Toniolo and Linder, Equation (2)

Percentage Accurate: 83.6% → 98.8%

Time: 8.0s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{0.5}}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot \frac{1}{{t\_1}^{-2}}}}\right)\\ \end{array} \]

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)))))) 0.0)
     (asin (/ (* (fabs l) (sqrt 0.5)) (fabs t)))
     (asin (sqrt (/ t_2 (+ 1.0 (* 2.0 (/ 1.0 (pow t_1 -2.0))))))))))

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)))))) <= 0.0) {
		tmp = asin(((fabs(l) * sqrt(0.5)) / fabs(t)));
	} else {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * (1.0 / pow(t_1, -2.0)))))));
	}
	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) :: 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)))))) <= 0.0d0) then
        tmp = asin(((abs(l) * sqrt(0.5d0)) / abs(t)))
    else
        tmp = asin(sqrt((t_2 / (1.0d0 + (2.0d0 * (1.0d0 / (t_1 ** (-2.0d0))))))))
    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)))))) <= 0.0) {
		tmp = Math.asin(((Math.abs(l) * Math.sqrt(0.5)) / Math.abs(t)));
	} else {
		tmp = Math.asin(Math.sqrt((t_2 / (1.0 + (2.0 * (1.0 / Math.pow(t_1, -2.0)))))));
	}
	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)))))) <= 0.0:
		tmp = math.asin(((math.fabs(l) * math.sqrt(0.5)) / math.fabs(t)))
	else:
		tmp = math.asin(math.sqrt((t_2 / (1.0 + (2.0 * (1.0 / math.pow(t_1, -2.0)))))))
	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)))))) <= 0.0)
		tmp = asin(Float64(Float64(abs(l) * sqrt(0.5)) / abs(t)));
	else
		tmp = asin(sqrt(Float64(t_2 / Float64(1.0 + Float64(2.0 * Float64(1.0 / (t_1 ^ -2.0)))))));
	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)))))) <= 0.0)
		tmp = asin(((abs(l) * sqrt(0.5)) / abs(t)));
	else
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * (1.0 / (t_1 ^ -2.0)))))));
	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], 0.0], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[(1.0 / N[Power[t$95$1, -2.0], $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))))))) < 0.0

   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 71.9%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 25.2%

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

   6. Applied rewrites 25.2%

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

   7. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{-1}{2} \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{t}\right) \]
   8. 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 32.6%

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

   9. Applied rewrites 32.6%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 32.4%

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

   if 0.0 < (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. remove-double-neg N/A

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

      3. pow-neg N/A

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

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

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

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

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

      6. metadata-eval 83.6%

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

   3. Applied rewrites 83.6%

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

Alternative 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{0.5}}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{\mathsf{fma}\left(\frac{\left|t\right| + \left|t\right|}{\left|\ell\right|}, t\_1, 1\right)}}\right)\\ \end{array} \]

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)))))) 0.0)
     (asin (/ (* (fabs l) (sqrt 0.5)) (fabs t)))
     (asin (sqrt (/ t_2 (fma (/ (+ (fabs t) (fabs t)) (fabs l)) t_1 1.0)))))))

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)))))) <= 0.0) {
		tmp = asin(((fabs(l) * sqrt(0.5)) / fabs(t)));
	} else {
		tmp = asin(sqrt((t_2 / fma(((fabs(t) + fabs(t)) / fabs(l)), t_1, 1.0))));
	}
	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)))))) <= 0.0)
		tmp = asin(Float64(Float64(abs(l) * sqrt(0.5)) / abs(t)));
	else
		tmp = asin(sqrt(Float64(t_2 / fma(Float64(Float64(abs(t) + abs(t)) / abs(l)), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(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], 0.0], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.0

   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 71.9%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 25.2%

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

   6. Applied rewrites 25.2%

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

   7. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{-1}{2} \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{t}\right) \]
   8. 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 32.6%

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

   9. Applied rewrites 32.6%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 32.4%

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

   if 0.0 < (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. 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%

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

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 5 \cdot 10^{-63}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{0.5}}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot \left(\left(-\left|t\right|\right) \cdot \frac{t\_1}{-\left|\ell\right|}\right)}}\right)\\ \end{array} \]

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)))))) 5e-63)
     (asin (/ (* (fabs l) (sqrt 0.5)) (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)))))) <= 5e-63) {
		tmp = asin(((fabs(l) * sqrt(0.5)) / 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)))))) <= 5d-63) then
        tmp = asin(((abs(l) * sqrt(0.5d0)) / 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)))))) <= 5e-63) {
		tmp = Math.asin(((Math.abs(l) * Math.sqrt(0.5)) / 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)))))) <= 5e-63:
		tmp = math.asin(((math.fabs(l) * math.sqrt(0.5)) / 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)))))) <= 5e-63)
		tmp = asin(Float64(Float64(abs(l) * sqrt(0.5)) / 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)))))) <= 5e-63)
		tmp = asin(((abs(l) * sqrt(0.5)) / 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], 5e-63], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[0.5], $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))))))) < 5.0000000000000002e-63

   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 71.9%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 25.2%

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

   6. Applied rewrites 25.2%

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

   7. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{-1}{2} \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{t}\right) \]
   8. 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 32.6%

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

   9. Applied rewrites 32.6%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 32.4%

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

   if 5.0000000000000002e-63 < (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 80.5%

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

   3. Applied rewrites 80.5%

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

Alternative 4: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ \mathbf{if}\;1 + 2 \cdot {t\_1}^{2} \leq 10^{+173}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{Om}{Omc} - -1\right) \cdot \frac{\frac{Om}{Omc} - 1}{\mathsf{fma}\left(\frac{-2 \cdot \left|t\right|}{\left|\ell\right|}, t\_1, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{0.5}}{\left|t\right|}\right)\\ \end{array} \]

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))) 1e+173)
     (asin
      (sqrt
       (*
        (- (/ Om Omc) -1.0)
        (/ (- (/ Om Omc) 1.0) (fma (/ (* -2.0 (fabs t)) (fabs l)) t_1 -1.0)))))
     (asin (/ (* (fabs l) (sqrt 0.5)) (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))) <= 1e+173) {
		tmp = asin(sqrt((((Om / Omc) - -1.0) * (((Om / Omc) - 1.0) / fma(((-2.0 * fabs(t)) / fabs(l)), t_1, -1.0)))));
	} else {
		tmp = asin(((fabs(l) * sqrt(0.5)) / 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))) <= 1e+173)
		tmp = asin(sqrt(Float64(Float64(Float64(Om / Omc) - -1.0) * Float64(Float64(Float64(Om / Omc) - 1.0) / fma(Float64(Float64(-2.0 * abs(t)) / abs(l)), t_1, -1.0)))));
	else
		tmp = asin(Float64(Float64(abs(l) * sqrt(0.5)) / 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], 1e+173], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(Om / Omc), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(N[(-2.0 * N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[0.5], $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)))) < 1e173

   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 71.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 83.5%

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

   5. Applied rewrites 83.5%

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

   if 1e173 < (+.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 71.9%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 25.2%

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

   6. Applied rewrites 25.2%

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

   7. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{-1}{2} \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{t}\right) \]
   8. 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 32.6%

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

   9. Applied rewrites 32.6%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 32.4%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (1.0 + (2.0 * math.pow((math.fabs(t) / math.fabs(l)), 2.0))) <= 2.0:
		tmp = math.asin(math.sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)))
	else:
		tmp = math.asin(((math.fabs(l) * math.sqrt(0.5)) / math.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))) <= 2.0)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / 1.0)));
	else
		tmp = asin(Float64(Float64(abs(l) * sqrt(0.5)) / abs(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((1.0 + (2.0 * ((abs(t) / abs(l)) ^ 2.0))) <= 2.0)
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	else
		tmp = asin(((abs(l) * sqrt(0.5)) / abs(t)));
	end
	tmp_2 = 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], 2.0], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[0.5], $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)))) < 2

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.0%

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

   6. Applied rewrites 50.0%

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

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

   1. Initial program 83.6%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 71.9%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 25.2%

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

   6. Applied rewrites 25.2%

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

   7. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{-1}{2} \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{t}\right) \]
   8. 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 32.6%

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

   9. Applied rewrites 32.6%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 32.4%

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

Alternative 6: 94.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{\left|t\right|}{\left|\ell\right|}\right)}^{2} \leq 1.000002:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{0.5}}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ (fabs t) (fabs l)) 2.0))) 1.000002)
   (asin (sqrt (/ (- 1.0 (* Om (/ Om (* Omc Omc)))) 1.0)))
   (asin (/ (* (fabs l) (sqrt 0.5)) (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))) <= 1.000002) {
		tmp = asin(sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)));
	} else {
		tmp = asin(((fabs(l) * sqrt(0.5)) / fabs(t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((1.0d0 + (2.0d0 * ((abs(t) / abs(l)) ** 2.0d0))) <= 1.000002d0) then
        tmp = asin(sqrt(((1.0d0 - (om * (om / (omc * omc)))) / 1.0d0)))
    else
        tmp = asin(((abs(l) * sqrt(0.5d0)) / abs(t)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if ((1.0 + (2.0 * Math.pow((Math.abs(t) / Math.abs(l)), 2.0))) <= 1.000002) {
		tmp = Math.asin(Math.sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)));
	} else {
		tmp = Math.asin(((Math.abs(l) * Math.sqrt(0.5)) / Math.abs(t)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if (1.0 + (2.0 * math.pow((math.fabs(t) / math.fabs(l)), 2.0))) <= 1.000002:
		tmp = math.asin(math.sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)))
	else:
		tmp = math.asin(((math.fabs(l) * math.sqrt(0.5)) / math.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))) <= 1.000002)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc)))) / 1.0)));
	else
		tmp = asin(Float64(Float64(abs(l) * sqrt(0.5)) / abs(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if ((1.0 + (2.0 * ((abs(t) / abs(l)) ^ 2.0))) <= 1.000002)
		tmp = asin(sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) / 1.0)));
	else
		tmp = asin(((abs(l) * sqrt(0.5)) / abs(t)));
	end
	tmp_2 = 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], 1.000002], 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[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[0.5], $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)))) < 1.00000200000000006

   1. Initial program 83.6%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 50.0%

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

   6. Applied rewrites 46.9%

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

   if 1.00000200000000006 < (+.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 71.9%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 25.2%

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

   6. Applied rewrites 25.2%

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

   7. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{-1}{2} \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{t}\right) \]
   8. 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 32.6%

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

   9. Applied rewrites 32.6%

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

   10. Taylor expanded in Om around 0

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

   12. Applied rewrites 32.4%

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

Alternative 7: 50.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

double code(double t, double l, double Om, double Omc) {
	return asin(((fabs(l) * sqrt(0.5)) / fabs(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(((abs(l) * sqrt(0.5d0)) / abs(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $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. 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 71.9%

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

4. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-/.f64 25.2%

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

6. Applied rewrites 25.2%

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

7. Taylor expanded in l around 0

   \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{-1}{2} \cdot \left(\left(1 + \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} - 1\right)\right)}}{t}\right) \]
8. 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 32.6%

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

9. Applied rewrites 32.6%

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

10. Taylor expanded in Om around 0

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

12. Applied rewrites 32.4%

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

Reproduce

herbie shell --seed 2025184 
(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)))))))