distribute-lft-neg-out N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

   3. Applied rewrites 66.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

      6. div-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      8. mult-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      9. lift-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

      13. mult-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

      15. div-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      17. associate-*l* N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

      19. lower-*.f64 83.9

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   5. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      6. lower-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   8. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \sqrt{\frac{-1}{2} \cdot \frac{\frac{{Om}^{2}}{{Omc}^{2}} - 1}{{t}^{2}}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot \frac{\frac{{Om}^{2}}{{Omc}^{2}} - 1}{{t}^{2}}} \cdot \color{blue}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{2} \cdot \frac{\frac{{Om}^{2}}{{Omc}^{2}} - 1}{{t}^{2}}} \cdot \color{blue}{\ell}\right) \]

   10. Applied rewrites 33.0%

       \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}{t \cdot t} \cdot -0.5} \cdot \ell\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-27}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t\_m \cdot t\_m}{l\_m \cdot l\_m}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      1e-27)
   (asin (sqrt (/ -1.0 (fma (/ (* t_m t_m) (* l_m l_m)) -2.0 -1.0))))
   (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0)))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 1e-27) {
		tmp = asin(sqrt((-1.0 / fma(((t_m * t_m) / (l_m * l_m)), -2.0, -1.0))));
	} else {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 1e-27)
		tmp = asin(sqrt(Float64(-1.0 / fma(Float64(Float64(t_m * t_m) / Float64(l_m * l_m)), -2.0, -1.0))));
	else
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[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$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-27], N[ArcSin[N[Sqrt[N[(-1.0 / N[(N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision]), $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 (asin.f64 (sqrt.f64 (/.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))))))) < 1e-27

   1. Initial program 83.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right) \]

      2. frac-2neg N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      5. sub-negate-rev N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} - 1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      6. sub-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

      12. +-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      15. distribute-lft-neg-out N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

   3. Applied rewrites 66.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

      6. div-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      8. mult-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      9. lift-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

      13. mult-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

      15. div-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      17. associate-*l* N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

      19. lower-*.f64 83.9

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   5. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      6. lower-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   8. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{-1}{-2 \cdot \frac{{t}^{2}}{{\ell}^{2}} - 1}}}\right) \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\color{blue}{-2 \cdot \frac{{t}^{2}}{{\ell}^{2}} - 1}}}\right) \]

      2. sub-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{-2 \cdot \frac{{t}^{2}}{{\ell}^{2}} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

      3. *-commutative N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\frac{{t}^{2}}{{\ell}^{2}} \cdot -2 + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)}}\right) \]

      4. metadata-eval N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\frac{{t}^{2}}{{\ell}^{2}} \cdot -2 + -1}}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{{t}^{2}}{{\ell}^{2}}, \color{blue}{-2}, -1\right)}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{{t}^{2}}{{\ell}^{2}}, -2, -1\right)}}\right) \]

      7. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t \cdot t}{{\ell}^{2}}, -2, -1\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t \cdot t}{{\ell}^{2}}, -2, -1\right)}}\right) \]

      9. pow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -2, -1\right)}}\right) \]

      10. lift-*.f64 66.4

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{-1}{\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -2, -1\right)}}\right) \]

   10. Applied rewrites 66.4%

       \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{-1}{\mathsf{fma}\left(\frac{t \cdot t}{\ell \cdot \ell}, -2, -1\right)}}}\right) \]

   if 1e-27 < (asin.f64 (sqrt.f64 (/.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 83.9%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right) \]

      2. frac-2neg N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

      4. lift--.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      5. sub-negate-rev N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} - 1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      6. sub-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      7. lift-pow.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

      11. lift-+.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

      12. +-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

      13. distribute-neg-in N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      15. distribute-lft-neg-out N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

      16. metadata-eval N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

      17. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

   3. Applied rewrites 66.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

      4. times-frac N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

      6. div-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      8. mult-flip N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      9. lift-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

      11. *-commutative N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

      12. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

      13. mult-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

      15. div-flip N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

      17. associate-*l* N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

      19. lower-*.f64 83.9

          \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   5. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

      4. associate-/l* N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

      6. lower-/.f64 83.9

         \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   8. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-1}}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 51.3%

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-1}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 51.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{-1}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (sqrt (/ (fma (/ Om Omc) (/ Om Omc) -1.0) -1.0))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / -1.0)));
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / -1.0)))
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.9%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right) \]

   2. frac-2neg N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   5. sub-negate-rev N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} - 1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   6. sub-flip N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   7. lift-pow.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   8. unpow2 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   11. lift-+.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

   12. +-commutative N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

   15. distribute-lft-neg-out N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

   16. metadata-eval N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

   17. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

3. Applied rewrites 66.9%

   \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

   4. times-frac N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

   6. div-flip N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   8. mult-flip N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   9. lift-/.f64 83.9

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

   11. *-commutative N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

   13. mult-flip N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

   14. lift-/.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

   15. div-flip N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

   16. lift-/.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

   17. associate-*l* N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

   18. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

   19. lower-*.f64 83.9

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

5. Applied rewrites 83.9%

   \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

   3. associate-*l/ N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

   4. associate-/l* N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   6. lower-/.f64 83.9

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

7. Applied rewrites 83.9%

   \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

8. Taylor expanded in t around 0

   \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-1}}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 51.3%

    \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-1}}}\right) \]
11. Add Preprocessing

Alternative 9: 45.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (sqrt (- 1.0 (/ (* Om Om) (* Omc Omc))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin(sqrt((1.0 - ((Om * Om) / (Omc * Omc)))));
}

Fortran

t_m =     private
l_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(t_m, l_m, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(sqrt((1.0d0 - ((om * om) / (omc * omc)))))
end function

Java

t_m = Math.abs(t);
l_m = Math.abs(l);
public static double code(double t_m, double l_m, double Om, double Omc) {
	return Math.asin(Math.sqrt((1.0 - ((Om * Om) / (Omc * Omc)))));
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	return math.asin(math.sqrt((1.0 - ((Om * Om) / (Omc * Omc)))))

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(Float64(1.0 - Float64(Float64(Om * Om) / Float64(Omc * Omc)))))
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(sqrt((1.0 - ((Om * Om) / (Omc * Omc)))));
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.9%

   \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}}\right) \]

   2. frac-2neg N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{neg}\left(\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}}\right) \]

   4. lift--.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{neg}\left(\color{blue}{\left(1 - {\left(\frac{Om}{Omc}\right)}^{2}\right)}\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   5. sub-negate-rev N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} - 1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   6. sub-flip N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2} + \left(\mathsf{neg}\left(1\right)\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   7. lift-pow.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{{\left(\frac{Om}{Omc}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   8. unpow2 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\frac{Om}{Omc} \cdot \frac{Om}{Omc}} + \left(\mathsf{neg}\left(1\right)\right)}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   9. metadata-eval N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc} + \color{blue}{-1}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\color{blue}{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}}{\mathsf{neg}\left(\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right)}}\right) \]

   11. lift-+.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}\right)}}\right) \]

   12. +-commutative N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{neg}\left(\color{blue}{\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2} + 1\right)}\right)}}\right) \]

   13. distribute-neg-in N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(\color{blue}{2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

   15. distribute-lft-neg-out N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}} + \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

   16. metadata-eval N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\mathsf{neg}\left(2\right)\right) \cdot {\left(\frac{t}{\ell}\right)}^{2} + \color{blue}{-1}}}\right) \]

   17. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(2\right), {\left(\frac{t}{\ell}\right)}^{2}, -1\right)}}}\right) \]

3. Applied rewrites 66.9%

   \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\ell \cdot \ell}, -1\right)}}}\right) \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t \cdot t}{\ell \cdot \ell}}, -1\right)}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\color{blue}{t \cdot t}}{\ell \cdot \ell}, -1\right)}}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t \cdot t}{\color{blue}{\ell \cdot \ell}}, -1\right)}}\right) \]

   4. times-frac N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell} \cdot \frac{t}{\ell}}, -1\right)}}\right) \]

   5. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{t}{\ell}} \cdot \frac{t}{\ell}, -1\right)}}\right) \]

   6. div-flip N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   7. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   8. mult-flip N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   9. lift-/.f64 83.9

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}}, -1\right)}}\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{-2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}} + -1}}}\right) \]

   11. *-commutative N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}} \cdot -2} + -1}}\right) \]

   12. lift-/.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{\frac{t}{\ell}}{\frac{\ell}{t}}} \cdot -2 + -1}}\right) \]

   13. mult-flip N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot -2 + -1}}\right) \]

   14. lift-/.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \frac{1}{\color{blue}{\frac{\ell}{t}}}\right) \cdot -2 + -1}}\right) \]

   15. div-flip N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

   16. lift-/.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\left(\frac{t}{\ell} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot -2 + -1}}\right) \]

   17. associate-*l* N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\frac{t}{\ell} \cdot \left(\frac{t}{\ell} \cdot -2\right)} + -1}}\right) \]

   18. lower-fma.f64 N/A

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]

   19. lower-*.f64 83.9

       \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

5. Applied rewrites 83.9%

   \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot -2, -1\right)}}}\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell} \cdot -2}, -1\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t}{\ell}} \cdot -2, -1\right)}}\right) \]

   3. associate-*l/ N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{\frac{t \cdot -2}{\ell}}, -1\right)}}\right) \]

   4. associate-/l* N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

   6. lower-/.f64 83.9

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, t \cdot \color{blue}{\frac{-2}{\ell}}, -1\right)}}\right) \]

7. Applied rewrites 83.9%

   \[\leadsto \sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(\frac{t}{\ell}, \color{blue}{t \cdot \frac{-2}{\ell}}, -1\right)}}\right) \]

8. Taylor expanded in t around 0

   \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{-1 \cdot \left(\frac{{Om}^{2}}{{Omc}^{2}} - 1\right)}}\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{neg}\left(\left(\frac{{Om}^{2}}{{Omc}^{2}} - 1\right)\right)}\right) \]

   2. sub-negate-rev N/A

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{{Om}^{2}}{{Omc}^{2}}}}\right) \]

   3. lower--.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \color{blue}{\frac{{Om}^{2}}{{Omc}^{2}}}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{{Om}^{2}}{\color{blue}{{Omc}^{2}}}}\right) \]

   5. unpow2 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{{\color{blue}{Omc}}^{2}}}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{{\color{blue}{Omc}}^{2}}}\right) \]

   7. pow2 N/A

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot \color{blue}{Omc}}}\right) \]

   8. lift-*.f64 45.2

      \[\leadsto \sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot \color{blue}{Omc}}}\right) \]

10. Applied rewrites 45.2%

    \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))))))