Toniolo and Linder, Equation (2)

Percentage Accurate: 84.3% → 95.8%

Time: 8.4s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 95.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t\_m + t\_m}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t\_m} \cdot \sqrt{\left(\left(l\_m \cdot l\_m\right) \cdot \left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+152)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ (+ t_m t_m) l_m) (/ t_m l_m) 1.0))))
   (asin
    (*
     (/ 1.0 t_m)
     (sqrt (* (* (* l_m l_m) (- 1.0 (* (/ Om Omc) (/ Om Omc)))) 0.5))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+152) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma(((t_m + t_m) / l_m), (t_m / l_m), 1.0))));
	} else {
		tmp = asin(((1.0 / t_m) * sqrt((((l_m * l_m) * (1.0 - ((Om / Omc) * (Om / Omc)))) * 0.5))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 2e+152)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(Float64(t_m + t_m) / l_m), Float64(t_m / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(1.0 / t_m) * sqrt(Float64(Float64(Float64(l_m * l_m) * Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * 0.5))));
	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[(t$95$m / l$95$m), $MachinePrecision], 2e+152], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e152

   1. Initial program 84.3%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 84.3

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

   3. Applied rewrites 84.3%

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

   if 2.0000000000000001e152 < (/.f64 t l)

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 31.2

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

   4. Applied rewrites 31.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5 \cdot \left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)}}{t}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 34.9

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

   8. Applied rewrites 34.9%

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

Alternative 2: 93.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{t\_m + t\_m}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{elif}\;\frac{t\_m}{l\_m} \leq \infty:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5 \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 \cdot {l\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 2e+152)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma (/ (+ t_m t_m) l_m) (/ t_m l_m) 1.0))))
   (if (<= (/ t_m l_m) INFINITY)
     (asin
      (/ (* l_m (sqrt (* 0.5 (- 1.0 (/ (pow Om 2.0) (pow Omc 2.0)))))) t_m))
     (asin (/ (sqrt (* 0.5 (pow l_m 2.0))) t_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 ((t_m / l_m) <= 2e+152) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma(((t_m + t_m) / l_m), (t_m / l_m), 1.0))));
	} else if ((t_m / l_m) <= ((double) INFINITY)) {
		tmp = asin(((l_m * sqrt((0.5 * (1.0 - (pow(Om, 2.0) / pow(Omc, 2.0)))))) / t_m));
	} else {
		tmp = asin((sqrt((0.5 * pow(l_m, 2.0))) / t_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(t_m / l_m) <= 2e+152)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(Float64(t_m + t_m) / l_m), Float64(t_m / l_m), 1.0))));
	elseif (Float64(t_m / l_m) <= Inf)
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(0.5 * Float64(1.0 - Float64((Om ^ 2.0) / (Omc ^ 2.0)))))) / t_m));
	else
		tmp = asin(Float64(sqrt(Float64(0.5 * (l_m ^ 2.0))) / 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_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+152], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], Infinity], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(0.5 * N[(1.0 - N[(N[Power[Om, 2.0], $MachinePrecision] / N[Power[Omc, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(0.5 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < 2.0000000000000001e152

   1. Initial program 84.3%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 84.3

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

   3. Applied rewrites 84.3%

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

   if 2.0000000000000001e152 < (/.f64 t l) < +inf.0

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 31.2

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 43.2

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

   7. Applied rewrites 43.2%

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

   if +inf.0 < (/.f64 t l)

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 31.2

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in Om around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 34.8

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{0.5 \cdot {\ell}^{2}}}{t}\right) \]

   7. Applied rewrites 34.8%

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

Alternative 3: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\frac{Omc}{Omc}}{\mathsf{fma}\left(\frac{t\_m + t\_m}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{t\_m} \cdot \sqrt{\left(\left(l\_m \cdot l\_m\right) \cdot \left(1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

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))) 5e+304)
   (asin (sqrt (/ (/ Omc Omc) (fma (/ (+ t_m t_m) l_m) (/ t_m l_m) 1.0))))
   (asin
    (*
     (/ 1.0 t_m)
     (sqrt (* (* (* l_m l_m) (- 1.0 (* (/ Om Omc) (/ Om Omc)))) 0.5))))))

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))) <= 5e+304) {
		tmp = asin(sqrt(((Omc / Omc) / fma(((t_m + t_m) / l_m), (t_m / l_m), 1.0))));
	} else {
		tmp = asin(((1.0 / t_m) * sqrt((((l_m * l_m) * (1.0 - ((Om / Omc) * (Om / Omc)))) * 0.5))));
	}
	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))) <= 5e+304)
		tmp = asin(sqrt(Float64(Float64(Omc / Omc) / fma(Float64(Float64(t_m + t_m) / l_m), Float64(t_m / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(1.0 / t_m) * sqrt(Float64(Float64(Float64(l_m * l_m) * Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * 0.5))));
	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], 5e+304], N[ArcSin[N[Sqrt[N[(N[(Omc / Omc), $MachinePrecision] / N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(1.0 / t$95$m), $MachinePrecision] * N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $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)))) < 4.9999999999999997e304

   1. Initial program 84.3%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 84.3

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

   3. Applied rewrites 84.3%

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

      1. lift--.f64 N/A

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

      2. *-inverses N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-div N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-lft-identity N/A

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

      20. lift-*.f64 N/A

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

      21. sub-to-fraction-rev N/A

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

      22. lower--.f64 N/A

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

      23. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 83.5%

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

   if 4.9999999999999997e304 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 31.2

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

   4. Applied rewrites 31.2%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5 \cdot \left({\ell}^{2} \cdot \left(1 - \frac{{Om}^{2}}{{Omc}^{2}}\right)\right)}}{t}\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 31.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower-*.f64 N/A

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

   6. Applied rewrites 33.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 34.9

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

   8. Applied rewrites 34.9%

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

Alternative 4: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 5 \cdot 10^{+304}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\frac{Omc}{Omc}}{\mathsf{fma}\left(\frac{t\_m + t\_m}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 \cdot {l\_m}^{2}}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))) 5e+304)
   (asin (sqrt (/ (/ Omc Omc) (fma (/ (+ t_m t_m) l_m) (/ t_m l_m) 1.0))))
   (asin (/ (sqrt (* 0.5 (pow l_m 2.0))) t_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))) <= 5e+304) {
		tmp = asin(sqrt(((Omc / Omc) / fma(((t_m + t_m) / l_m), (t_m / l_m), 1.0))));
	} else {
		tmp = asin((sqrt((0.5 * pow(l_m, 2.0))) / t_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))) <= 5e+304)
		tmp = asin(sqrt(Float64(Float64(Omc / Omc) / fma(Float64(Float64(t_m + t_m) / l_m), Float64(t_m / l_m), 1.0))));
	else
		tmp = asin(Float64(sqrt(Float64(0.5 * (l_m ^ 2.0))) / 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_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+304], N[ArcSin[N[Sqrt[N[(N[(Omc / Omc), $MachinePrecision] / N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(0.5 * N[Power[l$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$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)))) < 4.9999999999999997e304

   1. Initial program 84.3%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 84.3

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

   3. Applied rewrites 84.3%

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

      1. lift--.f64 N/A

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

      2. *-inverses N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. sub-div N/A

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

      11. *-lft-identity N/A

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/r* N/A

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

      16. lower-/.f64 N/A

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

      17. lift--.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-lft-identity N/A

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

      20. lift-*.f64 N/A

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

      21. sub-to-fraction-rev N/A

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

      22. lower--.f64 N/A

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

      23. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 83.5%

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

   if 4.9999999999999997e304 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 31.2

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in Om around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 34.8

         \[\leadsto \sin^{-1} \left(\frac{\sqrt{0.5 \cdot {\ell}^{2}}}{t}\right) \]

   7. Applied rewrites 34.8%

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

Alternative 5: 83.5% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(Float64(Float64(Omc / Omc) / fma(Float64(Float64(t_m + t_m) / l_m), Float64(t_m / l_m), 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[(Omc / Omc), $MachinePrecision] / N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.3%

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

   1. remove-double-neg N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. remove-double-neg N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. distribute-rgt-out N/A

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

   15. *-commutative N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. div-add-rev N/A

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

   21. lower-/.f64 N/A

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

   22. lower-+.f64 84.3

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

3. Applied rewrites 84.3%

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

   1. lift--.f64 N/A

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

   2. *-inverses N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sub-div N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-lft-identity N/A

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

   20. lift-*.f64 N/A

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

   21. sub-to-fraction-rev N/A

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

   22. lower--.f64 N/A

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

   23. lower-/.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 83.5%

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

Alternative 6: 72.2% accurate, 2.0× speedup

Math

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

FPCore

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

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(Float64(Float64(Omc / Omc) / fma(Float64(t_m + t_m), Float64(t_m / Float64(l_m * l_m)), 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[(Omc / Omc), $MachinePrecision] / N[(N[(t$95$m + t$95$m), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.3%

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

   1. remove-double-neg N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. remove-double-neg N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. distribute-rgt-out N/A

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

   15. *-commutative N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. div-add-rev N/A

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

   21. lower-/.f64 N/A

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

   22. lower-+.f64 84.3

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

3. Applied rewrites 84.3%

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

   1. lift--.f64 N/A

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

   2. *-inverses N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sub-div N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-lft-identity N/A

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

   20. lift-*.f64 N/A

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

   21. sub-to-fraction-rev N/A

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

   22. lower--.f64 N/A

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

   23. lower-/.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 83.5%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. lift-/.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. count-2 N/A

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

   7. associate-/l* N/A

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

   8. lift-/.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. frac-times N/A

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

   13. lift-*.f64 N/A

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

   14. associate-*r/ N/A

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

   15. lift-/.f64 N/A

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

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

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lift-/.f64 N/A

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

   20. associate-*r/ N/A

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

   21. lift-*.f64 N/A

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

10. Applied rewrites 72.2%

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

Alternative 7: 67.0% accurate, 2.1× speedup

Math

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

FPCore

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

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(Float64(Omc / Float64(fma(Float64(t_m + t_m), Float64(t_m / Float64(l_m * l_m)), 1.0) * 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[(Omc / N[(N[(N[(t$95$m + t$95$m), $MachinePrecision] * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.3%

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

   1. remove-double-neg N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. remove-double-neg N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. distribute-rgt-out N/A

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

   15. *-commutative N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. div-add-rev N/A

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

   21. lower-/.f64 N/A

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

   22. lower-+.f64 84.3

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

3. Applied rewrites 84.3%

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

   1. lift--.f64 N/A

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

   2. *-inverses N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sub-div N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-lft-identity N/A

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

   20. lift-*.f64 N/A

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

   21. sub-to-fraction-rev N/A

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

   22. lower--.f64 N/A

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

   23. lower-/.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 83.5%

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

10. Applied rewrites 67.0%

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

Alternative 8: 51.6% accurate, 2.5× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (sqrt (/ (- Omc (* (/ Om Omc) Om)) (* 1.0 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(((Omc - ((Om / Omc) * Om)) / (1.0 * 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(((omc - ((om / omc) * om)) / (1.0d0 * 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(((Omc - ((Om / Omc) * Om)) / (1.0 * 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(((Omc - ((Om / Omc) * Om)) / (1.0 * Omc))))

Julia

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

MATLAB

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

Derivation

1. Initial program 84.3%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 51.6%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 51.6

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

6. Applied rewrites 51.6%

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

8. Applied rewrites 51.6%

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

Alternative 9: 51.0% accurate, 4.0× speedup

Math

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

FPCore

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

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(((omc / omc) / 1.0d0)))
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(((Omc / Omc) / 1.0)));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 84.3%

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

   1. remove-double-neg N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-neg-in N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. remove-double-neg N/A

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

   8. lift-*.f64 N/A

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

   9. count-2-rev N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. distribute-rgt-out N/A

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

   15. *-commutative N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. lift-/.f64 N/A

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

   20. div-add-rev N/A

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

   21. lower-/.f64 N/A

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

   22. lower-+.f64 84.3

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

3. Applied rewrites 84.3%

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

   1. lift--.f64 N/A

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

   2. *-inverses N/A

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

   3. lift-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. frac-times N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. sub-div N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f64 N/A

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

   13. lift--.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/r* N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-lft-identity N/A

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

   20. lift-*.f64 N/A

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

   21. sub-to-fraction-rev N/A

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

   22. lower--.f64 N/A

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

   23. lower-/.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 83.5%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 51.0%

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

Reproduce

herbie shell --seed 2025156 
(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)))))))