Toniolo and Linder, Equation (2)

Percentage Accurate: 84.1% → 98.9%

Time: 8.7s

Alternatives: 11

Speedup: 0.7×

Specification

Math

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

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: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(\frac{\left({0.5}^{0.25} \cdot {0.5}^{0.25}\right) \cdot l\_m}{t\_m} \cdot \sqrt{-\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0)))))) 0.0)
     (asin
      (*
       (/ (* (* (pow 0.5 0.25) (pow 0.5 0.25)) l_m) t_m)
       (sqrt (- (fma (/ Om Omc) (/ Om Omc) -1.0)))))
     (asin (sqrt (/ t_1 (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, 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((t_m / l_m), 2.0)))))) <= 0.0) {
		tmp = asin(((((pow(0.5, 0.25) * pow(0.5, 0.25)) * l_m) / t_m) * sqrt(-fma((Om / Omc), (Om / Omc), -1.0))));
	} else {
		tmp = asin(sqrt((t_1 / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, 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(t_m / l_m) ^ 2.0)))))) <= 0.0)
		tmp = asin(Float64(Float64(Float64(Float64((0.5 ^ 0.25) * (0.5 ^ 0.25)) * l_m) / t_m) * sqrt(Float64(-fma(Float64(Om / Omc), Float64(Om / Omc), -1.0)))));
	else
		tmp = asin(sqrt(Float64(t_1 / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, 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[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0], N[ArcSin[N[(N[(N[(N[(N[Power[0.5, 0.25], $MachinePrecision] * N[Power[0.5, 0.25], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[(-N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $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 57.4%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 76.9%

      \[\leadsto \sin^{-1} \left(\frac{\left({0.5}^{0.25} \cdot {0.5}^{0.25}\right) \cdot \ell}{t} \cdot \sqrt{-\mathsf{fma}\left(\color{blue}{\frac{Om}{Omc}}, \frac{Om}{Omc}, -1\right)}\right) \]

   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 98.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-146}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{--1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0)))))) 1e-146)
     (asin (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- -1.0))))
     (asin (sqrt (/ t_1 (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, 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((t_m / l_m), 2.0)))))) <= 1e-146) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt(-(-1.0))));
	} else {
		tmp = asin(sqrt((t_1 / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, 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(t_m / l_m) ^ 2.0)))))) <= 1e-146)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(-(-1.0)))));
	else
		tmp = asin(sqrt(Float64(t_1 / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, 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[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-146], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[(--1.0)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $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))))))) < 1.00000000000000003e-146

   1. Initial program 59.1%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 75.1%

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

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

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

Alternative 3: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right) \cdot \sqrt{-\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2 \cdot \frac{t\_m}{l\_m}}{l\_m}, t\_m, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.0001)
   (asin
    (* (* (/ l_m t_m) (sqrt 0.5)) (sqrt (- (fma (/ Om Omc) (/ Om Omc) -1.0)))))
   (asin (sqrt (/ 1.0 (fma (/ (* 2.0 (/ t_m l_m)) l_m) t_m 1.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.0001) {
		tmp = asin((((l_m / t_m) * sqrt(0.5)) * sqrt(-fma((Om / Omc), (Om / Omc), -1.0))));
	} else {
		tmp = asin(sqrt((1.0 / fma(((2.0 * (t_m / l_m)) / l_m), t_m, 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, 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_m / l_m) ^ 2.0)))))) <= 0.0001)
		tmp = asin(Float64(Float64(Float64(l_m / t_m) * sqrt(0.5)) * sqrt(Float64(-fma(Float64(Om / Omc), Float64(Om / Omc), -1.0)))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(Float64(2.0 * Float64(t_m / l_m)) / l_m), t_m, 1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, 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$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0001], N[ArcSin[N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$m + 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))))))) < 1.00000000000000005e-4

   1. Initial program 76.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 67.7%

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

   if 1.00000000000000005e-4 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 97.9%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 97.6%

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

Alternative 4: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot \sqrt{-\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2 \cdot \frac{t\_m}{l\_m}}{l\_m}, t\_m, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.0001)
   (asin
    (* (* l_m (/ (sqrt 0.5) t_m)) (sqrt (- (fma (/ Om Omc) (/ Om Omc) -1.0)))))
   (asin (sqrt (/ 1.0 (fma (/ (* 2.0 (/ t_m l_m)) l_m) t_m 1.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.0001) {
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * sqrt(-fma((Om / Omc), (Om / Omc), -1.0))));
	} else {
		tmp = asin(sqrt((1.0 / fma(((2.0 * (t_m / l_m)) / l_m), t_m, 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, 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_m / l_m) ^ 2.0)))))) <= 0.0001)
		tmp = asin(Float64(Float64(l_m * Float64(sqrt(0.5) / t_m)) * sqrt(Float64(-fma(Float64(Om / Omc), Float64(Om / Omc), -1.0)))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(Float64(2.0 * Float64(t_m / l_m)) / l_m), t_m, 1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, 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$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0001], N[ArcSin[N[(N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$m + 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))))))) < 1.00000000000000005e-4

   1. Initial program 76.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 67.8%

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

   if 1.00000000000000005e-4 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 97.9%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 88.6

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

   5. Applied rewrites 88.6%

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

   7. Applied rewrites 97.6%

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

Alternative 5: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-72}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{--1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{2 \cdot \frac{t\_m}{l\_m}}{l\_m}, t\_m, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      1e-72)
   (asin (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- -1.0))))
   (asin (sqrt (/ 1.0 (fma (/ (* 2.0 (/ t_m l_m)) l_m) t_m 1.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 1e-72) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt(-(-1.0))));
	} else {
		tmp = asin(sqrt((1.0 / fma(((2.0 * (t_m / l_m)) / l_m), t_m, 1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, 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_m / l_m) ^ 2.0)))))) <= 1e-72)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(-(-1.0)))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(Float64(2.0 * Float64(t_m / l_m)) / l_m), t_m, 1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, 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$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-72], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[(--1.0)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$m + 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))))))) < 9.9999999999999997e-73

   1. Initial program 67.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 74.0%

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

   if 9.9999999999999997e-73 < (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 98.3%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 97.7%

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

Alternative 6: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{--1}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(\left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right) \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{t\_m}{-l\_m}, \frac{t\_m}{l\_m}, 1\right) \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (sqrt (- -1.0))))
   (if (<=
        (asin
         (sqrt
          (/
           (- 1.0 (pow (/ Om Omc) 2.0))
           (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
        0.05)
     (asin (* (* (/ l_m t_m) (sqrt 0.5)) t_1))
     (asin (* (fma (/ t_m (- l_m)) (/ t_m l_m) 1.0) t_1)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = sqrt(-(-1.0));
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.05) {
		tmp = asin((((l_m / t_m) * sqrt(0.5)) * t_1));
	} else {
		tmp = asin((fma((t_m / -l_m), (t_m / l_m), 1.0) * t_1));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = sqrt(Float64(-(-1.0)))
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 0.05)
		tmp = asin(Float64(Float64(Float64(l_m / t_m) * sqrt(0.5)) * t_1));
	else
		tmp = asin(Float64(fma(Float64(t_m / Float64(-l_m)), Float64(t_m / l_m), 1.0) * t_1));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[Sqrt[(--1.0)], $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.05], N[ArcSin[N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(t$95$m / (-l$95$m)), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $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))))))) < 0.050000000000000003

   1. Initial program 76.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 67.3%

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

   if 0.050000000000000003 < (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 97.9%

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

   3. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-lft-identity N/A

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

      13. metadata-eval N/A

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

      14. fp-cancel-sign-sub-inv N/A

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

      15. +-commutative N/A

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

      16. mul-1-neg N/A

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

      17. metadata-eval N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 97.5%

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

   10. Applied rewrites 97.5%

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

Alternative 7: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 0.05:\\ \;\;\;\;\sin^{-1} \left(\left(\frac{l\_m}{t\_m} \cdot \sqrt{0.5}\right) \cdot \sqrt{--1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{Om}{Omc}, -0.5 \cdot \frac{Om}{Omc}, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      0.05)
   (asin (* (* (/ l_m t_m) (sqrt 0.5)) (sqrt (- -1.0))))
   (asin (fma (/ Om Omc) (* -0.5 (/ Om Omc)) 1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.05) {
		tmp = asin((((l_m / t_m) * sqrt(0.5)) * sqrt(-(-1.0))));
	} else {
		tmp = asin(fma((Om / Omc), (-0.5 * (Om / Omc)), 1.0));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, 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_m / l_m) ^ 2.0)))))) <= 0.05)
		tmp = asin(Float64(Float64(Float64(l_m / t_m) * sqrt(0.5)) * sqrt(Float64(-(-1.0)))));
	else
		tmp = asin(fma(Float64(Om / Omc), Float64(-0.5 * Float64(Om / Omc)), 1.0));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, 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$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.05], N[ArcSin[N[(N[(N[(l$95$m / t$95$m), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(--1.0)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(Om / Omc), $MachinePrecision] * N[(-0.5 * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $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))))))) < 0.050000000000000003

   1. Initial program 76.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. +-commutative N/A

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

      11. mul-1-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-out N/A

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

      14. lower-neg.f64 N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. times-frac N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-/.f64 67.5

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

   5. Applied rewrites 67.5%

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

   7. Applied rewrites 67.5%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 67.3%

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

   if 0.050000000000000003 < (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 97.9%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 91.1%

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

   10. Applied rewrites 97.4%

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

Alternative 8: 74.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 1000:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{Om}{Omc}, -0.5 \cdot \frac{Om}{Omc}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{l\_m \cdot l\_m}{t\_m} \cdot \frac{0.5}{t\_m}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))) 1000.0)
   (asin (fma (/ Om Omc) (* -0.5 (/ Om Omc)) 1.0))
   (asin (sqrt (* (/ (* l_m l_m) t_m) (/ 0.5 t_m))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((1.0 + (2.0 * pow((t_m / l_m), 2.0))) <= 1000.0) {
		tmp = asin(fma((Om / Omc), (-0.5 * (Om / Omc)), 1.0));
	} else {
		tmp = asin(sqrt((((l_m * l_m) / t_m) * (0.5 / t_m))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))) <= 1000.0)
		tmp = asin(fma(Float64(Om / Omc), Float64(-0.5 * Float64(Om / Omc)), 1.0));
	else
		tmp = asin(sqrt(Float64(Float64(Float64(l_m * l_m) / t_m) * Float64(0.5 / t_m))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1000.0], N[ArcSin[N[(N[(Om / Omc), $MachinePrecision] * N[(-0.5 * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(l$95$m * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(0.5 / t$95$m), $MachinePrecision]), $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)))) < 1e3

   1. Initial program 97.9%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 88.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 90.5%

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

   10. Applied rewrites 96.7%

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

   if 1e3 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 76.6%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 60.5%

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

Alternative 9: 58.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.9 \cdot 10^{+177}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(\frac{Om}{Omc}, -0.5 \cdot \frac{Om}{Omc}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5}{Omc} \cdot \frac{Om \cdot Om}{Omc}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.9e+177)
   (asin (fma (/ Om Omc) (* -0.5 (/ Om Omc)) 1.0))
   (asin (* (/ -0.5 Omc) (/ (* Om Om) Omc)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.9e+177) {
		tmp = asin(fma((Om / Omc), (-0.5 * (Om / Omc)), 1.0));
	} else {
		tmp = asin(((-0.5 / Omc) * ((Om * Om) / Omc)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.9e+177)
		tmp = asin(fma(Float64(Om / Omc), Float64(-0.5 * Float64(Om / Omc)), 1.0));
	else
		tmp = asin(Float64(Float64(-0.5 / Omc) * Float64(Float64(Om * Om) / Omc)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.9e+177], N[ArcSin[N[(N[(Om / Omc), $MachinePrecision] * N[(-0.5 * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(-0.5 / Omc), $MachinePrecision] * N[(N[(Om * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.8999999999999999e177

   1. Initial program 90.3%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 54.2%

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

   10. Applied rewrites 57.9%

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

   if 1.8999999999999999e177 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 54.3%

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

Alternative 10: 58.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.9 \cdot 10^{+177}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{--1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{-0.5}{Omc} \cdot \frac{Om \cdot Om}{Omc}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.9e+177)
   (asin (sqrt (- -1.0)))
   (asin (* (/ -0.5 Omc) (/ (* Om Om) Omc)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.9e+177) {
		tmp = asin(sqrt(-(-1.0)));
	} else {
		tmp = asin(((-0.5 / Omc) * ((Om * Om) / Omc)));
	}
	return tmp;
}

Fortran

l_m =     private
t_m =     private
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_m, l_m, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 1.9d+177) then
        tmp = asin(sqrt(-(-1.0d0)))
    else
        tmp = asin((((-0.5d0) / omc) * ((om * om) / omc)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 1.9e+177) {
		tmp = Math.asin(Math.sqrt(-(-1.0)));
	} else {
		tmp = Math.asin(((-0.5 / Omc) * ((Om * Om) / Omc)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 1.9e+177:
		tmp = math.asin(math.sqrt(-(-1.0)))
	else:
		tmp = math.asin(((-0.5 / Omc) * ((Om * Om) / Omc)))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 1.9e+177)
		tmp = asin(sqrt(Float64(-(-1.0))));
	else
		tmp = asin(Float64(Float64(-0.5 / Omc) * Float64(Float64(Om * Om) / Omc)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 1.9e+177)
		tmp = asin(sqrt(-(-1.0)));
	else
		tmp = asin(((-0.5 / Omc) * ((Om * Om) / Omc)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 1.9e+177], N[ArcSin[N[Sqrt[(--1.0)], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(-0.5 / Omc), $MachinePrecision] * N[(N[(Om * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.8999999999999999e177

   1. Initial program 90.3%

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

   3. Taylor expanded in t around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-out N/A

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

      8. lower-neg.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 57.9

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

   5. Applied rewrites 57.9%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{--1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 57.8%

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

   if 1.8999999999999999e177 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in Om around inf

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

   11. Applied rewrites 54.3%

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

Alternative 11: 50.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \sin^{-1} \left(\sqrt{--1}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc) :precision binary64 (asin (sqrt (- -1.0))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin(sqrt(-(-1.0)));
}

Fortran

l_m =     private
t_m =     private
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_m, l_m, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(sqrt(-(-1.0d0)))
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	return Math.asin(Math.sqrt(-(-1.0)));
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	return math.asin(math.sqrt(-(-1.0)))

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	return asin(sqrt(Float64(-(-1.0))))
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(sqrt(-(-1.0)));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := N[ArcSin[N[Sqrt[(--1.0)], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in t around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. +-commutative N/A

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

   5. mul-1-neg N/A

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

   6. metadata-eval N/A

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

   7. distribute-neg-out N/A

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

   8. lower-neg.f64 N/A

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

   9. unpow2 N/A

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

   10. unpow2 N/A

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

   11. times-frac N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 50.9

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

5. Applied rewrites 50.9%

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

6. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \left(\sqrt{--1}\right) \]
7. Step-by-step derivation

8. Applied rewrites 50.8%

   \[\leadsto \sin^{-1} \left(\sqrt{--1}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(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)))))))