Toniolo and Linder, Equation (2)

Percentage Accurate: 84.3% → 97.4%

Time: 9.0s

Alternatives: 10

Speedup: 0.9×

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: 84.3% 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: 97.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (pow (/ (fabs t) l) 2.0)))))) 0.0)
     (asin
      (/
       (* (fabs l) (sqrt (* (- 1.0 (* (/ Om (* Omc Omc)) Om)) 0.5)))
       (fabs t)))
     (asin
      (sqrt
       (/
        t_1
        (fma (/ (+ (fabs t) (fabs t)) l) (/ 1.0 (/ l (fabs t))) 1.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if (asin(sqrt((t_1 / (1.0 + (2.0 * pow((fabs(t) / l), 2.0)))))) <= 0.0) {
		tmp = asin(((fabs(l) * sqrt(((1.0 - ((Om / (Omc * Omc)) * Om)) * 0.5))) / fabs(t)));
	} else {
		tmp = asin(sqrt((t_1 / fma(((fabs(t) + fabs(t)) / l), (1.0 / (l / fabs(t))), 1.0))));
	}
	return tmp;
}

Julia

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

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[Power[N[(N[Abs[t], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(1.0 - N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision] * N[(1.0 / N[(l / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.5%

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

   4. Applied rewrites 21.5%

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

   6. Applied rewrites 29.8%

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

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

   1. Initial program 84.3%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 84.3%

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

   3. Applied rewrites 84.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 84.3%

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

   5. Applied rewrites 84.3%

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

Alternative 2: 97.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{\left(1 - \frac{Om}{Omc \cdot Omc} \cdot Om\right) \cdot 0.5}}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{\mathsf{fma}\left(\frac{\left|t\right| + \left|t\right|}{\ell}, t\_1, 1\right)}}\right)\\ \end{array} \]

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)))))) 2e-144)
     (asin
      (/
       (* (fabs l) (sqrt (* (- 1.0 (* (/ Om (* Omc Omc)) Om)) 0.5)))
       (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)))))) <= 2e-144) {
		tmp = asin(((fabs(l) * sqrt(((1.0 - ((Om / (Omc * Omc)) * Om)) * 0.5))) / 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)))))) <= 2e-144)
		tmp = asin(Float64(Float64(abs(l) * sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Float64(Omc * Omc)) * Om)) * 0.5))) / 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], 2e-144], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(1.0 - N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $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))))))) < 1.9999999999999999e-144

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.5%

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

   4. Applied rewrites 21.5%

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

   6. Applied rewrites 29.8%

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

   if 1.9999999999999999e-144 < (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 84.3%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 84.3%

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

   3. Applied rewrites 84.3%

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

Alternative 3: 96.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\ell}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_2}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\sin^{-1} \left(\frac{\left|\ell\right| \cdot \sqrt{\left(1 - \frac{Om}{Omc \cdot Omc} \cdot Om\right) \cdot 0.5}}{\left|t\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_2}{\mathsf{fma}\left(\left|t\right| + \left|t\right|, \frac{t\_1}{\ell}, 1\right)}}\right)\\ \end{array} \]

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)))))) 2e-56)
     (asin
      (/
       (* (fabs l) (sqrt (* (- 1.0 (* (/ Om (* Omc Omc)) Om)) 0.5)))
       (fabs t)))
     (asin (sqrt (/ t_2 (fma (+ (fabs t) (fabs t)) (/ t_1 l) 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)))))) <= 2e-56) {
		tmp = asin(((fabs(l) * sqrt(((1.0 - ((Om / (Omc * Omc)) * Om)) * 0.5))) / fabs(t)));
	} else {
		tmp = asin(sqrt((t_2 / fma((fabs(t) + fabs(t)), (t_1 / l), 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)))))) <= 2e-56)
		tmp = asin(Float64(Float64(abs(l) * sqrt(Float64(Float64(1.0 - Float64(Float64(Om / Float64(Omc * Omc)) * Om)) * 0.5))) / abs(t)));
	else
		tmp = asin(sqrt(Float64(t_2 / fma(Float64(abs(t) + abs(t)), Float64(t_1 / l), 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], 2e-56], N[ArcSin[N[(N[(N[Abs[l], $MachinePrecision] * N[Sqrt[N[(N[(1.0 - N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision] + 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))))))) < 2.0000000000000001e-56

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.5%

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

   4. Applied rewrites 21.5%

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

   6. Applied rewrites 29.8%

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

   if 2.0000000000000001e-56 < (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 84.3%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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 84.3%

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

   3. Applied rewrites 84.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow-flip N/A

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

      7. lift-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lift-/.f64 N/A

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

      13. associate-/l* N/A

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

      14. count-2 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-*l/ N/A

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

      17. associate-/l* N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 81.0%

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

   5. Applied rewrites 81.0%

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

Alternative 4: 95.0% accurate, 0.9× speedup

Math

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

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 (pow (/ Om Omc) 2.0)) 1.0)))
   (asin
    (/
     (* (fabs l) (sqrt (* (- 1.0 (* (/ Om (* Omc Omc)) Om)) 0.5)))
     (fabs t)))))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 51.2%

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

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

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.5%

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

   4. Applied rewrites 21.5%

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

   6. Applied rewrites 29.8%

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

Alternative 5: 92.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t, l, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = om / (omc * omc)
    if ((1.0d0 + (2.0d0 * ((abs(t) / l) ** 2.0d0))) <= 2.0d0) then
        tmp = asin(sqrt(((1.0d0 - (om * t_1)) / 1.0d0)))
    else
        tmp = asin(((abs(l) * sqrt(((1.0d0 - (t_1 * om)) * 0.5d0))) / abs(t)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	t_1 = Om / (Omc * Omc);
	tmp = 0.0;
	if ((1.0 + (2.0 * ((abs(t) / l) ^ 2.0))) <= 2.0)
		tmp = asin(sqrt(((1.0 - (Om * t_1)) / 1.0)));
	else
		tmp = asin(((abs(l) * sqrt(((1.0 - (t_1 * Om)) * 0.5))) / abs(t)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 51.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. +-commutative N/A

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

      15. sub-flip N/A

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

      16. lift--.f64 45.4%

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

      17. *-lft-identity N/A

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

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

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

      19. lower-unsound-/.f32 N/A

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

      20. lower-/.f32 N/A

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

      21. lift-pow.f64 N/A

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

      22. unpow2 N/A

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

      23. associate-/l* N/A

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

      24. lower-*.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. *-lft-identity 48.4%

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

      27. lift-pow.f64 N/A

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

      28. unpow2 N/A

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

      29. lower-*.f64 48.4%

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

   6. Applied rewrites 48.4%

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

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

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.5%

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

   4. Applied rewrites 21.5%

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

   6. Applied rewrites 29.8%

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

Alternative 6: 80.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1

   1. Initial program 84.3%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 51.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. +-commutative N/A

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

      15. sub-flip N/A

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

      16. lift--.f64 45.4%

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

      17. *-lft-identity N/A

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

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

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

      19. lower-unsound-/.f32 N/A

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

      20. lower-/.f32 N/A

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

      21. lift-pow.f64 N/A

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

      22. unpow2 N/A

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

      23. associate-/l* N/A

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

      24. lower-*.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. *-lft-identity 48.4%

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

      27. lift-pow.f64 N/A

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

      28. unpow2 N/A

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

      29. lower-*.f64 48.4%

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

   6. Applied rewrites 48.4%

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

   if 1 < (/.f64 t l)

   1. Initial program 84.3%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 21.5%

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

   4. Applied rewrites 21.5%

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

   5. Taylor expanded in Om around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 24.2%

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

   7. Applied rewrites 24.2%

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

Alternative 7: 60.7% accurate, 0.7× speedup

Math

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

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) l) (- t))))
   (asin (sqrt (/ (- 1.0 (* Om (/ Om (* Omc Omc)))) 1.0)))))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_, l_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t / 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] * l), $MachinePrecision] / (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $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 84.3%

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

   2. Taylor expanded in t around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 21.1%

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

   4. Applied rewrites 21.1%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 27.9%

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

   7. Applied rewrites 27.9%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 31.4%

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

       1. lift-asin.f64 N/A

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

       2. asin-acos N/A

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

       3. lower--.f64 N/A

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

       4. mult-flip N/A

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

       5. metadata-eval N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 N/A

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

   12. Applied rewrites 14.6%

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

   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 84.3%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 51.2%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. frac-times N/A

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

      9. unpow2 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. +-commutative N/A

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

      15. sub-flip N/A

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

      16. lift--.f64 45.4%

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

      17. *-lft-identity N/A

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

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

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

      19. lower-unsound-/.f32 N/A

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

      20. lower-/.f32 N/A

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

      21. lift-pow.f64 N/A

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

      22. unpow2 N/A

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

      23. associate-/l* N/A

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

      24. lower-*.f64 N/A

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

      25. lower-/.f64 N/A

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

      26. *-lft-identity 48.4%

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

      27. lift-pow.f64 N/A

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

      28. unpow2 N/A

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

      29. lower-*.f64 48.4%

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

   6. Applied rewrites 48.4%

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

Alternative 8: 31.0% 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 84.3%

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

2. Taylor expanded in t around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 21.1%

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

4. Applied rewrites 21.1%

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

5. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 27.9%

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

7. Applied rewrites 27.9%

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

8. Taylor expanded in Om around 0

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

10. Applied rewrites 31.4%

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

    1. lift-asin.f64 N/A

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

    2. asin-acos N/A

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

    3. lower--.f64 N/A

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

    4. mult-flip N/A

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

    5. metadata-eval N/A

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

    6. lower-*.f64 N/A

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

    7. lower-PI.f64 N/A

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

12. Applied rewrites 14.6%

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

Alternative 9: 23.8% 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 84.3%

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

2. Taylor expanded in t around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 21.1%

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

4. Applied rewrites 21.1%

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

5. Taylor expanded in Om around 0

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 23.8%

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

7. Applied rewrites 23.8%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lower-/.f64 N/A

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

9. Applied rewrites 23.8%

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

Alternative 10: 14.7% accurate, 3.6× speedup

Math

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

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(Float64(-Float64(sqrt(0.5) * abs(l))) / 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[(N[Sqrt[0.5], $MachinePrecision] * N[Abs[l], $MachinePrecision]), $MachinePrecision]) / t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.3%

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

2. Taylor expanded in t around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 21.1%

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

4. Applied rewrites 21.1%

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

5. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 27.9%

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

7. Applied rewrites 27.9%

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

8. Taylor expanded in Om around 0

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

10. Applied rewrites 31.4%

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

    1. lift-*.f64 N/A

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

    2. mul-1-neg N/A

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

    3. lift-/.f64 N/A

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

    4. distribute-neg-frac N/A

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

    5. lower-/.f64 N/A

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

12. Applied rewrites 31.4%

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

Reproduce

herbie shell --seed 2025192 
(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)))))))