Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.6%

Time: 10.3s

Alternatives: 12

Speedup: 1.2×

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: 84.0% 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: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 5 \cdot 10^{+87}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(t\_m, \frac{2 \cdot \frac{t\_m}{l\_m}}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))) 5e+87)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (fma t_m (/ (* 2.0 (/ t_m l_m)) l_m) 1.0))))
   (asin
    (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((1.0 + (2.0 * pow((t_m / l_m), 2.0))) <= 5e+87) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma(t_m, ((2.0 * (t_m / l_m)) / l_m), 1.0))));
	} else {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))) <= 5e+87)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(t_m, Float64(Float64(2.0 * Float64(t_m / l_m)) / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+87], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 4.9999999999999998e87

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 98.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.8

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

   6. Applied rewrites 98.8%

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

   if 4.9999999999999998e87 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 68.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 72.6

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

   5. Applied rewrites 72.6%

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

Alternative 2: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{0.5 \cdot l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{2 \cdot l\_m}{\mathsf{fma}\left(4, \frac{t\_m}{l\_m} \cdot t\_m, 2 \cdot l\_m\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      4e-16)
   (asin (sqrt (* (/ (* 0.5 l_m) t_m) (/ l_m t_m))))
   (asin (sqrt (/ (* 2.0 l_m) (fma 4.0 (* (/ t_m l_m) t_m) (* 2.0 l_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 4e-16) {
		tmp = asin(sqrt((((0.5 * l_m) / t_m) * (l_m / t_m))));
	} else {
		tmp = asin(sqrt(((2.0 * l_m) / fma(4.0, ((t_m / l_m) * t_m), (2.0 * l_m)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 4e-16)
		tmp = asin(sqrt(Float64(Float64(Float64(0.5 * l_m) / t_m) * Float64(l_m / t_m))));
	else
		tmp = asin(sqrt(Float64(Float64(2.0 * l_m) / fma(4.0, Float64(Float64(t_m / l_m) * t_m), Float64(2.0 * l_m)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 4e-16], N[ArcSin[N[Sqrt[N[(N[(N[(0.5 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(4.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] + N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 3.9999999999999999e-16

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 48.4

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

   5. Applied rewrites 48.4%

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

   7. Applied rewrites 39.6%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 56.5%

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

   12. Applied rewrites 73.3%

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

   if 3.9999999999999999e-16 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 98.7

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

   4. Applied rewrites 98.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. frac-add N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 98.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.6

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 98.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in Om around 0

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

       1. lower-*.f64 98.5

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

   11. Applied rewrites 98.5%

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

Alternative 3: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{0.5 \cdot l\_m}{t\_m} \cdot \frac{l\_m}{t\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.05)
   (asin (sqrt (* (/ (* 0.5 l_m) t_m) (/ l_m t_m))))
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.05) {
		tmp = asin(sqrt((((0.5 * l_m) / t_m) * (l_m / t_m))));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.05d0) then
        tmp = asin(sqrt((((0.5d0 * l_m) / t_m) * (l_m / t_m))))
    else
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 0.05) {
		tmp = Math.asin(Math.sqrt((((0.5 * l_m) / t_m) * (l_m / t_m))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 0.05:
		tmp = math.asin(math.sqrt((((0.5 * l_m) / t_m) * (l_m / t_m))))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.05)
		tmp = asin(sqrt(Float64(Float64(Float64(0.5 * l_m) / t_m) * Float64(l_m / t_m))));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 0.05)
		tmp = asin(sqrt((((0.5 * l_m) / t_m) * (l_m / t_m))));
	else
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.05], N[ArcSin[N[Sqrt[N[(N[(N[(0.5 * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.050000000000000003

   1. Initial program 73.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   7. Applied rewrites 38.3%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 54.8%

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

   12. Applied rewrites 73.0%

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

   if 0.050000000000000003 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.1

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

   5. Applied rewrites 97.1%

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

Alternative 4: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(\sqrt{l\_m \cdot \left(\frac{l\_m}{t\_m} \cdot \frac{0.5}{t\_m}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.05)
   (asin (sqrt (* l_m (* (/ l_m t_m) (/ 0.5 t_m)))))
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.05) {
		tmp = asin(sqrt((l_m * ((l_m / t_m) * (0.5 / t_m)))));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.05d0) then
        tmp = asin(sqrt((l_m * ((l_m / t_m) * (0.5d0 / t_m)))))
    else
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 0.05) {
		tmp = Math.asin(Math.sqrt((l_m * ((l_m / t_m) * (0.5 / t_m)))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 0.05:
		tmp = math.asin(math.sqrt((l_m * ((l_m / t_m) * (0.5 / t_m)))))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.05)
		tmp = asin(sqrt(Float64(l_m * Float64(Float64(l_m / t_m) * Float64(0.5 / t_m)))));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 0.05)
		tmp = asin(sqrt((l_m * ((l_m / t_m) * (0.5 / t_m)))));
	else
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.05], N[ArcSin[N[Sqrt[N[(l$95$m * N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[(0.5 / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.050000000000000003

   1. Initial program 73.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.1

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

   5. Applied rewrites 47.1%

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

   7. Applied rewrites 38.3%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 54.8%

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

   12. Applied rewrites 67.3%

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

   if 0.050000000000000003 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.1

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

   5. Applied rewrites 97.1%

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

Alternative 5: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{0.5}{t\_m} \cdot \frac{l\_m \cdot l\_m}{t\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.005)
   (asin (sqrt (* (/ 0.5 t_m) (/ (* l_m l_m) t_m))))
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = asin(sqrt(((0.5 / t_m) * ((l_m * l_m) / t_m))));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.005d0) then
        tmp = asin(sqrt(((0.5d0 / t_m) * ((l_m * l_m) / t_m))))
    else
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = Math.asin(Math.sqrt(((0.5 / t_m) * ((l_m * l_m) / t_m))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 0.005:
		tmp = math.asin(math.sqrt(((0.5 / t_m) * ((l_m * l_m) / t_m))))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt(Float64(Float64(0.5 / t_m) * Float64(Float64(l_m * l_m) / t_m))));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt(((0.5 / t_m) * ((l_m * l_m) / t_m))));
	else
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(N[(0.5 / t$95$m), $MachinePrecision] * N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.0050000000000000001

   1. Initial program 73.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 55.2%

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

   if 0.0050000000000000001 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{0.5}{t} \cdot \frac{\ell \cdot \ell}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{l\_m}{t\_m \cdot t\_m} \cdot 0.5\right) \cdot l\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.005)
   (asin (sqrt (* (* (/ l_m (* t_m t_m)) 0.5) l_m)))
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.005d0) then
        tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5d0) * l_m)))
    else
        tmp = asin(sqrt((1.0d0 - ((om / omc) * (om / omc)))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = Math.asin(Math.sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 0.005:
		tmp = math.asin(math.sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)))
	else:
		tmp = math.asin(math.sqrt((1.0 - ((Om / Omc) * (Om / Omc)))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt(Float64(Float64(Float64(l_m / Float64(t_m * t_m)) * 0.5) * l_m)));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)));
	else
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(N[(N[(l$95$m / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.0050000000000000001

   1. Initial program 73.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 52.9%

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

   if 0.0050000000000000001 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.4

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

   5. Applied rewrites 96.4%

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

Alternative 7: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{l\_m}{t\_m \cdot t\_m} \cdot 0.5\right) \cdot l\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.005)
   (asin (sqrt (* (* (/ l_m (* t_m t_m)) 0.5) l_m)))
   (asin (sqrt 1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)));
	} else {
		tmp = asin(sqrt(1.0));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.005d0) then
        tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5d0) * l_m)))
    else
        tmp = asin(sqrt(1.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = Math.asin(Math.sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)));
	} else {
		tmp = Math.asin(Math.sqrt(1.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 0.005:
		tmp = math.asin(math.sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)))
	else:
		tmp = math.asin(math.sqrt(1.0))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt(Float64(Float64(Float64(l_m / Float64(t_m * t_m)) * 0.5) * l_m)));
	else
		tmp = asin(sqrt(1.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt((((l_m / (t_m * t_m)) * 0.5) * l_m)));
	else
		tmp = asin(sqrt(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(N[(N[(l$95$m / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.0050000000000000001

   1. Initial program 73.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 53.9%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 52.9%

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

   if 0.0050000000000000001 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 96.3%

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

Alternative 8: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\left(l\_m \cdot l\_m\right) \cdot 0.5}{t\_m \cdot t\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.005)
   (asin (sqrt (/ (* (* l_m l_m) 0.5) (* t_m t_m))))
   (asin (sqrt 1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = asin(sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))));
	} else {
		tmp = asin(sqrt(1.0));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.005d0) then
        tmp = asin(sqrt((((l_m * l_m) * 0.5d0) / (t_m * t_m))))
    else
        tmp = asin(sqrt(1.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = Math.asin(Math.sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))));
	} else {
		tmp = Math.asin(Math.sqrt(1.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 0.005:
		tmp = math.asin(math.sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))))
	else:
		tmp = math.asin(math.sqrt(1.0))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt(Float64(Float64(Float64(l_m * l_m) * 0.5) / Float64(t_m * t_m))));
	else
		tmp = asin(sqrt(1.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt((((l_m * l_m) * 0.5) / (t_m * t_m))));
	else
		tmp = asin(sqrt(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.0050000000000000001

   1. Initial program 73.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 38.6%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 55.2%

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

   12. Applied rewrites 47.2%

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

   if 0.0050000000000000001 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 96.3%

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

Alternative 9: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(l\_m \cdot l\_m\right) \cdot \frac{0.5}{t\_m \cdot t\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.005)
   (asin (sqrt (* (* l_m l_m) (/ 0.5 (* t_m t_m)))))
   (asin (sqrt 1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = asin(sqrt(((l_m * l_m) * (0.5 / (t_m * t_m)))));
	} else {
		tmp = asin(sqrt(1.0));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.005d0) then
        tmp = asin(sqrt(((l_m * l_m) * (0.5d0 / (t_m * t_m)))))
    else
        tmp = asin(sqrt(1.0d0))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 0.005) {
		tmp = Math.asin(Math.sqrt(((l_m * l_m) * (0.5 / (t_m * t_m)))));
	} else {
		tmp = Math.asin(Math.sqrt(1.0));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 0.005:
		tmp = math.asin(math.sqrt(((l_m * l_m) * (0.5 / (t_m * t_m)))))
	else:
		tmp = math.asin(math.sqrt(1.0))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt(Float64(Float64(l_m * l_m) * Float64(0.5 / Float64(t_m * t_m)))));
	else
		tmp = asin(sqrt(1.0));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 0.005)
		tmp = asin(sqrt(((l_m * l_m) * (0.5 / (t_m * t_m)))));
	else
		tmp = asin(sqrt(1.0));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[(N[(l$95$m * l$95$m), $MachinePrecision] * N[(0.5 / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.0050000000000000001

   1. Initial program 73.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 47.4

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

   5. Applied rewrites 47.4%

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

   7. Applied rewrites 38.6%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 55.2%

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

   12. Applied rewrites 47.2%

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

   if 0.0050000000000000001 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 44.5%

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

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 96.3%

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

Alternative 10: 98.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 10000000000000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{2 \cdot l\_m}{\mathsf{fma}\left(4, \frac{t\_m}{l\_m} \cdot t\_m, 2 \cdot l\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))) 10000000000000.0)
   (asin (sqrt (/ (* 2.0 l_m) (fma 4.0 (* (/ t_m l_m) t_m) (* 2.0 l_m)))))
   (asin
    (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((1.0 + (2.0 * pow((t_m / l_m), 2.0))) <= 10000000000000.0) {
		tmp = asin(sqrt(((2.0 * l_m) / fma(4.0, ((t_m / l_m) * t_m), (2.0 * l_m)))));
	} else {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))) <= 10000000000000.0)
		tmp = asin(sqrt(Float64(Float64(2.0 * l_m) / fma(4.0, Float64(Float64(t_m / l_m) * t_m), Float64(2.0 * l_m)))));
	else
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 10000000000000.0], N[ArcSin[N[Sqrt[N[(N[(2.0 * l$95$m), $MachinePrecision] / N[(4.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision] + N[(2.0 * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 1e13

   1. Initial program 98.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. frac-add N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 98.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.6

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 98.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in Om around 0

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

       1. lower-*.f64 98.5

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

   11. Applied rewrites 98.5%

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

   if 1e13 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 72.2%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 69.7

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

   5. Applied rewrites 69.7%

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

Alternative 11: 57.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.15 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{\left(-Omc\right) \cdot Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.15e+206)
   (asin (sqrt 1.0))
   (asin (sqrt (/ (* Om Om) (* (- Omc) Omc))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.15e+206) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin(sqrt(((Om * Om) / (-Omc * Omc))));
	}
	return tmp;
}

Fortran

l_m =     private
t_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
    real(8) :: tmp
    if ((t_m / l_m) <= 1.15d+206) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin(sqrt(((om * om) / (-omc * omc))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.15e+206) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin(Math.sqrt(((Om * Om) / (-Omc * Omc))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1.15e+206:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin(math.sqrt(((Om * Om) / (-Omc * Omc))))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.15e+206)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(Float64(Float64(Om * Om) / Float64(Float64(-Omc) * Omc))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1.15e+206)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(sqrt(((Om * Om) / (-Omc * Omc))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.15e+206], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Om * Om), $MachinePrecision] / N[((-Omc) * Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.15000000000000008e206

   1. Initial program 89.5%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 25.1%

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

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 53.8%

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

   if 1.15000000000000008e206 < (/.f64 t l)

   1. Initial program 50.6%

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

   3. Taylor expanded in Om around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 22.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 1.3%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 24.4%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 1.15 \cdot 10^{+206}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Om \cdot Om}{\left(-Omc\right) \cdot Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

l_m =     private
t_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in Om around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 33.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 22.5%

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

9. Taylor expanded in Om around 0

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

11. Applied rewrites 48.2%

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

Reproduce

herbie shell --seed 2025011 
(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)))))))