Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 98.7%

Time: 7.5s

Alternatives: 7

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: 83.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

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

FPCore

l_m = (fabs.f64 l)
(FPCore (t 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 l_m) 2.0)))))) 0.0)
     (asin (* (/ (sqrt 0.5) (fabs t)) l_m))
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t l_m) (/ l_m t))))))))))

C

l_m = fabs(l);
double code(double t, 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 / l_m), 2.0)))))) <= 0.0) {
		tmp = asin(((sqrt(0.5) / fabs(t)) * l_m));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l_m) / (l_m / t)))))));
	}
	return tmp;
}

Fortran

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, l_m, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t
    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 / l_m) ** 2.0d0)))))) <= 0.0d0) then
        tmp = asin(((sqrt(0.5d0) / abs(t)) * l_m))
    else
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l_m) / (l_m / t)))))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
public static double code(double t, 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 / l_m), 2.0)))))) <= 0.0) {
		tmp = Math.asin(((Math.sqrt(0.5) / Math.abs(t)) * l_m));
	} else {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l_m) / (l_m / t)))))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
def code(t, 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 / l_m), 2.0)))))) <= 0.0:
		tmp = math.asin(((math.sqrt(0.5) / math.fabs(t)) * l_m))
	else:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l_m) / (l_m / t)))))))
	return tmp

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, 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 / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l$95$m), $MachinePrecision] / N[(l$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 29.3

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

   4. Applied rewrites 29.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 33.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 33.8

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

   11. Applied rewrites 47.9%

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

   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 83.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 83.8

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

   3. Applied rewrites 83.8%

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

Alternative 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, 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 / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[(t / l$95$m), $MachinePrecision] * N[(N[(t / l$95$m), $MachinePrecision] * -2.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 29.3

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

   4. Applied rewrites 29.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 33.8%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 33.8

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

   11. Applied rewrites 47.9%

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

   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 83.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

Alternative 3: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{l\_m}\right)}^{2}}}\right) \leq 2 \cdot 10^{-46}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}\right)}}{\left|t\right|} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{\frac{t}{l\_m} \cdot t}{l\_m}, -1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l_m) 2.0))))))
      2e-46)
   (asin (* (/ (sqrt (* 0.5 (- 1.0 (/ (* (/ Om Omc) Om) Omc)))) (fabs t)) l_m))
   (asin
    (sqrt
     (/
      (fma (/ Om Omc) (/ Om Omc) -1.0)
      (fma -2.0 (/ (* (/ t l_m) t) l_m) -1.0))))))

C

l_m = fabs(l);
double code(double t, 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 / l_m), 2.0)))))) <= 2e-46) {
		tmp = asin(((sqrt((0.5 * (1.0 - (((Om / Omc) * Om) / Omc)))) / fabs(t)) * l_m));
	} else {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / fma(-2.0, (((t / l_m) * t) / l_m), -1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, 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 / l_m) ^ 2.0)))))) <= 2e-46)
		tmp = asin(Float64(Float64(sqrt(Float64(0.5 * Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)))) / abs(t)) * l_m));
	else
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / fma(-2.0, Float64(Float64(Float64(t / l_m) * t) / l_m), -1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, 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 / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2e-46], N[ArcSin[N[(N[(N[Sqrt[N[(0.5 * N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-2.0 * N[(N[(N[(t / l$95$m), $MachinePrecision] * t), $MachinePrecision] / l$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 2.00000000000000005e-46

   1. Initial program 83.8%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 29.3

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

   4. Applied rewrites 29.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.8

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

   8. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 48.2

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

   10. Applied rewrites 48.2%

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

   if 2.00000000000000005e-46 < (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 83.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 81.0

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

   5. Applied rewrites 81.0%

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

Alternative 4: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{l\_m}\right)}^{2}}}\right) \leq 10^{-35}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}\right)}}{\left|t\right|} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc}, \frac{Om}{Omc}, -1\right)}{\mathsf{fma}\left(-2, \frac{t}{\frac{l\_m}{t} \cdot l\_m}, -1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
(FPCore (t l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t l_m) 2.0))))))
      1e-35)
   (asin (* (/ (sqrt (* 0.5 (- 1.0 (/ (* (/ Om Omc) Om) Omc)))) (fabs t)) l_m))
   (asin
    (sqrt
     (/
      (fma (/ Om Omc) (/ Om Omc) -1.0)
      (fma -2.0 (/ t (* (/ l_m t) l_m)) -1.0))))))

C

l_m = fabs(l);
double code(double t, 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 / l_m), 2.0)))))) <= 1e-35) {
		tmp = asin(((sqrt((0.5 * (1.0 - (((Om / Omc) * Om) / Omc)))) / fabs(t)) * l_m));
	} else {
		tmp = asin(sqrt((fma((Om / Omc), (Om / Omc), -1.0) / fma(-2.0, (t / ((l_m / t) * l_m)), -1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
function code(t, 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 / l_m) ^ 2.0)))))) <= 1e-35)
		tmp = asin(Float64(Float64(sqrt(Float64(0.5 * Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)))) / abs(t)) * l_m));
	else
		tmp = asin(sqrt(Float64(fma(Float64(Om / Omc), Float64(Om / Omc), -1.0) / fma(-2.0, Float64(t / Float64(Float64(l_m / t) * l_m)), -1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, 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 / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-35], N[ArcSin[N[(N[(N[Sqrt[N[(0.5 * N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision] * l$95$m), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision] + -1.0), $MachinePrecision] / N[(-2.0 * N[(t / N[(N[(l$95$m / t), $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 1.00000000000000001e-35

   1. Initial program 83.8%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 29.3

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

   4. Applied rewrites 29.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.8

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

   8. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 48.2

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

   10. Applied rewrites 48.2%

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

   if 1.00000000000000001e-35 < (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 83.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.1%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. frac-times N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. Initial program 83.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.1%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 51.9%

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

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

   1. Initial program 83.8%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 29.3

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

   4. Applied rewrites 29.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.8

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

   8. Applied rewrites 45.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 48.2

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

   10. Applied rewrites 48.2%

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

Alternative 6: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.8%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. sub-flip N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. distribute-neg-in N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   3. Applied rewrites 66.1%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 51.9%

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

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

   1. Initial program 83.8%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 29.3

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

   4. Applied rewrites 29.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 31.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 31.8

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

   8. Applied rewrites 45.3%

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

   9. Taylor expanded in Om around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-fabs.f64 47.9

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

   11. Applied rewrites 47.9%

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

Alternative 7: 47.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
code[t_, l$95$m_, Om_, Omc_] := N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 83.8%

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

2. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 29.3

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

4. Applied rewrites 29.3%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f64 N/A

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

6. Applied rewrites 31.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 31.8

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

8. Applied rewrites 45.3%

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

9. Taylor expanded in Om around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sqrt.f64 N/A

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

    4. lower-fabs.f64 47.9

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

11. Applied rewrites 47.9%

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

Reproduce

herbie shell --seed 2025151 
(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)))))))