Toniolo and Linder, Equation (2)

Percentage Accurate: 84.1% → 97.4%

Time: 6.6s

Alternatives: 10

Speedup: 1.1×

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: 97.4% accurate, 0.5× speedup

Math

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

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)))))) 2e-151)
     (asin (* l_m (/ (sqrt (* (- 1.0 (* Om (/ Om (* Omc Omc)))) 0.5)) t_m)))
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ 1.0 (pow (/ t_m l_m) -2.0))))))))))

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)))))) <= 2e-151) {
		tmp = asin((l_m * (sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) * 0.5)) / t_m)));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / pow((t_m / l_m), -2.0)))))));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if (asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0)))))) <= 2d-151) then
        tmp = asin((l_m * (sqrt(((1.0d0 - (om * (om / (omc * omc)))) * 0.5d0)) / t_m)))
    else
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * (1.0d0 / ((t_m / l_m) ** (-2.0d0))))))))
    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 t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if (Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0)))))) <= 2e-151) {
		tmp = Math.asin((l_m * (Math.sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) * 0.5)) / t_m)));
	} else {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / Math.pow((t_m / l_m), -2.0)))))));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if math.asin(math.sqrt((t_1 / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0)))))) <= 2e-151:
		tmp = math.asin((l_m * (math.sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) * 0.5)) / t_m)))
	else:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * (1.0 / math.pow((t_m / l_m), -2.0)))))))
	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)))))) <= 2e-151)
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc)))) * 0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(1.0 / (Float64(t_m / l_m) ^ -2.0)))))));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if (asin(sqrt((t_1 / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0)))))) <= 2e-151)
		tmp = asin((l_m * (sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) * 0.5)) / t_m)));
	else
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * (1.0 / ((t_m / l_m) ^ -2.0)))))));
	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_] := 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], 2e-151], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(1.0 / N[Power[N[(t$95$m / l$95$m), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 1.9999999999999999e-151

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 46.8

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

   7. Applied rewrites 46.8%

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

   if 1.9999999999999999e-151 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      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. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 84.1

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

   3. Applied rewrites 84.1%

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

Alternative 2: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_1}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 4 \cdot 10^{-61}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{\left(1 - Om \cdot \frac{Om}{Omc \cdot Omc}\right) \cdot 0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{fma}\left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}, 2, 1\right)}}\right)\\ \end{array} \end{array} \]

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)))))) 4e-61)
     (asin (* l_m (/ (sqrt (* (- 1.0 (* Om (/ Om (* Omc Omc)))) 0.5)) t_m)))
     (asin (sqrt (/ t_1 (fma (* (/ t_m l_m) (/ t_m l_m)) 2.0 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)))))) <= 4e-61) {
		tmp = asin((l_m * (sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) * 0.5)) / t_m)));
	} else {
		tmp = asin(sqrt((t_1 / fma(((t_m / l_m) * (t_m / l_m)), 2.0, 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)))))) <= 4e-61)
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc)))) * 0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(t_1 / fma(Float64(Float64(t_m / l_m) * Float64(t_m / l_m)), 2.0, 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], 4e-61], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0 + 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))))))) < 4.0000000000000002e-61

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 46.8

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

   7. Applied rewrites 46.8%

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

   if 4.0000000000000002e-61 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      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. metadata-eval N/A

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

      4. pow-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-/.f64 84.1

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

   3. Applied rewrites 84.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 84.1

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

   5. Applied rewrites 84.1%

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

Alternative 3: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{\left(1 - Om \cdot \frac{Om}{Omc \cdot Omc}\right) \cdot 0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}, 2, 1\right)}}\right)\\ \end{array} \end{array} \]

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))))))
      2e-65)
   (asin (* l_m (/ (sqrt (* (- 1.0 (* Om (/ Om (* Omc Omc)))) 0.5)) t_m)))
   (asin (sqrt (/ 1.0 (fma (* t_m (/ (/ t_m l_m) l_m)) 2.0 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)))))) <= 2e-65) {
		tmp = asin((l_m * (sqrt(((1.0 - (Om * (Om / (Omc * Omc)))) * 0.5)) / t_m)));
	} else {
		tmp = asin(sqrt((1.0 / fma((t_m * ((t_m / l_m) / l_m)), 2.0, 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)))))) <= 2e-65)
		tmp = asin(Float64(l_m * Float64(sqrt(Float64(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc)))) * 0.5)) / t_m)));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m * Float64(Float64(t_m / l_m) / l_m)), 2.0, 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], 2e-65], N[ArcSin[N[(l$95$m * N[(N[Sqrt[N[(N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0 + 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))))))) < 1.99999999999999985e-65

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in l around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 46.8

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

   7. Applied rewrites 46.8%

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

   if 1.99999999999999985e-65 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 84.1%

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

   2. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 80.5

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

   8. Applied rewrites 80.5%

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

Alternative 4: 89.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 10^{+303}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(t\_m \cdot \frac{\frac{t\_m}{l\_m}}{l\_m}, 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(l\_m \cdot l\_m\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))) 1e+303)
   (asin (sqrt (/ 1.0 (fma (* t_m (/ (/ t_m l_m) l_m)) 2.0 1.0))))
   (asin (/ (sqrt (* (* l_m l_m) 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
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))) <= 1e+303) {
		tmp = asin(sqrt((1.0 / fma((t_m * ((t_m / l_m) / l_m)), 2.0, 1.0))));
	} else {
		tmp = asin((sqrt(((l_m * l_m) * 0.5)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
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))) <= 1e+303)
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m * Float64(Float64(t_m / l_m) / l_m)), 2.0, 1.0))));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(l_m * l_m) * 0.5)) / t_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_] := If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+303], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m * N[(N[(t$95$m / l$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $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)))) < 1e303

   1. Initial program 84.1%

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

   2. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 80.5

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

   8. Applied rewrites 80.5%

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

   if 1e303 < (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

Alternative 5: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-107}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - 2 \cdot \left(t\_m \cdot \frac{t\_m}{l\_m \cdot l\_m}\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(l\_m \cdot l\_m\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (* 2.0 (pow (/ t_m l_m) 2.0))))
   (if (<= t_1 4e-107)
     (asin (sqrt (- 1.0 (* 2.0 (* t_m (/ t_m (* l_m l_m)))))))
     (if (<= t_1 2e-8)
       (asin (sqrt (- 1.0 (/ (* Om Om) (* Omc Omc)))))
       (asin (/ (sqrt (* (* l_m l_m) 0.5)) t_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 = 2.0 * pow((t_m / l_m), 2.0);
	double tmp;
	if (t_1 <= 4e-107) {
		tmp = asin(sqrt((1.0 - (2.0 * (t_m * (t_m / (l_m * l_m)))))));
	} else if (t_1 <= 2e-8) {
		tmp = asin(sqrt((1.0 - ((Om * Om) / (Omc * Omc)))));
	} else {
		tmp = asin((sqrt(((l_m * l_m) * 0.5)) / t_m));
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * ((t_m / l_m) ** 2.0d0)
    if (t_1 <= 4d-107) then
        tmp = asin(sqrt((1.0d0 - (2.0d0 * (t_m * (t_m / (l_m * l_m)))))))
    else if (t_1 <= 2d-8) then
        tmp = asin(sqrt((1.0d0 - ((om * om) / (omc * omc)))))
    else
        tmp = asin((sqrt(((l_m * l_m) * 0.5d0)) / t_m))
    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 t_1 = 2.0 * Math.pow((t_m / l_m), 2.0);
	double tmp;
	if (t_1 <= 4e-107) {
		tmp = Math.asin(Math.sqrt((1.0 - (2.0 * (t_m * (t_m / (l_m * l_m)))))));
	} else if (t_1 <= 2e-8) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om * Om) / (Omc * Omc)))));
	} else {
		tmp = Math.asin((Math.sqrt(((l_m * l_m) * 0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	t_1 = 2.0 * math.pow((t_m / l_m), 2.0)
	tmp = 0
	if t_1 <= 4e-107:
		tmp = math.asin(math.sqrt((1.0 - (2.0 * (t_m * (t_m / (l_m * l_m)))))))
	elif t_1 <= 2e-8:
		tmp = math.asin(math.sqrt((1.0 - ((Om * Om) / (Omc * Omc)))))
	else:
		tmp = math.asin((math.sqrt(((l_m * l_m) * 0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))
	tmp = 0.0
	if (t_1 <= 4e-107)
		tmp = asin(sqrt(Float64(1.0 - Float64(2.0 * Float64(t_m * Float64(t_m / Float64(l_m * l_m)))))));
	elseif (t_1 <= 2e-8)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om * Om) / Float64(Omc * Omc)))));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(l_m * l_m) * 0.5)) / t_m));
	end
	return tmp
end

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = 2.0 * ((t_m / l_m) ^ 2.0);
	tmp = 0.0;
	if (t_1 <= 4e-107)
		tmp = asin(sqrt((1.0 - (2.0 * (t_m * (t_m / (l_m * l_m)))))));
	elseif (t_1 <= 2e-8)
		tmp = asin(sqrt((1.0 - ((Om * Om) / (Omc * Omc)))));
	else
		tmp = asin((sqrt(((l_m * l_m) * 0.5)) / t_m));
	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_] := Block[{t$95$1 = N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-107], N[ArcSin[N[Sqrt[N[(1.0 - N[(2.0 * N[(t$95$m * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e-8], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 4e-107

   1. Initial program 84.1%

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

   2. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in t around 0

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

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

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

      2. metadata-eval N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 45.3

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

   7. Applied rewrites 45.3%

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

   if 4e-107 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 2e-8

   1. Initial program 84.1%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 44.6

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

   4. Applied rewrites 44.6%

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

   if 2e-8 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

Alternative 6: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-320}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-2, \frac{t\_m \cdot t\_m}{l\_m \cdot l\_m}, 1\right)}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(l\_m \cdot l\_m\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (* 2.0 (pow (/ t_m l_m) 2.0))))
   (if (<= t_1 2e-320)
     (asin (sqrt (fma -2.0 (/ (* t_m t_m) (* l_m l_m)) 1.0)))
     (if (<= t_1 2e-8)
       (asin (sqrt (- 1.0 (* Om (/ Om (* Omc Omc))))))
       (asin (/ (sqrt (* (* l_m l_m) 0.5)) t_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 = 2.0 * pow((t_m / l_m), 2.0);
	double tmp;
	if (t_1 <= 2e-320) {
		tmp = asin(sqrt(fma(-2.0, ((t_m * t_m) / (l_m * l_m)), 1.0)));
	} else if (t_1 <= 2e-8) {
		tmp = asin(sqrt((1.0 - (Om * (Om / (Omc * Omc))))));
	} else {
		tmp = asin((sqrt(((l_m * l_m) * 0.5)) / t_m));
	}
	return tmp;
}

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))
	tmp = 0.0
	if (t_1 <= 2e-320)
		tmp = asin(sqrt(fma(-2.0, Float64(Float64(t_m * t_m) / Float64(l_m * l_m)), 1.0)));
	elseif (t_1 <= 2e-8)
		tmp = asin(sqrt(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc))))));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(l_m * l_m) * 0.5)) / t_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[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-320], N[ArcSin[N[Sqrt[N[(-2.0 * N[(N[(t$95$m * t$95$m), $MachinePrecision] / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e-8], N[ArcSin[N[Sqrt[N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 1.99998e-320

   1. Initial program 84.1%

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

   2. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 23.4

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

   7. Applied rewrites 23.4%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 42.1

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

   10. Applied rewrites 42.1%

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

   if 1.99998e-320 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 2e-8

   1. Initial program 84.1%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 47.8

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

   6. Applied rewrites 47.8%

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

   if 2e-8 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

Alternative 7: 80.9% accurate, 0.7× speedup

Math

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

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 (* (* l_m l_m) 0.5)) t_m))
   (asin (sqrt (/ 1.0 (fma (* t_m (/ t_m (* l_m l_m))) 2.0 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(((l_m * l_m) * 0.5)) / t_m));
	} else {
		tmp = asin(sqrt((1.0 / fma((t_m * (t_m / (l_m * l_m))), 2.0, 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(sqrt(Float64(Float64(l_m * l_m) * 0.5)) / t_m));
	else
		tmp = asin(sqrt(Float64(1.0 / fma(Float64(t_m * Float64(t_m / Float64(l_m * l_m))), 2.0, 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[Sqrt[N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 / N[(N[(t$95$m * N[(t$95$m / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

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

   1. Initial program 84.1%

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

   2. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 72.2

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

   6. Applied rewrites 72.2%

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

Alternative 8: 80.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{Om}{Omc \cdot Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{\left(l\_m \cdot l\_m\right) \cdot 0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (* 2.0 (pow (/ t_m l_m) 2.0)) 2e-8)
   (asin (sqrt (- 1.0 (* Om (/ Om (* Omc Omc))))))
   (asin (/ (sqrt (* (* l_m l_m) 0.5)) t_m))))

C

t_m = fabs(t);
l_m = fabs(l);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((2.0 * pow((t_m / l_m), 2.0)) <= 2e-8) {
		tmp = asin(sqrt((1.0 - (Om * (Om / (Omc * Omc))))));
	} else {
		tmp = asin((sqrt(((l_m * l_m) * 0.5)) / t_m));
	}
	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 ((2.0d0 * ((t_m / l_m) ** 2.0d0)) <= 2d-8) then
        tmp = asin(sqrt((1.0d0 - (om * (om / (omc * omc))))))
    else
        tmp = asin((sqrt(((l_m * l_m) * 0.5d0)) / t_m))
    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 ((2.0 * Math.pow((t_m / l_m), 2.0)) <= 2e-8) {
		tmp = Math.asin(Math.sqrt((1.0 - (Om * (Om / (Omc * Omc))))));
	} else {
		tmp = Math.asin((Math.sqrt(((l_m * l_m) * 0.5)) / t_m));
	}
	return tmp;
}

Python

t_m = math.fabs(t)
l_m = math.fabs(l)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (2.0 * math.pow((t_m / l_m), 2.0)) <= 2e-8:
		tmp = math.asin(math.sqrt((1.0 - (Om * (Om / (Omc * Omc))))))
	else:
		tmp = math.asin((math.sqrt(((l_m * l_m) * 0.5)) / t_m))
	return tmp

Julia

t_m = abs(t)
l_m = abs(l)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)) <= 2e-8)
		tmp = asin(sqrt(Float64(1.0 - Float64(Om * Float64(Om / Float64(Omc * Omc))))));
	else
		tmp = asin(Float64(sqrt(Float64(Float64(l_m * l_m) * 0.5)) / t_m));
	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 ((2.0 * ((t_m / l_m) ^ 2.0)) <= 2e-8)
		tmp = asin(sqrt((1.0 - (Om * (Om / (Omc * Omc))))));
	else
		tmp = asin((sqrt(((l_m * l_m) * 0.5)) / t_m));
	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[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e-8], N[ArcSin[N[Sqrt[N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))) < 2e-8

   1. Initial program 84.1%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. pow2 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 47.8

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

   6. Applied rewrites 47.8%

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

   if 2e-8 < (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))

   1. Initial program 84.1%

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

   2. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 31.3

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in Om around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 35.5

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

   7. Applied rewrites 35.5%

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

Alternative 9: 35.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\frac{\sqrt{\left(l\_m \cdot l\_m\right) \cdot 0.5}}{t\_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (/ (sqrt (* (* l_m l_m) 0.5)) t_m)))

C

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

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((sqrt(((l_m * l_m) * 0.5d0)) / t_m))
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((Math.sqrt(((l_m * l_m) * 0.5)) / t_m));
}

Python

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

Julia

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

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin((sqrt(((l_m * l_m) * 0.5)) / t_m));
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[N[(N[Sqrt[N[(N[(l$95$m * l$95$m), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in t around inf

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 31.3

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

4. Applied rewrites 31.3%

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

5. Taylor expanded in Om around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 35.5

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

7. Applied rewrites 35.5%

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

Alternative 10: 13.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(-l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (- (* l_m (/ (sqrt 0.5) t_m)))))

C

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

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(-(l_m * (sqrt(0.5d0) / t_m)))
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(-(l_m * (Math.sqrt(0.5) / t_m)));
}

Python

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

Julia

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

MATLAB

t_m = abs(t);
l_m = abs(l);
function tmp = code(t_m, l_m, Om, Omc)
	tmp = asin(-(l_m * (sqrt(0.5) / t_m)));
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[(-N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]

Derivation

1. Initial program 84.1%

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

2. Taylor expanded in t around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

4. Applied rewrites 12.1%

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

5. Taylor expanded in l around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. pow2 N/A

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

   9. associate-/l* N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 12.7

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

7. Applied rewrites 12.7%

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

8. Taylor expanded in Om around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 13.5

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

10. Applied rewrites 13.5%

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

Reproduce

herbie shell --seed 2025138 
(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)))))))