Toniolo and Linder, Equation (2)

Percentage Accurate: 83.4% → 98.9%

Time: 6.1s

Alternatives: 7

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: 83.4% 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, 0.4× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0)))) (t_2 (pow (/ Om Omc) 2.0)))
   (if (<= (asin (sqrt (/ (- 1.0 t_2) t_1))) 0.0)
     (asin (* (/ (* (sqrt 0.5) l_m) t_m) (fma -0.5 t_2 1.0)))
     (asin (sqrt (/ (/ (- 1.0 (* t_2 t_2)) (+ 1.0 t_2)) t_1))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = 1.0 + (2.0 * pow((t_m / l_m), 2.0));
	double t_2 = pow((Om / Omc), 2.0);
	double tmp;
	if (asin(sqrt(((1.0 - t_2) / t_1))) <= 0.0) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * fma(-0.5, t_2, 1.0)));
	} else {
		tmp = asin(sqrt((((1.0 - (t_2 * t_2)) / (1.0 + t_2)) / t_1)));
	}
	return tmp;
}

Julia

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.5 * t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(1.0 - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$1), $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 46.9%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in Om around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

   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 98.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lift-pow.f64 N/A

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

      15. lift-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 2: 98.9% accurate, 0.6× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (pow (/ Om Omc) 2.0)) (t_2 (- 1.0 t_1)))
   (if (<= (asin (sqrt (/ t_2 (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0)))))) 0.0)
     (asin (* (/ (* (sqrt 0.5) l_m) t_m) (fma -0.5 t_1 1.0)))
     (asin (sqrt (/ t_2 (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m))))))))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = pow((Om / Omc), 2.0);
	double t_2 = 1.0 - t_1;
	double tmp;
	if (asin(sqrt((t_2 / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 0.0) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * fma(-0.5, t_1, 1.0)));
	} else {
		tmp = asin(sqrt((t_2 / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	}
	return tmp;
}

Julia

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - t$95$1), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(t$95$2 / 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[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[(-0.5 * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$2 / N[(1.0 + N[(2.0 * N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 46.9%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in Om around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. times-frac N/A

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

      6. unpow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-/.f64 67.5

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

   8. Applied rewrites 67.5%

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

   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 98.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 3: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-17}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \frac{t\_m}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(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-17)
     (asin (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt t_1)))
     (asin (sqrt (/ 1.0 (fma (* 2.0 (/ t_m l_m)) (/ t_m l_m) 1.0)))))))

C

t_m = fabs(t);
l_m = fabs(l);
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-17) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt(t_1)));
	} else {
		tmp = asin(sqrt((1.0 / fma((2.0 * (t_m / l_m)), (t_m / l_m), 1.0))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
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-17)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(t_1)));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 * Float64(t_m / l_m)), Float64(t_m / l_m), 1.0))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $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-17], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $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))))))) < 1.00000000000000007e-17

   1. Initial program 67.4%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-/.f64 61.9

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

   7. Applied rewrites 61.9%

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

   if 1.00000000000000007e-17 < (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

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

   5. Applied rewrites 97.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 97.6

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

   7. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 97.6

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

   9. Applied rewrites 97.6%

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

Alternative 4: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-17}:\\ \;\;\;\;\sin^{-1} \left(\left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2 \cdot \frac{t\_m}{l\_m}, \frac{t\_m}{l\_m}, 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(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-17)
   (asin
    (* (* l_m (/ (sqrt 0.5) t_m)) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
   (asin (sqrt (/ 1.0 (fma (* 2.0 (/ t_m l_m)) (/ t_m l_m) 1.0))))))

C

t_m = fabs(t);
l_m = fabs(l);
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-17) {
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else {
		tmp = asin(sqrt((1.0 / fma((2.0 * (t_m / l_m)), (t_m / l_m), 1.0))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
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-17)
		tmp = asin(Float64(Float64(l_m * Float64(sqrt(0.5) / t_m)) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 * Float64(t_m / l_m)), Float64(t_m / l_m), 1.0))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $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-17], N[ArcSin[N[(N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $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))))))) < 1.00000000000000007e-17

   1. Initial program 67.4%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 52.9

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

   5. Applied rewrites 52.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 52.9

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

   7. Applied rewrites 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 61.8

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

   9. Applied rewrites 61.8%

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

   if 1.00000000000000007e-17 < (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

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

   5. Applied rewrites 97.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 97.6

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

   7. Applied rewrites 97.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 97.6

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

   9. Applied rewrites 97.6%

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

Alternative 5: 98.1% accurate, 0.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(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))))))
      0.0)
   (asin (/ (* (sqrt 0.5) l_m) t_m))
   (asin (sqrt (/ 1.0 (fma (* 2.0 (/ t_m l_m)) (/ t_m l_m) 1.0))))))

C

t_m = fabs(t);
l_m = fabs(l);
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)))))) <= 0.0) {
		tmp = asin(((sqrt(0.5) * l_m) / t_m));
	} else {
		tmp = asin(sqrt((1.0 / fma((2.0 * (t_m / l_m)), (t_m / l_m), 1.0))));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
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)))))) <= 0.0)
		tmp = asin(Float64(Float64(sqrt(0.5) * l_m) / t_m));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(2.0 * Float64(t_m / l_m)), Float64(t_m / l_m), 1.0))));
	end
	return tmp
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $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], 0.0], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(2.0 * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / l$95$m), $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 46.9%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

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

   1. Initial program 98.6%

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

   3. Taylor expanded in Om around 0

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

   5. Applied rewrites 97.5%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 97.5

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

   9. Applied rewrites 97.5%

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

Alternative 6: 97.0% accurate, 0.7× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(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))))))
      0.5)
   (asin (/ (* (sqrt 0.5) l_m) t_m))
   (asin 1.0)))

C

t_m = fabs(t);
l_m = fabs(l);
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)))))) <= 0.5) {
		tmp = asin(((sqrt(0.5) * l_m) / t_m));
	} else {
		tmp = asin(1.0);
	}
	return tmp;
}

Fortran

t_m =     private
l_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 (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 0.5d0) then
        tmp = asin(((sqrt(0.5d0) * l_m) / t_m))
    else
        tmp = asin(1.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

t_m = abs(t)
l_m = abs(l)
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)))))) <= 0.5)
		tmp = asin(Float64(Float64(sqrt(0.5) * l_m) / t_m));
	else
		tmp = asin(1.0);
	end
	return tmp
end

MATLAB

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

Wolfram

t_m = N[Abs[t], $MachinePrecision]
l_m = N[Abs[l], $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], 0.5], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[1.0], $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))))))) < 0.5

   1. Initial program 68.2%

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

   3. Taylor expanded in t around 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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

   if 0.5 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 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

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 96.7%

      \[\leadsto \sin^{-1} 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 50.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

t_m =     private
l_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(1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.6%

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

3. Taylor expanded in Om around 0

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 64.5

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

5. Applied rewrites 64.5%

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

6. Taylor expanded in t around 0

   \[\leadsto \sin^{-1} 1 \]
7. Step-by-step derivation

8. Applied rewrites 55.1%

   \[\leadsto \sin^{-1} 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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)))))))