Toniolo and Linder, Equation (2)

Percentage Accurate: 84.4% → 97.2%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

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: 84.4% 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: 97.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (fabs t) 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
      (/
       (* (sqrt (* -0.5 (fma (/ Om (* Omc Omc)) Om -1.0))) (fabs l))
       (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) / 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(((sqrt((-0.5 * fma((Om / (Omc * Omc)), Om, -1.0))) * fabs(l)) / fabs(t)));
	} else {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * (1.0 / pow(t_1, -2.0)))))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / 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(sqrt(Float64(-0.5 * fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0))) * abs(l)) / 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

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $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[Sqrt[N[(-0.5 * N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $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 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.0%

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

   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 23.8%

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

   6. Applied rewrites 23.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-sqrt.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. lift-fabs.f64 N/A

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

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

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

   8. Applied rewrites 28.5%

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

   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 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 84.4%

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

   3. Applied rewrites 84.4%

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

Alternative 2: 97.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{-0.5 \cdot \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)} \cdot \left|\ell\right|}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{\mathsf{fma}\left(\frac{\left|t\right| + \left|t\right|}{\ell}, t\_1, 1\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (fabs t) 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-151)
     (asin
      (/
       (* (sqrt (* -0.5 (fma (/ Om (* Omc Omc)) Om -1.0))) (fabs l))
       (fabs t)))
     (asin (sqrt (/ t_2 (fma (/ (+ (fabs t) (fabs t)) l) t_1 1.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / 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-151) {
		tmp = asin(((sqrt((-0.5 * fma((Om / (Omc * Omc)), Om, -1.0))) * fabs(l)) / fabs(t)));
	} else {
		tmp = asin(sqrt((t_2 / fma(((fabs(t) + fabs(t)) / l), t_1, 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / 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-151)
		tmp = asin(Float64(Float64(sqrt(Float64(-0.5 * fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0))) * abs(l)) / abs(t)));
	else
		tmp = asin(sqrt(Float64(t_2 / fma(Float64(Float64(abs(t) + abs(t)) / l), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $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-151], N[ArcSin[N[(N[(N[Sqrt[N[(-0.5 * N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $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] / l), $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))))))) < 5.00000000000000003e-151

   1. Initial program 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.0%

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

   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 23.8%

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

   6. Applied rewrites 23.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-sqrt.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. lift-fabs.f64 N/A

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

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

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

   8. Applied rewrites 28.5%

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

   if 5.00000000000000003e-151 < (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 84.4%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. distribute-rgt-out N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 84.4%

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

Alternative 3: 97.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 5 \cdot 10^{-151}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{-0.5 \cdot \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)} \cdot \left|\ell\right|}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{Om}{Omc} - -1\right) \cdot \frac{\frac{Om}{Omc} - 1}{\mathsf{fma}\left(\frac{-2 \cdot \left|t\right|}{\ell}, t\_1, -1\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (fabs t) l)))
   (if (<=
        (asin
         (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow t_1 2.0))))))
        5e-151)
     (asin
      (/
       (* (sqrt (* -0.5 (fma (/ Om (* Omc Omc)) Om -1.0))) (fabs l))
       (fabs t)))
     (asin
      (sqrt
       (*
        (- (/ Om Omc) -1.0)
        (/ (- (/ Om Omc) 1.0) (fma (/ (* -2.0 (fabs t)) l) t_1 -1.0))))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / l;
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow(t_1, 2.0)))))) <= 5e-151) {
		tmp = asin(((sqrt((-0.5 * fma((Om / (Omc * Omc)), Om, -1.0))) * fabs(l)) / fabs(t)));
	} else {
		tmp = asin(sqrt((((Om / Omc) - -1.0) * (((Om / Omc) - 1.0) / fma(((-2.0 * fabs(t)) / l), t_1, -1.0)))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (t_1 ^ 2.0)))))) <= 5e-151)
		tmp = asin(Float64(Float64(sqrt(Float64(-0.5 * fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0))) * abs(l)) / abs(t)));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(Om / Omc) - -1.0) * Float64(Float64(Float64(Om / Omc) - 1.0) / fma(Float64(Float64(-2.0 * abs(t)) / l), t_1, -1.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 5e-151], N[ArcSin[N[(N[(N[Sqrt[N[(-0.5 * N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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] / l), $MachinePrecision] * t$95$1 + -1.0), $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))))))) < 5.00000000000000003e-151

   1. Initial program 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.0%

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

   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 23.8%

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

   6. Applied rewrites 23.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-sqrt.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. lift-fabs.f64 N/A

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

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

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

   8. Applied rewrites 28.5%

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

   if 5.00000000000000003e-151 < (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 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.0%

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

   5. Applied rewrites 84.3%

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

Alternative 4: 95.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ \mathbf{if}\;1 + 2 \cdot {t\_1}^{2} \leq 10^{+122}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(\frac{Om}{Omc} - -1\right) \cdot \frac{\frac{Om}{Omc} - 1}{\mathsf{fma}\left(-2 \cdot \left|t\right|, \frac{t\_1}{\ell}, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{-0.5 \cdot \mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)} \cdot \left|\ell\right|}{\left|t\right|}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (/ (fabs t) l)))
   (if (<= (+ 1.0 (* 2.0 (pow t_1 2.0))) 1e+122)
     (asin
      (sqrt
       (*
        (- (/ Om Omc) -1.0)
        (/ (- (/ Om Omc) 1.0) (fma (* -2.0 (fabs t)) (/ t_1 l) -1.0)))))
     (asin
      (/
       (* (sqrt (* -0.5 (fma (/ Om (* Omc Omc)) Om -1.0))) (fabs l))
       (fabs t))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / l;
	double tmp;
	if ((1.0 + (2.0 * pow(t_1, 2.0))) <= 1e+122) {
		tmp = asin(sqrt((((Om / Omc) - -1.0) * (((Om / Omc) - 1.0) / fma((-2.0 * fabs(t)), (t_1 / l), -1.0)))));
	} else {
		tmp = asin(((sqrt((-0.5 * fma((Om / (Omc * Omc)), Om, -1.0))) * fabs(l)) / fabs(t)));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / l)
	tmp = 0.0
	if (Float64(1.0 + Float64(2.0 * (t_1 ^ 2.0))) <= 1e+122)
		tmp = asin(sqrt(Float64(Float64(Float64(Om / Omc) - -1.0) * Float64(Float64(Float64(Om / Omc) - 1.0) / fma(Float64(-2.0 * abs(t)), Float64(t_1 / l), -1.0)))));
	else
		tmp = asin(Float64(Float64(sqrt(Float64(-0.5 * fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0))) * abs(l)) / abs(t)));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[N[(1.0 + N[(2.0 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+122], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[(N[(Om / Omc), $MachinePrecision] - 1.0), $MachinePrecision] / N[(N[(-2.0 * N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[N[(-0.5 * N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $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)))) < 1.00000000000000001e122

   1. Initial program 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 81.5%

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

   5. Applied rewrites 81.5%

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

   if 1.00000000000000001e122 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.0%

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

   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 23.8%

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

   6. Applied rewrites 23.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-sqrt.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. lift-fabs.f64 N/A

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

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

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

   8. Applied rewrites 28.5%

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

Alternative 5: 94.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (Float64(1.0 + Float64(2.0 * (Float64(abs(t) / 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(sqrt(Float64(-0.5 * fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0))) * abs(l)) / abs(t)));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(N[Abs[t], $MachinePrecision] / l), $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[Sqrt[N[(-0.5 * N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[l], $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 84.4%

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

   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 52.3%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\color{blue}{1}}}\right) \]
   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 52.3%

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

   6. Applied rewrites 52.3%

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

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

   1. Initial program 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 73.0%

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

   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 23.8%

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

   6. Applied rewrites 23.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lower-unsound-sqrt.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

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

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

      12. lift-fabs.f64 N/A

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

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

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

   8. Applied rewrites 28.5%

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

Alternative 6: 64.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
      0.0)
   (- (* PI 0.5) (acos (/ (* (sqrt 0.5) (fabs l)) (- t))))
   (asin (sqrt (/ (- 1.0 (/ (* (/ Om Omc) Om) Omc)) 1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0)))))) <= 0.0) {
		tmp = (((double) M_PI) * 0.5) - acos(((sqrt(0.5) * fabs(l)) / -t));
	} else {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	}
	return tmp;
}

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t / l), 2.0)))))) <= 0.0) {
		tmp = (Math.PI * 0.5) - Math.acos(((Math.sqrt(0.5) * Math.abs(l)) / -t));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t / l), 2.0)))))) <= 0.0:
		tmp = (math.pi * 0.5) - math.acos(((math.sqrt(0.5) * math.fabs(l)) / -t))
	else:
		tmp = math.asin(math.sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0], N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.4%

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

   2. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 20.7%

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

   4. Applied rewrites 20.7%

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

   6. Applied rewrites 28.8%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 30.4%

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

   11. Applied rewrites 14.0%

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

   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 84.4%

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

   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 52.3%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\color{blue}{1}}}\right) \]
   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 52.3%

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

   6. Applied rewrites 52.3%

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

Alternative 7: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \leq 10^{-146}:\\ \;\;\;\;\pi \cdot 0.5 - \cos^{-1} \left(\frac{\sqrt{0.5} \cdot \left|\ell\right|}{-t}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}{-1}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0))))))
      1e-146)
   (- (* PI 0.5) (acos (/ (* (sqrt 0.5) (fabs l)) (- t))))
   (asin (sqrt (/ (fma (/ Om (* Omc Omc)) Om -1.0) -1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0)))))) <= 1e-146) {
		tmp = (((double) M_PI) * 0.5) - acos(((sqrt(0.5) * fabs(l)) / -t));
	} else {
		tmp = asin(sqrt((fma((Om / (Omc * Omc)), Om, -1.0) / -1.0)));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t / l) ^ 2.0)))))) <= 1e-146)
		tmp = Float64(Float64(pi * 0.5) - acos(Float64(Float64(sqrt(0.5) * abs(l)) / Float64(-t))));
	else
		tmp = asin(sqrt(Float64(fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0) / -1.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-146], N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcCos[N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision] / (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 1.00000000000000003e-146

   1. Initial program 84.4%

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

   2. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 20.7%

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

   4. Applied rewrites 20.7%

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

   6. Applied rewrites 28.8%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 30.4%

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

   11. Applied rewrites 14.0%

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

   if 1.00000000000000003e-146 < (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 84.4%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. metadata-eval N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. lift-/.f64 N/A

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

      11. mult-flip N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. frac-times N/A

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

      17. *-commutative N/A

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

      18. *-rgt-identity N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

   3. Applied rewrites 68.8%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 49.3%

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

Alternative 8: 30.4% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (- (* PI 0.5) (acos (/ (* (sqrt 0.5) (fabs l)) (- t)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

2. Taylor expanded in t around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 20.7%

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

4. Applied rewrites 20.7%

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

6. Applied rewrites 28.8%

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

7. Taylor expanded in Om around 0

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

9. Applied rewrites 30.4%

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

11. Applied rewrites 14.0%

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

Alternative 9: 23.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin((sqrt(((l * l) * 0.5d0)) / -t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

2. Taylor expanded in t around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 20.7%

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

4. Applied rewrites 20.7%

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

6. Applied rewrites 28.8%

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

7. Taylor expanded in Om around 0

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

9. Applied rewrites 30.4%

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

    1. lift-*.f64 N/A

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

    2. lift-fabs.f64 N/A

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

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

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

    4. pow2 N/A

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

    5. lift-pow.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. sqrt-unprod N/A

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

    8. lower-sqrt.f64 N/A

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

    9. lower-*.f64 23.4%

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

    10. lift-pow.f64 N/A

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

    11. pow2 N/A

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

    12. lower-*.f64 23.4%

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

11. Applied rewrites 23.4%

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

Alternative 10: 14.0% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin((sqrt(0.5d0) * (abs(l) / -t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

2. Taylor expanded in t around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 20.7%

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

4. Applied rewrites 20.7%

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

6. Applied rewrites 28.8%

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

7. Taylor expanded in Om around 0

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

9. Applied rewrites 30.4%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. *-commutative N/A

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

    4. associate-/l* N/A

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

    5. lower-*.f64 N/A

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

    6. lower-/.f64 30.4%

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

11. Applied rewrites 30.4%

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

Reproduce

herbie shell --seed 2025179 
(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)))))))