Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.8%

Time: 8.1s

Alternatives: 12

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.6% accurate, 1.0× speedup

Math

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

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: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ \mathbf{if}\;2 \cdot {t\_1}^{2} \leq 5.8 \cdot 10^{+276}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\frac{\left|t\right| + \left|t\right|}{\left|\ell\right|}, t\_1, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(-1 \cdot \frac{-1 \cdot \left(\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}\right)}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (let* ((t_1 (/ (fabs t) (fabs l))))
  (if (<= (* 2.0 (pow t_1 2.0)) 5.8e+276)
    (asin
     (sqrt
      (/
       (- 1.0 (pow (/ Om Omc) 2.0))
       (fma (/ (+ (fabs t) (fabs t)) (fabs l)) t_1 1.0))))
    (asin
     (*
      -1.0
      (/
       (*
        -1.0
        (*
         (fabs l)
         (sqrt (* -0.5 (* (+ 1.0 (/ Om Omc)) (- (/ Om Omc) 1.0))))))
       (fabs t)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / fabs(l);
	double tmp;
	if ((2.0 * pow(t_1, 2.0)) <= 5.8e+276) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma(((fabs(t) + fabs(t)) / fabs(l)), t_1, 1.0))));
	} else {
		tmp = asin((-1.0 * ((-1.0 * (fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0)))))) / fabs(t))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / abs(l))
	tmp = 0.0
	if (Float64(2.0 * (t_1 ^ 2.0)) <= 5.8e+276)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(Float64(abs(t) + abs(t)) / abs(l)), t_1, 1.0))));
	else
		tmp = asin(Float64(-1.0 * Float64(Float64(-1.0 * Float64(abs(l) * sqrt(Float64(-0.5 * Float64(Float64(1.0 + Float64(Om / Omc)) * Float64(Float64(Om / Omc) - 1.0)))))) / abs(t))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], 5.8e+276], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(-1.0 * N[(N[(-1.0 * N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(N[(1.0 + N[(Om / Omc), $MachinePrecision]), $MachinePrecision] * N[(N[(Om / Omc), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 5.7999999999999999e276

   1. Initial program 83.6%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 83.6%

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

   3. Applied rewrites 83.6%

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

   if 5.7999999999999999e276 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 25.4%

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

   7. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 32.0%

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

   9. Applied rewrites 32.0%

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

Alternative 2: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{Om}{Omc} - 1\\ t_2 := \frac{\left|t\right|}{\left|\ell\right|}\\ \mathbf{if}\;2 \cdot {t\_2}^{2} \leq 5.8 \cdot 10^{+276}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{Om}{Omc} - -1\right) \cdot \frac{t\_1}{\mathsf{fma}\left(\frac{-2 \cdot \left|t\right|}{\left|\ell\right|}, t\_2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(-1 \cdot \frac{-1 \cdot \left(\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot t\_1\right)}\right)}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (let* ((t_1 (- (/ Om Omc) 1.0)) (t_2 (/ (fabs t) (fabs l))))
  (if (<= (* 2.0 (pow t_2 2.0)) 5.8e+276)
    (asin
     (sqrt
      (*
       (- (/ Om Omc) -1.0)
       (/ t_1 (fma (/ (* -2.0 (fabs t)) (fabs l)) t_2 -1.0)))))
    (asin
     (*
      -1.0
      (/
       (*
        -1.0
        (* (fabs l) (sqrt (* -0.5 (* (+ 1.0 (/ Om Omc)) t_1)))))
       (fabs t)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = (Om / Omc) - 1.0;
	double t_2 = fabs(t) / fabs(l);
	double tmp;
	if ((2.0 * pow(t_2, 2.0)) <= 5.8e+276) {
		tmp = asin(sqrt((((Om / Omc) - -1.0) * (t_1 / fma(((-2.0 * fabs(t)) / fabs(l)), t_2, -1.0)))));
	} else {
		tmp = asin((-1.0 * ((-1.0 * (fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * t_1))))) / fabs(t))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(Float64(Om / Omc) - 1.0)
	t_2 = Float64(abs(t) / abs(l))
	tmp = 0.0
	if (Float64(2.0 * (t_2 ^ 2.0)) <= 5.8e+276)
		tmp = asin(sqrt(Float64(Float64(Float64(Om / Omc) - -1.0) * Float64(t_1 / fma(Float64(Float64(-2.0 * abs(t)) / abs(l)), t_2, -1.0)))));
	else
		tmp = asin(Float64(-1.0 * Float64(Float64(-1.0 * Float64(abs(l) * sqrt(Float64(-0.5 * Float64(Float64(1.0 + Float64(Om / Omc)) * t_1))))) / abs(t))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / Omc), $MachinePrecision] - 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision], 5.8e+276], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] - -1.0), $MachinePrecision] * N[(t$95$1 / N[(N[(N[(-2.0 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$2 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(-1.0 * N[(N[(-1.0 * N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(N[(1.0 + N[(Om / Omc), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 5.7999999999999999e276

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

   3. Applied rewrites 72.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. count-2 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lift-*.f64 83.5%

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

   5. Applied rewrites 83.5%

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

   if 5.7999999999999999e276 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 25.4%

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

   7. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 32.0%

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

   9. Applied rewrites 32.0%

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

Alternative 3: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_2 := 2 \cdot {t\_1}^{2}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{1}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+229}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om \cdot Om}{Omc}}{Omc}}{\mathsf{fma}\left(\frac{\left|t\right| + \left|t\right|}{\left|\ell\right|}, t\_1, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(-1 \cdot \frac{-1 \cdot \left(\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}\right)}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (let* ((t_1 (/ (fabs t) (fabs l))) (t_2 (* 2.0 (pow t_1 2.0))))
  (if (<= t_2 5e-20)
    (asin (sqrt (/ (- 1.0 (/ (* (/ Om Omc) Om) Omc)) 1.0)))
    (if (<= t_2 2e+229)
      (asin
       (sqrt
        (/
         (- 1.0 (/ (/ (* Om Om) Omc) Omc))
         (fma (/ (+ (fabs t) (fabs t)) (fabs l)) t_1 1.0))))
      (asin
       (*
        -1.0
        (/
         (*
          -1.0
          (*
           (fabs l)
           (sqrt (* -0.5 (* (+ 1.0 (/ Om Omc)) (- (/ Om Omc) 1.0))))))
         (fabs t))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / fabs(l);
	double t_2 = 2.0 * pow(t_1, 2.0);
	double tmp;
	if (t_2 <= 5e-20) {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	} else if (t_2 <= 2e+229) {
		tmp = asin(sqrt(((1.0 - (((Om * Om) / Omc) / Omc)) / fma(((fabs(t) + fabs(t)) / fabs(l)), t_1, 1.0))));
	} else {
		tmp = asin((-1.0 * ((-1.0 * (fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0)))))) / fabs(t))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / abs(l))
	t_2 = Float64(2.0 * (t_1 ^ 2.0))
	tmp = 0.0
	if (t_2 <= 5e-20)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / 1.0)));
	elseif (t_2 <= 2e+229)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om * Om) / Omc) / Omc)) / fma(Float64(Float64(abs(t) + abs(t)) / abs(l)), t_1, 1.0))));
	else
		tmp = asin(Float64(-1.0 * Float64(Float64(-1.0 * Float64(abs(l) * sqrt(Float64(-0.5 * Float64(Float64(1.0 + Float64(Om / Omc)) * Float64(Float64(Om / Omc) - 1.0)))))) / abs(t))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-20], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+229], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om * Om), $MachinePrecision] / Omc), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(-1.0 * N[(N[(-1.0 * N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(N[(1.0 + N[(Om / Omc), $MachinePrecision]), $MachinePrecision] * N[(N[(Om / Omc), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 4.9999999999999999e-20

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 4.9999999999999999e-20 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 2e229

   1. Initial program 83.6%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 83.6%

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

   3. Applied rewrites 83.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 78.2%

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 78.2%

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

   5. Applied rewrites 78.2%

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

   if 2e229 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 25.4%

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

   7. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 32.0%

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

   9. Applied rewrites 32.0%

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

Alternative 4: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_2 := 2 \cdot {t\_1}^{2}\\ \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-24}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{1}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}{\mathsf{fma}\left(\frac{-2 \cdot \left|t\right|}{\left|\ell\right|}, t\_1, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(-1 \cdot \frac{-1 \cdot \left(\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}\right)}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (let* ((t_1 (/ (fabs t) (fabs l))) (t_2 (* 2.0 (pow t_1 2.0))))
  (if (<= t_2 5e-24)
    (asin (sqrt (/ (- 1.0 (/ (* (/ Om Omc) Om) Omc)) 1.0)))
    (if (<= t_2 2e+20)
      (asin
       (sqrt
        (/
         (fma (/ Om (* Omc Omc)) Om -1.0)
         (fma (/ (* -2.0 (fabs t)) (fabs l)) t_1 -1.0))))
      (asin
       (*
        -1.0
        (/
         (*
          -1.0
          (*
           (fabs l)
           (sqrt (* -0.5 (* (+ 1.0 (/ Om Omc)) (- (/ Om Omc) 1.0))))))
         (fabs t))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / fabs(l);
	double t_2 = 2.0 * pow(t_1, 2.0);
	double tmp;
	if (t_2 <= 5e-24) {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	} else if (t_2 <= 2e+20) {
		tmp = asin(sqrt((fma((Om / (Omc * Omc)), Om, -1.0) / fma(((-2.0 * fabs(t)) / fabs(l)), t_1, -1.0))));
	} else {
		tmp = asin((-1.0 * ((-1.0 * (fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0)))))) / fabs(t))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / abs(l))
	t_2 = Float64(2.0 * (t_1 ^ 2.0))
	tmp = 0.0
	if (t_2 <= 5e-24)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / 1.0)));
	elseif (t_2 <= 2e+20)
		tmp = asin(sqrt(Float64(fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0) / fma(Float64(Float64(-2.0 * abs(t)) / abs(l)), t_1, -1.0))));
	else
		tmp = asin(Float64(-1.0 * Float64(Float64(-1.0 * Float64(abs(l) * sqrt(Float64(-0.5 * Float64(Float64(1.0 + Float64(Om / Omc)) * Float64(Float64(Om / Omc) - 1.0)))))) / abs(t))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 5e-24], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$2, 2e+20], N[ArcSin[N[Sqrt[N[(N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision] / N[(N[(N[(-2.0 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(-1.0 * N[(N[(-1.0 * N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(N[(1.0 + N[(Om / Omc), $MachinePrecision]), $MachinePrecision] * N[(N[(Om / Omc), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 4.9999999999999998e-24

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 4.9999999999999998e-24 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 2e20

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

   3. Applied rewrites 72.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. count-2 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lift-*.f64 83.5%

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 67.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 78.5%

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

   9. Applied rewrites 78.5%

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

   if 2e20 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

   3. Applied rewrites 72.1%

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

   4. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

   6. Applied rewrites 25.4%

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

   7. Taylor expanded in l around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 32.0%

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

   9. Applied rewrites 32.0%

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

Alternative 5: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_2 := \frac{Om}{Omc \cdot Omc}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{1}}\right)\\ \mathbf{elif}\;t\_1 \leq 3 \cdot 10^{+114}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(t\_2, Om, -1\right)}{\mathsf{fma}\left(\frac{-2 \cdot \left|t\right|}{\left|\ell\right|}, t\_1, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \frac{\sqrt{\frac{0.5 \cdot \left(1 - Om \cdot t\_2\right)}{\left|t\right|}}}{\sqrt{\left|t\right|}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (let* ((t_1 (/ (fabs t) (fabs l))) (t_2 (/ Om (* Omc Omc))))
  (if (<= t_1 2e-12)
    (asin (sqrt (/ (- 1.0 (/ (* (/ Om Omc) Om) Omc)) 1.0)))
    (if (<= t_1 3e+114)
      (asin
       (sqrt
        (/
         (fma t_2 Om -1.0)
         (fma (/ (* -2.0 (fabs t)) (fabs l)) t_1 -1.0))))
      (asin
       (*
        (fabs l)
        (/
         (sqrt (/ (* 0.5 (- 1.0 (* Om t_2))) (fabs t)))
         (sqrt (fabs t)))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / fabs(l);
	double t_2 = Om / (Omc * Omc);
	double tmp;
	if (t_1 <= 2e-12) {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	} else if (t_1 <= 3e+114) {
		tmp = asin(sqrt((fma(t_2, Om, -1.0) / fma(((-2.0 * fabs(t)) / fabs(l)), t_1, -1.0))));
	} else {
		tmp = asin((fabs(l) * (sqrt(((0.5 * (1.0 - (Om * t_2))) / fabs(t))) / sqrt(fabs(t)))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / abs(l))
	t_2 = Float64(Om / Float64(Omc * Omc))
	tmp = 0.0
	if (t_1 <= 2e-12)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / 1.0)));
	elseif (t_1 <= 3e+114)
		tmp = asin(sqrt(Float64(fma(t_2, Om, -1.0) / fma(Float64(Float64(-2.0 * abs(t)) / abs(l)), t_1, -1.0))));
	else
		tmp = asin(Float64(abs(l) * Float64(sqrt(Float64(Float64(0.5 * Float64(1.0 - Float64(Om * t_2))) / abs(t))) / sqrt(abs(t)))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-12], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 3e+114], N[ArcSin[N[Sqrt[N[(N[(t$95$2 * Om + -1.0), $MachinePrecision] / N[(N[(N[(-2.0 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[(N[Sqrt[N[(N[(0.5 * N[(1.0 - N[(Om * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < 2e-12

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 2e-12 < (/.f64 t l) < 3e114

   1. Initial program 83.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. add-flip N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

   3. Applied rewrites 72.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. count-2 N/A

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

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

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

      17. metadata-eval N/A

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

      18. lift-*.f64 83.5%

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 67.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-*.f64 N/A

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

      10. times-frac N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 78.5%

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

   9. Applied rewrites 78.5%

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

   if 3e114 < (/.f64 t l)

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.7%

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

   4. Applied rewrites 21.7%

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

   6. Applied rewrites 15.0%

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

Alternative 6: 94.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (if (<= (/ (fabs t) (fabs l)) 0.5)
  (asin (sqrt (/ (- 1.0 (/ (* (/ Om Omc) Om) Omc)) 1.0)))
  (asin
   (*
    (fabs l)
    (/
     (sqrt (/ (* 0.5 (- 1.0 (* Om (/ Om (* Omc Omc))))) (fabs t)))
     (sqrt (fabs t)))))))

C

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

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
    real(8) :: tmp
    if ((abs(t) / abs(l)) <= 0.5d0) then
        tmp = asin(sqrt(((1.0d0 - (((om / omc) * om) / omc)) / 1.0d0)))
    else
        tmp = asin((abs(l) * (sqrt(((0.5d0 * (1.0d0 - (om * (om / (omc * omc))))) / abs(t))) / sqrt(abs(t)))))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 0.5], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[(N[Sqrt[N[(N[(0.5 * N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.5

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

   if 0.5 < (/.f64 t l)

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.7%

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

   4. Applied rewrites 21.7%

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

   6. Applied rewrites 15.0%

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

Alternative 7: 94.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / abs(l)) ** 2.0d0)))))) <= 1.0d0) then
        tmp = asin(((sqrt((0.5d0 * (1.0d0 - (om * (om / (omc * omc)))))) / abs(t)) * abs(l)))
    else
        tmp = asin(sqrt(((1.0d0 - (((om / omc) * om) / omc)) / 1.0d0)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, 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 / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1.0], N[ArcSin[N[(N[(N[Sqrt[N[(0.5 * N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.7%

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

   4. Applied rewrites 21.7%

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

   6. Applied rewrites 30.0%

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

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

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

Alternative 8: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\left|\ell\right|}\right)}^{2}}}\right) \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \sqrt{\frac{\frac{0.5}{t}}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{1}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (if (<=
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (pow (/ t (fabs l)) 2.0))))))
     0.0002)
  (asin (* (fabs l) (sqrt (/ (/ 0.5 t) t))))
  (asin (sqrt (/ (- 1.0 (/ (* (/ Om Omc) Om) Omc)) 1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / fabs(l)), 2.0)))))) <= 0.0002) {
		tmp = asin((fabs(l) * sqrt(((0.5 / t) / t))));
	} else {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / abs(l)) ** 2.0d0)))))) <= 0.0002d0) then
        tmp = asin((abs(l) * sqrt(((0.5d0 / t) / t))))
    else
        tmp = asin(sqrt(((1.0d0 - (((om / omc) * om) / omc)) / 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, 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 / Math.abs(l)), 2.0)))))) <= 0.0002) {
		tmp = Math.asin((Math.abs(l) * Math.sqrt(((0.5 / t) / t))));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / math.fabs(l)), 2.0)))))) <= 0.0002:
		tmp = math.asin((math.fabs(l) * math.sqrt(((0.5 / t) / t))))
	else:
		tmp = math.asin(math.sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)))
	return tmp

Julia

function code(t, l, 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 / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin(Float64(abs(l) * sqrt(Float64(Float64(0.5 / t) / t))));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin((abs(l) * sqrt(((0.5 / t) / t))));
	else
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, 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 / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0002], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(0.5 / t), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 2.0000000000000001e-4

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.7%

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 24.4%

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

   7. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 24.6%

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

   9. Applied rewrites 24.6%

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

   if 2.0000000000000001e-4 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.2%

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

   6. Applied rewrites 50.2%

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

Alternative 9: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\left|\ell\right|}\right)}^{2}}}\right) \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \sqrt{\frac{\frac{0.5}{t}}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Omc - Om \cdot \frac{Om}{Omc}}{1 \cdot Omc}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (if (<=
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (pow (/ t (fabs l)) 2.0))))))
     0.0002)
  (asin (* (fabs l) (sqrt (/ (/ 0.5 t) t))))
  (asin (sqrt (/ (- Omc (* Om (/ Om Omc))) (* 1.0 Omc))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / fabs(l)), 2.0)))))) <= 0.0002) {
		tmp = asin((fabs(l) * sqrt(((0.5 / t) / t))));
	} else {
		tmp = asin(sqrt(((Omc - (Om * (Om / Omc))) / (1.0 * Omc))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / abs(l)) ** 2.0d0)))))) <= 0.0002d0) then
        tmp = asin((abs(l) * sqrt(((0.5d0 / t) / t))))
    else
        tmp = asin(sqrt(((omc - (om * (om / omc))) / (1.0d0 * omc))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, 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 / Math.abs(l)), 2.0)))))) <= 0.0002) {
		tmp = Math.asin((Math.abs(l) * Math.sqrt(((0.5 / t) / t))));
	} else {
		tmp = Math.asin(Math.sqrt(((Omc - (Om * (Om / Omc))) / (1.0 * Omc))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / math.fabs(l)), 2.0)))))) <= 0.0002:
		tmp = math.asin((math.fabs(l) * math.sqrt(((0.5 / t) / t))))
	else:
		tmp = math.asin(math.sqrt(((Omc - (Om * (Om / Omc))) / (1.0 * Omc))))
	return tmp

Julia

function code(t, l, 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 / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin(Float64(abs(l) * sqrt(Float64(Float64(0.5 / t) / t))));
	else
		tmp = asin(sqrt(Float64(Float64(Omc - Float64(Om * Float64(Om / Omc))) / Float64(1.0 * Omc))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin((abs(l) * sqrt(((0.5 / t) / t))));
	else
		tmp = asin(sqrt(((Omc - (Om * (Om / Omc))) / (1.0 * Omc))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, 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 / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0002], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(0.5 / t), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Omc - N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 * Omc), $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))))))) < 2.0000000000000001e-4

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.7%

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 24.4%

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

   7. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 24.6%

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

   9. Applied rewrites 24.6%

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

   if 2.0000000000000001e-4 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. +-commutative N/A

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

      15. sub-flip N/A

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

      16. lift-/.f64 N/A

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

      17. sub-to-fraction N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/r* N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 25.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 50.2%

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

   8. Applied rewrites 50.2%

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

Alternative 10: 82.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\left|\ell\right|}\right)}^{2}}}\right) \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \sqrt{\frac{\frac{0.5}{t}}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\frac{Omc}{Omc}}{1}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (if (<=
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (pow (/ t (fabs l)) 2.0))))))
     0.0002)
  (asin (* (fabs l) (sqrt (/ (/ 0.5 t) t))))
  (asin (sqrt (/ (/ Omc Omc) 1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / fabs(l)), 2.0)))))) <= 0.0002) {
		tmp = asin((fabs(l) * sqrt(((0.5 / t) / t))));
	} else {
		tmp = asin(sqrt(((Omc / Omc) / 1.0)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / abs(l)) ** 2.0d0)))))) <= 0.0002d0) then
        tmp = asin((abs(l) * sqrt(((0.5d0 / t) / t))))
    else
        tmp = asin(sqrt(((omc / omc) / 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, 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 / Math.abs(l)), 2.0)))))) <= 0.0002) {
		tmp = Math.asin((Math.abs(l) * Math.sqrt(((0.5 / t) / t))));
	} else {
		tmp = Math.asin(Math.sqrt(((Omc / Omc) / 1.0)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / math.fabs(l)), 2.0)))))) <= 0.0002:
		tmp = math.asin((math.fabs(l) * math.sqrt(((0.5 / t) / t))))
	else:
		tmp = math.asin(math.sqrt(((Omc / Omc) / 1.0)))
	return tmp

Julia

function code(t, l, 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 / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin(Float64(abs(l) * sqrt(Float64(Float64(0.5 / t) / t))));
	else
		tmp = asin(sqrt(Float64(Float64(Omc / Omc) / 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin((abs(l) * sqrt(((0.5 / t) / t))));
	else
		tmp = asin(sqrt(((Omc / Omc) / 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, 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 / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0002], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(0.5 / t), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Omc / Omc), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 2.0000000000000001e-4

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.7%

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 24.4%

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

   7. Applied rewrites 24.4%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 24.6%

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

   9. Applied rewrites 24.6%

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

   if 2.0000000000000001e-4 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. +-commutative N/A

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

      15. sub-flip N/A

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

      16. lift-/.f64 N/A

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

      17. sub-to-fraction N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/r* N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 25.9%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 49.6%

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

Alternative 11: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\left|\ell\right|}\right)}^{2}}}\right) \leq 0.0002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{0.5}{t \cdot t}} \cdot \left|\ell\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\frac{Omc}{Omc}}{1}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (if (<=
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (pow (/ t (fabs l)) 2.0))))))
     0.0002)
  (asin (* (sqrt (/ 0.5 (* t t))) (fabs l)))
  (asin (sqrt (/ (/ Omc Omc) 1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / fabs(l)), 2.0)))))) <= 0.0002) {
		tmp = asin((sqrt((0.5 / (t * t))) * fabs(l)));
	} else {
		tmp = asin(sqrt(((Omc / Omc) / 1.0)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / abs(l)) ** 2.0d0)))))) <= 0.0002d0) then
        tmp = asin((sqrt((0.5d0 / (t * t))) * abs(l)))
    else
        tmp = asin(sqrt(((omc / omc) / 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, 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 / Math.abs(l)), 2.0)))))) <= 0.0002) {
		tmp = Math.asin((Math.sqrt((0.5 / (t * t))) * Math.abs(l)));
	} else {
		tmp = Math.asin(Math.sqrt(((Omc / Omc) / 1.0)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / math.fabs(l)), 2.0)))))) <= 0.0002:
		tmp = math.asin((math.sqrt((0.5 / (t * t))) * math.fabs(l)))
	else:
		tmp = math.asin(math.sqrt(((Omc / Omc) / 1.0)))
	return tmp

Julia

function code(t, l, 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 / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin(Float64(sqrt(Float64(0.5 / Float64(t * t))) * abs(l)));
	else
		tmp = asin(sqrt(Float64(Float64(Omc / Omc) / 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / abs(l)) ^ 2.0)))))) <= 0.0002)
		tmp = asin((sqrt((0.5 / (t * t))) * abs(l)));
	else
		tmp = asin(sqrt(((Omc / Omc) / 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, 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 / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0002], N[ArcSin[N[(N[Sqrt[N[(0.5 / N[(t * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Omc / Omc), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 2.0000000000000001e-4

   1. Initial program 83.6%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.7%

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 24.4%

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

   7. Applied rewrites 24.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 24.4%

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 24.4%

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

   9. Applied rewrites 24.4%

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

   if 2.0000000000000001e-4 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. +-commutative N/A

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

      15. sub-flip N/A

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

      16. lift-/.f64 N/A

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

      17. sub-to-fraction N/A

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

      18. lift-pow.f64 N/A

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

      19. unpow2 N/A

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

      20. associate-/r* N/A

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

      21. lower-/.f64 N/A

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

   6. Applied rewrites 25.9%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 49.6%

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

Alternative 12: 49.6% accurate, 4.2× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (asin (sqrt (/ (/ Omc Omc) 1.0))))

C

double code(double t, double l, double Om, double Omc) {
	return asin(sqrt(((Omc / Omc) / 1.0)));
}

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(((omc / omc) / 1.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[Sqrt[N[(N[(Omc / Omc), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.6%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 50.2%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. frac-times N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-pow.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. +-commutative N/A

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

   15. sub-flip N/A

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

   16. lift-/.f64 N/A

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

   17. sub-to-fraction N/A

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

   18. lift-pow.f64 N/A

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

   19. unpow2 N/A

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

   20. associate-/r* N/A

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

   21. lower-/.f64 N/A

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

6. Applied rewrites 25.9%

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

7. Taylor expanded in Om around 0

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

9. Applied rewrites 49.6%

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

Reproduce

herbie shell --seed 2025212 
(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)))))))