Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.2%

Time: 8.6s

Alternatives: 11

Speedup: 1.4×

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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 10^{-82}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot \left(\left(-\left|t\right|\right) \cdot \frac{t\_1}{-\left|\ell\right|}\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (fabs t) (fabs l))) (t_2 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (asin (sqrt (/ t_2 (+ 1.0 (* 2.0 (pow t_1 2.0)))))) 1e-82)
     (asin
      (/
       (* (fabs l) (sqrt (* -0.5 (* (+ 1.0 (/ Om Omc)) (- (/ Om Omc) 1.0)))))
       (fabs t)))
     (asin
      (sqrt (/ t_2 (+ 1.0 (* 2.0 (* (- (fabs t)) (/ t_1 (- (fabs l))))))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / fabs(l);
	double t_2 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if (asin(sqrt((t_2 / (1.0 + (2.0 * pow(t_1, 2.0)))))) <= 1e-82) {
		tmp = asin(((fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / fabs(t)));
	} else {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * (-fabs(t) * (t_1 / -fabs(l))))))));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs(t) / abs(l)
    t_2 = 1.0d0 - ((om / omc) ** 2.0d0)
    if (asin(sqrt((t_2 / (1.0d0 + (2.0d0 * (t_1 ** 2.0d0)))))) <= 1d-82) then
        tmp = asin(((abs(l) * sqrt(((-0.5d0) * ((1.0d0 + (om / omc)) * ((om / omc) - 1.0d0))))) / abs(t)))
    else
        tmp = asin(sqrt((t_2 / (1.0d0 + (2.0d0 * (-abs(t) * (t_1 / -abs(l))))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = Math.abs(t) / Math.abs(l);
	double t_2 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if (Math.asin(Math.sqrt((t_2 / (1.0 + (2.0 * Math.pow(t_1, 2.0)))))) <= 1e-82) {
		tmp = Math.asin(((Math.abs(l) * Math.sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / Math.abs(t)));
	} else {
		tmp = Math.asin(Math.sqrt((t_2 / (1.0 + (2.0 * (-Math.abs(t) * (t_1 / -Math.abs(l))))))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = math.fabs(t) / math.fabs(l)
	t_2 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if math.asin(math.sqrt((t_2 / (1.0 + (2.0 * math.pow(t_1, 2.0)))))) <= 1e-82:
		tmp = math.asin(((math.fabs(l) * math.sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / math.fabs(t)))
	else:
		tmp = math.asin(math.sqrt((t_2 / (1.0 + (2.0 * (-math.fabs(t) * (t_1 / -math.fabs(l))))))))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / abs(l))
	t_2 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (asin(sqrt(Float64(t_2 / Float64(1.0 + Float64(2.0 * (t_1 ^ 2.0)))))) <= 1e-82)
		tmp = asin(Float64(Float64(abs(l) * sqrt(Float64(-0.5 * Float64(Float64(1.0 + Float64(Om / Omc)) * Float64(Float64(Om / Omc) - 1.0))))) / abs(t)));
	else
		tmp = asin(sqrt(Float64(t_2 / Float64(1.0 + Float64(2.0 * Float64(Float64(-abs(t)) * Float64(t_1 / Float64(-abs(l)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = abs(t) / abs(l);
	t_2 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if (asin(sqrt((t_2 / (1.0 + (2.0 * (t_1 ^ 2.0)))))) <= 1e-82)
		tmp = asin(((abs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / abs(t)));
	else
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * (-abs(t) * (t_1 / -abs(l))))))));
	end
	tmp_2 = 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[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-82], N[ArcSin[N[(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] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[((-N[Abs[t], $MachinePrecision]) * N[(t$95$1 / (-N[Abs[l], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 1e-82

   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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 23.4%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 30.6%

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

   11. Applied rewrites 30.6%

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

   if 1e-82 < (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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-neg.f64 80.6%

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

   3. Applied rewrites 80.6%

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

Alternative 2: 98.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{Om}{Omc} - 1\\ t_2 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_3 := \left|\ell\right| + \left|\ell\right|\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {t\_2}^{2}}}\right) \leq 10^{-74}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot t\_1\right)}}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{Om}{Omc} - -1\right) \cdot \frac{-t\_1 \cdot t\_3}{t\_3 - \left(\left|t\right| \cdot -4\right) \cdot t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- (/ Om Omc) 1.0))
        (t_2 (/ (fabs t) (fabs l)))
        (t_3 (+ (fabs l) (fabs l))))
   (if (<=
        (asin
         (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow t_2 2.0))))))
        1e-74)
     (asin
      (/ (* (fabs l) (sqrt (* -0.5 (* (+ 1.0 (/ Om Omc)) t_1)))) (fabs t)))
     (asin
      (sqrt
       (*
        (- (/ Om Omc) -1.0)
        (/ (- (* t_1 t_3)) (- t_3 (* (* (fabs t) -4.0) t_2)))))))))

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 t_3 = fabs(l) + fabs(l);
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow(t_2, 2.0)))))) <= 1e-74) {
		tmp = asin(((fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * t_1)))) / fabs(t)));
	} else {
		tmp = asin(sqrt((((Om / Omc) - -1.0) * (-(t_1 * t_3) / (t_3 - ((fabs(t) * -4.0) * t_2))))));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (om / omc) - 1.0d0
    t_2 = abs(t) / abs(l)
    t_3 = abs(l) + abs(l)
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * (t_2 ** 2.0d0)))))) <= 1d-74) then
        tmp = asin(((abs(l) * sqrt(((-0.5d0) * ((1.0d0 + (om / omc)) * t_1)))) / abs(t)))
    else
        tmp = asin(sqrt((((om / omc) - (-1.0d0)) * (-(t_1 * t_3) / (t_3 - ((abs(t) * (-4.0d0)) * t_2))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = (Om / Omc) - 1.0;
	double t_2 = Math.abs(t) / Math.abs(l);
	double t_3 = Math.abs(l) + Math.abs(l);
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow(t_2, 2.0)))))) <= 1e-74) {
		tmp = Math.asin(((Math.abs(l) * Math.sqrt((-0.5 * ((1.0 + (Om / Omc)) * t_1)))) / Math.abs(t)));
	} else {
		tmp = Math.asin(Math.sqrt((((Om / Omc) - -1.0) * (-(t_1 * t_3) / (t_3 - ((Math.abs(t) * -4.0) * t_2))))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	t_1 = (Om / Omc) - 1.0
	t_2 = math.fabs(t) / math.fabs(l)
	t_3 = math.fabs(l) + math.fabs(l)
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow(t_2, 2.0)))))) <= 1e-74:
		tmp = math.asin(((math.fabs(l) * math.sqrt((-0.5 * ((1.0 + (Om / Omc)) * t_1)))) / math.fabs(t)))
	else:
		tmp = math.asin(math.sqrt((((Om / Omc) - -1.0) * (-(t_1 * t_3) / (t_3 - ((math.fabs(t) * -4.0) * t_2))))))
	return tmp

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(Float64(Om / Omc) - 1.0)
	t_2 = Float64(abs(t) / abs(l))
	t_3 = Float64(abs(l) + abs(l))
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (t_2 ^ 2.0)))))) <= 1e-74)
		tmp = asin(Float64(Float64(abs(l) * sqrt(Float64(-0.5 * Float64(Float64(1.0 + Float64(Om / Omc)) * t_1)))) / abs(t)));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(Om / Omc) - -1.0) * Float64(Float64(-Float64(t_1 * t_3)) / Float64(t_3 - Float64(Float64(abs(t) * -4.0) * t_2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = (Om / Omc) - 1.0;
	t_2 = abs(t) / abs(l);
	t_3 = abs(l) + abs(l);
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * (t_2 ^ 2.0)))))) <= 1e-74)
		tmp = asin(((abs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * t_1)))) / abs(t)));
	else
		tmp = asin(sqrt((((Om / Omc) - -1.0) * (-(t_1 * t_3) / (t_3 - ((abs(t) * -4.0) * t_2))))));
	end
	tmp_2 = 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]}, Block[{t$95$3 = N[(N[Abs[l], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision]}, 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[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-74], N[ArcSin[N[(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] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] - -1.0), $MachinePrecision] * N[((-N[(t$95$1 * t$95$3), $MachinePrecision]) / N[(t$95$3 - N[(N[(N[Abs[t], $MachinePrecision] * -4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 9.99999999999999958e-75

   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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 23.4%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 30.6%

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

   11. Applied rewrites 30.6%

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

   if 9.99999999999999958e-75 < (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. 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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l/ N/A

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

      5. frac-2neg N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. count-2-rev N/A

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

      12. lower-+.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 81.1%

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

   7. Applied rewrites 81.1%

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

Alternative 3: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{\left|t\right|}{\left|\ell\right|}\right)}^{2} \leq 2 \cdot 10^{+143}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\left(Omc - \frac{Om}{Omc} \cdot Om\right) \cdot \frac{\left|\ell\right|}{\mathsf{fma}\left(\frac{\left|t\right| + \left|t\right|}{\left|\ell\right|}, \left|t\right|, \left|\ell\right|\right)}}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ (fabs t) (fabs l)) 2.0))) 2e+143)
   (asin
    (sqrt
     (/
      (*
       (- Omc (* (/ Om Omc) Om))
       (/ (fabs l) (fma (/ (+ (fabs t) (fabs t)) (fabs l)) (fabs t) (fabs l))))
      Omc)))
   (asin
    (/
     (* (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 tmp;
	if ((1.0 + (2.0 * pow((fabs(t) / fabs(l)), 2.0))) <= 2e+143) {
		tmp = asin(sqrt((((Omc - ((Om / Omc) * Om)) * (fabs(l) / fma(((fabs(t) + fabs(t)) / fabs(l)), fabs(t), fabs(l)))) / Omc)));
	} else {
		tmp = asin(((fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / fabs(t)));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(1.0 + Float64(2.0 * (Float64(abs(t) / abs(l)) ^ 2.0))) <= 2e+143)
		tmp = asin(sqrt(Float64(Float64(Float64(Omc - Float64(Float64(Om / Omc) * Om)) * Float64(abs(l) / fma(Float64(Float64(abs(t) + abs(t)) / abs(l)), abs(t), abs(l)))) / Omc)));
	else
		tmp = asin(Float64(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_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+143], N[ArcSin[N[Sqrt[N[(N[(N[(Omc - N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[l], $MachinePrecision] / N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * N[Abs[t], $MachinePrecision] + N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(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] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 2e143

   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 78.1%

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

      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 78.1%

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

   5. Applied rewrites 78.1%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. sub-to-fraction N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. associate-*l/ N/A

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

      14. lower-/.f64 N/A

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

   7. Applied rewrites 79.9%

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

   if 2e143 < (+.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. 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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 23.4%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 30.6%

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

   11. Applied rewrites 30.6%

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

Alternative 4: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ \mathbf{if}\;1 + 2 \cdot {t\_1}^{2} \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\frac{-1}{\mathsf{fma}\left(\left|t\right| \cdot -4, t\_1, -2 \cdot \left|\ell\right|\right)}}}{\sqrt{\frac{0.5}{\left|\ell\right|}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{-0.5 \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (fabs t) (fabs l))))
   (if (<= (+ 1.0 (* 2.0 (pow t_1 2.0))) 4e+46)
     (asin
      (/
       (sqrt (/ -1.0 (fma (* (fabs t) -4.0) t_1 (* -2.0 (fabs l)))))
       (sqrt (/ 0.5 (fabs l)))))
     (asin
      (/
       (* (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 ((1.0 + (2.0 * pow(t_1, 2.0))) <= 4e+46) {
		tmp = asin((sqrt((-1.0 / fma((fabs(t) * -4.0), t_1, (-2.0 * fabs(l))))) / sqrt((0.5 / fabs(l)))));
	} else {
		tmp = asin(((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(1.0 + Float64(2.0 * (t_1 ^ 2.0))) <= 4e+46)
		tmp = asin(Float64(sqrt(Float64(-1.0 / fma(Float64(abs(t) * -4.0), t_1, Float64(-2.0 * abs(l))))) / sqrt(Float64(0.5 / abs(l)))));
	else
		tmp = asin(Float64(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[(1.0 + N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+46], N[ArcSin[N[(N[Sqrt[N[(-1.0 / N[(N[(N[Abs[t], $MachinePrecision] * -4.0), $MachinePrecision] * t$95$1 + N[(-2.0 * N[Abs[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(0.5 / N[Abs[l], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(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] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))) < 4e46

   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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   7. Applied rewrites 41.6%

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}{\mathsf{fma}\left(t \cdot -4, \frac{t}{\ell}, -2 \cdot \ell\right)}}}{\sqrt{\frac{0.5}{\ell}}}\right)} \]

   8. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{\frac{\color{blue}{-1}}{\mathsf{fma}\left(t \cdot -4, \frac{t}{\ell}, -2 \cdot \ell\right)}}}{\sqrt{\frac{0.5}{\ell}}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 43.7%

       \[\leadsto \sin^{-1} \left(\frac{\sqrt{\frac{\color{blue}{-1}}{\mathsf{fma}\left(t \cdot -4, \frac{t}{\ell}, -2 \cdot \ell\right)}}}{\sqrt{\frac{0.5}{\ell}}}\right) \]

   if 4e46 < (+.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. 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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 23.4%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 30.6%

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

   11. Applied rewrites 30.6%

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

Alternative 5: 97.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ (fabs t) (fabs l)) 2.0))))))
      5e-5)
   (asin
    (/
     (* (fabs l) (sqrt (* -0.5 (* (+ 1.0 (/ Om Omc)) (- (/ Om Omc) 1.0)))))
     (fabs t)))
   (asin (sqrt (/ (- Omc (* (/ 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((fabs(t) / fabs(l)), 2.0)))))) <= 5e-5) {
		tmp = asin(((fabs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / fabs(t)));
	} else {
		tmp = asin(sqrt(((Omc - ((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 * ((abs(t) / abs(l)) ** 2.0d0)))))) <= 5d-5) then
        tmp = asin(((abs(l) * sqrt(((-0.5d0) * ((1.0d0 + (om / omc)) * ((om / omc) - 1.0d0))))) / abs(t)))
    else
        tmp = asin(sqrt(((omc - ((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((Math.abs(t) / Math.abs(l)), 2.0)))))) <= 5e-5) {
		tmp = Math.asin(((Math.abs(l) * Math.sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / Math.abs(t)));
	} else {
		tmp = Math.asin(Math.sqrt(((Omc - ((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((math.fabs(t) / math.fabs(l)), 2.0)))))) <= 5e-5:
		tmp = math.asin(((math.fabs(l) * math.sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / math.fabs(t)))
	else:
		tmp = math.asin(math.sqrt(((Omc - ((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(abs(t) / abs(l)) ^ 2.0)))))) <= 5e-5)
		tmp = asin(Float64(Float64(abs(l) * sqrt(Float64(-0.5 * Float64(Float64(1.0 + Float64(Om / Omc)) * Float64(Float64(Om / Omc) - 1.0))))) / abs(t)));
	else
		tmp = asin(sqrt(Float64(Float64(Omc - Float64(Float64(Om / Omc) * Om)) / Float64(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 * ((abs(t) / abs(l)) ^ 2.0)))))) <= 5e-5)
		tmp = asin(((abs(l) * sqrt((-0.5 * ((1.0 + (Om / Omc)) * ((Om / Omc) - 1.0))))) / abs(t)));
	else
		tmp = asin(sqrt(((Omc - ((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[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 5e-5], N[ArcSin[N[(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] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(Omc - N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] / N[(Omc * 1.0), $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))))))) < 5.00000000000000024e-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. 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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 23.4%

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

   9. Taylor expanded in l around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-+.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-/.f64 30.6%

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

   11. Applied rewrites 30.6%

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

   if 5.00000000000000024e-5 < (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 51.8%

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

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.7%

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7%

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

   6. Applied rewrites 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-/.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 51.8%

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

Alternative 6: 94.9% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ (fabs t) (fabs l)) 0.002)
   (asin (sqrt (/ (- Omc (* (/ Om Omc) Om)) (* Omc 1.0))))
   (asin
    (/
     (* (sqrt (* -0.5 (fma (/ Om (* Omc Omc)) Om -1.0))) (fabs (fabs l)))
     (fabs t)))))

C

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

Julia

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

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 0.002], N[ArcSin[N[Sqrt[N[(N[(Omc - N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] / N[(Omc * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[N[(-0.5 * N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[N[Abs[l], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   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 51.8%

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

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.7%

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7%

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

   6. Applied rewrites 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-/.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 51.8%

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

   if 2e-3 < (/.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. 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 71.7%

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

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

      3. sub-flip-reverse N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. metadata-eval N/A

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

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

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/r* N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. metadata-eval N/A

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

      16. frac-sub N/A

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

      17. lower-/.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 23.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. lift-fabs.f64 N/A

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

      13. lower-unsound-*.f64 N/A

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

   10. Applied rewrites 29.0%

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

Alternative 7: 63.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left|t\right|}{\left|\ell\right|} \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Omc - \frac{Om}{Omc} \cdot Om}{Omc \cdot 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot 0.5 - \cos^{-1} \left(\frac{\sqrt{\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 0.5}}{-\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ (fabs t) (fabs l)) 2e+153)
   (asin (sqrt (/ (- Omc (* (/ Om Omc) Om)) (* Omc 1.0))))
   (-
    (* PI 0.5)
    (acos (/ (sqrt (* (* (fabs l) (fabs l)) 0.5)) (- (fabs t)))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((fabs(t) / fabs(l)) <= 2e+153) {
		tmp = asin(sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))));
	} else {
		tmp = (((double) M_PI) * 0.5) - acos((sqrt(((fabs(l) * fabs(l)) * 0.5)) / -fabs(t)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((Math.abs(t) / Math.abs(l)) <= 2e+153) {
		tmp = Math.asin(Math.sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))));
	} else {
		tmp = (Math.PI * 0.5) - Math.acos((Math.sqrt(((Math.abs(l) * Math.abs(l)) * 0.5)) / -Math.abs(t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (math.fabs(t) / math.fabs(l)) <= 2e+153:
		tmp = math.asin(math.sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))))
	else:
		tmp = (math.pi * 0.5) - math.acos((math.sqrt(((math.fabs(l) * math.fabs(l)) * 0.5)) / -math.fabs(t)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(abs(t) / abs(l)) <= 2e+153)
		tmp = asin(sqrt(Float64(Float64(Omc - Float64(Float64(Om / Omc) * Om)) / Float64(Omc * 1.0))));
	else
		tmp = Float64(Float64(pi * 0.5) - acos(Float64(sqrt(Float64(Float64(abs(l) * abs(l)) * 0.5)) / Float64(-abs(t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((abs(t) / abs(l)) <= 2e+153)
		tmp = asin(sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))));
	else
		tmp = (pi * 0.5) - acos((sqrt(((abs(l) * abs(l)) * 0.5)) / -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], 2e+153], N[ArcSin[N[Sqrt[N[(N[(Omc - N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] / N[(Omc * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[Sqrt[N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / (-N[Abs[t], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2e153

   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 51.8%

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

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.7%

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7%

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

   6. Applied rewrites 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-/.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 51.8%

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

   if 2e153 < (/.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 t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 21.1%

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

   4. Applied rewrites 21.1%

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

   5. Taylor expanded in Om around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 23.4%

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

   7. Applied rewrites 23.4%

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

      1. lift-asin.f64 N/A

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

      2. asin-acos N/A

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

      3. lower--.f64 N/A

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

      4. mult-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-acos.f64 13.8%

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

   9. Applied rewrites 13.8%

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

Alternative 8: 63.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left|t\right|}{\left|\ell\right|} \leq 10^{+190}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{Omc - \frac{Om}{Omc} \cdot Om}{Omc \cdot 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\frac{\left|t\right|}{-\sqrt{\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 0.5}}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ (fabs t) (fabs l)) 1e+190)
   (asin (sqrt (/ (- Omc (* (/ Om Omc) Om)) (* Omc 1.0))))
   (asin (/ 1.0 (/ (fabs t) (- (sqrt (* (* (fabs l) (fabs l)) 0.5))))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((fabs(t) / fabs(l)) <= 1e+190) {
		tmp = asin(sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))));
	} else {
		tmp = asin((1.0 / (fabs(t) / -sqrt(((fabs(l) * fabs(l)) * 0.5)))));
	}
	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)) <= 1d+190) then
        tmp = asin(sqrt(((omc - ((om / omc) * om)) / (omc * 1.0d0))))
    else
        tmp = asin((1.0d0 / (abs(t) / -sqrt(((abs(l) * abs(l)) * 0.5d0)))))
    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)) <= 1e+190) {
		tmp = Math.asin(Math.sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))));
	} else {
		tmp = Math.asin((1.0 / (Math.abs(t) / -Math.sqrt(((Math.abs(l) * Math.abs(l)) * 0.5)))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (math.fabs(t) / math.fabs(l)) <= 1e+190:
		tmp = math.asin(math.sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))))
	else:
		tmp = math.asin((1.0 / (math.fabs(t) / -math.sqrt(((math.fabs(l) * math.fabs(l)) * 0.5)))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(abs(t) / abs(l)) <= 1e+190)
		tmp = asin(sqrt(Float64(Float64(Omc - Float64(Float64(Om / Omc) * Om)) / Float64(Omc * 1.0))));
	else
		tmp = asin(Float64(1.0 / Float64(abs(t) / Float64(-sqrt(Float64(Float64(abs(l) * abs(l)) * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((abs(t) / abs(l)) <= 1e+190)
		tmp = asin(sqrt(((Omc - ((Om / Omc) * Om)) / (Omc * 1.0))));
	else
		tmp = asin((1.0 / (abs(t) / -sqrt(((abs(l) * abs(l)) * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 1e+190], N[ArcSin[N[Sqrt[N[(N[(Omc - N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] / N[(Omc * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 / N[(N[Abs[t], $MachinePrecision] / (-N[Sqrt[N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.0000000000000001e190

   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 51.8%

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

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.7%

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7%

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

   6. Applied rewrites 48.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lift-/.f64 N/A

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

      10. sub-to-fraction N/A

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

      11. associate-/l/ N/A

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

      12. lower-/.f64 N/A

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

   8. Applied rewrites 51.8%

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

   if 1.0000000000000001e190 < (/.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 t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 21.1%

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

   4. Applied rewrites 21.1%

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

   5. Taylor expanded in Om around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 23.4%

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

   7. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-neg.f64 23.3%

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

   9. Applied rewrites 23.3%

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

Alternative 9: 60.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{\left|t\right|}{\left|\ell\right|} \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\frac{\left|t\right|}{-\sqrt{\left(\left|\ell\right| \cdot \left|\ell\right|\right) \cdot 0.5}}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (/ (fabs t) (fabs l)) 5e+179)
   (asin (sqrt (/ (- 1.0 (* Om (/ Om (* Omc Omc)))) 1.0)))
   (asin (/ 1.0 (/ (fabs t) (- (sqrt (* (* (fabs l) (fabs l)) 0.5))))))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((fabs(t) / fabs(l)) <= 5e+179) {
		tmp = asin(sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)));
	} else {
		tmp = asin((1.0 / (fabs(t) / -sqrt(((fabs(l) * fabs(l)) * 0.5)))));
	}
	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)) <= 5d+179) then
        tmp = asin(sqrt(((1.0d0 - (om * (om / (omc * omc)))) / 1.0d0)))
    else
        tmp = asin((1.0d0 / (abs(t) / -sqrt(((abs(l) * abs(l)) * 0.5d0)))))
    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)) <= 5e+179) {
		tmp = Math.asin(Math.sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)));
	} else {
		tmp = Math.asin((1.0 / (Math.abs(t) / -Math.sqrt(((Math.abs(l) * Math.abs(l)) * 0.5)))));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (math.fabs(t) / math.fabs(l)) <= 5e+179:
		tmp = math.asin(math.sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)))
	else:
		tmp = math.asin((1.0 / (math.fabs(t) / -math.sqrt(((math.fabs(l) * math.fabs(l)) * 0.5)))))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(abs(t) / abs(l)) <= 5e+179)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc)))) / 1.0)));
	else
		tmp = asin(Float64(1.0 / Float64(abs(t) / Float64(-sqrt(Float64(Float64(abs(l) * abs(l)) * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((abs(t) / abs(l)) <= 5e+179)
		tmp = asin(sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)));
	else
		tmp = asin((1.0 / (abs(t) / -sqrt(((abs(l) * abs(l)) * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 5e+179], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(1.0 / N[(N[Abs[t], $MachinePrecision] / (-N[Sqrt[N[(N[(N[Abs[l], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5e179

   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 51.8%

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

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 48.7%

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 48.7%

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

   6. Applied rewrites 48.7%

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

   if 5e179 < (/.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 t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 21.1%

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

   4. Applied rewrites 21.1%

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

   5. Taylor expanded in Om around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 23.4%

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

   7. Applied rewrites 23.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. lower-unsound-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lower-neg.f64 23.3%

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

   9. Applied rewrites 23.3%

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

Alternative 10: 23.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double t, double l, double Om, double Omc) {
	return asin((1.0 / (t / -sqrt(((l * l) * 0.5)))));
}

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((1.0d0 / (t / -sqrt(((l * l) * 0.5d0)))))
end function

Java

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

Python

def code(t, l, Om, Omc):
	return math.asin((1.0 / (t / -math.sqrt(((l * l) * 0.5)))))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(1.0 / Float64(t / Float64(-sqrt(Float64(Float64(l * l) * 0.5))))))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin((1.0 / (t / -sqrt(((l * l) * 0.5)))));
end

Wolfram

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

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 21.1%

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

4. Applied rewrites 21.1%

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

5. Taylor expanded in Om around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 23.4%

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

7. Applied rewrites 23.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-flip N/A

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

   5. lower-unsound-/.f64 N/A

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

   6. mul-1-neg N/A

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

   7. lower-unsound-/.f64 N/A

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

   8. lower-neg.f64 23.3%

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

9. Applied rewrites 23.3%

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

Alternative 11: 23.3% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (asin (/ (sqrt (* (* l l) 0.5)) (- t))))

C

double code(double t, double l, double Om, double Omc) {
	return asin((sqrt(((l * l) * 0.5)) / -t));
}

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(((l * l) * 0.5d0)) / -t))
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	return Math.asin((Math.sqrt(((l * l) * 0.5)) / -t));
}

Python

def code(t, l, Om, Omc):
	return math.asin((math.sqrt(((l * l) * 0.5)) / -t))

Julia

function code(t, l, Om, Omc)
	return asin(Float64(sqrt(Float64(Float64(l * l) * 0.5)) / Float64(-t)))
end

MATLAB

function tmp = code(t, l, Om, Omc)
	tmp = asin((sqrt(((l * l) * 0.5)) / -t));
end

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[Sqrt[N[(N[(l * l), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / (-t)), $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 -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 21.1%

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

4. Applied rewrites 21.1%

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

5. Taylor expanded in Om around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 23.4%

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

7. Applied rewrites 23.4%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

9. Applied rewrites 23.4%

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

Reproduce

herbie shell --seed 2025183 
(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)))))))