Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.7%

Time: 7.5s

Alternatives: 5

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.0% 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

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

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) (/ l_m (fabs t))))
     (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (* (/ t l_m) (/ t l_m))))))))))

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) * (l_m / fabs(t))));
	} else {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l_m) * (t / l_m)))))));
	}
	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) * (l_m / abs(t))))
    else
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l_m) * (t / l_m)))))))
    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) * (l_m / Math.abs(t))));
	} else {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l_m) * (t / l_m)))))));
	}
	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) * (l_m / math.fabs(t))))
	else:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l_m) * (t / l_m)))))))
	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(sqrt(0.5) * Float64(l_m / abs(t))));
	else
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l_m) * Float64(t / l_m)))))));
	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) * (l_m / abs(t))));
	else
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l_m) * (t / l_m)))))));
	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[Sqrt[0.5], $MachinePrecision] * N[(l$95$m / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l$95$m), $MachinePrecision] * N[(t / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.0%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{\color{blue}{\left|t\right|}}\right) \]
   6. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lift-fabs.f64 48.9

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

   7. Applied rewrites 48.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fabs.f64 48.9

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

   9. Applied rewrites 48.9%

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

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

   1. Initial program 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 84.0

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

   3. Applied rewrites 84.0%

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

Alternative 2: 97.9% accurate, 0.7× speedup

Math

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

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.0%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{\color{blue}{\left|t\right|}}\right) \]
   6. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lift-fabs.f64 48.9

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

   7. Applied rewrites 48.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fabs.f64 48.9

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

   9. Applied rewrites 48.9%

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

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

   1. Initial program 84.0%

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

   2. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. frac-times N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. frac-times N/A

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

      11. unpow2 N/A

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

      12. unpow2 N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 65.0

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

   4. Applied rewrites 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 83.2

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

   6. Applied rewrites 83.2%

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

Alternative 3: 94.6% accurate, 0.7× speedup

Math

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

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))))))
      1.0)
   (asin (* (sqrt 0.5) (/ l_m (fabs t))))
   (asin (sqrt (- 1.0 (* Om (/ Om (* Omc Omc))))))))

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)))))) <= 1.0) {
		tmp = asin((sqrt(0.5) * (l_m / fabs(t))));
	} else {
		tmp = asin(sqrt((1.0 - (Om * (Om / (Omc * Omc))))));
	}
	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) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l_m) ** 2.0d0)))))) <= 1.0d0) then
        tmp = asin((sqrt(0.5d0) * (l_m / abs(t))))
    else
        tmp = asin(sqrt((1.0d0 - (om * (om / (omc * omc))))))
    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 tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t / l_m), 2.0)))))) <= 1.0) {
		tmp = Math.asin((Math.sqrt(0.5) * (l_m / Math.abs(t))));
	} else {
		tmp = Math.asin(Math.sqrt((1.0 - (Om * (Om / (Omc * Omc))))));
	}
	return tmp;
}

Python

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

MATLAB

l_m = abs(l);
function tmp_2 = code(t, l_m, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l_m) ^ 2.0)))))) <= 1.0)
		tmp = asin((sqrt(0.5) * (l_m / abs(t))));
	else
		tmp = asin(sqrt((1.0 - (Om * (Om / (Omc * Omc))))));
	end
	tmp_2 = 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], 1.0], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l$95$m / N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(Om * N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.0%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{\color{blue}{\left|t\right|}}\right) \]
   6. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lift-fabs.f64 48.9

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

   7. Applied rewrites 48.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-fabs.f64 48.9

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

   9. Applied rewrites 48.9%

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

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

   1. Initial program 84.0%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 45.1

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

   4. Applied rewrites 45.1%

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

      1. pow2 45.1

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 48.2

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

   6. Applied rewrites 48.2%

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

Alternative 4: 48.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

l_m = fabs(l);
double code(double t, double l_m, double Om, double Omc) {
	return asin((sqrt(0.5) * (l_m / 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((sqrt(0.5d0) * (l_m / 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((Math.sqrt(0.5) * (l_m / Math.abs(t))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.0%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 43.1%

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

5. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{\color{blue}{\left|t\right|}}\right) \]
6. 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. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lift-fabs.f64 48.9

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

7. Applied rewrites 48.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-fabs.f64 48.9

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

9. Applied rewrites 48.9%

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

Alternative 5: 31.3% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 84.0%

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

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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 43.1%

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

5. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \left(\frac{\ell \cdot \sqrt{\frac{1}{2}}}{\color{blue}{\left|t\right|}}\right) \]
6. 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. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lift-fabs.f64 48.9

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

7. Applied rewrites 48.9%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. unpow2 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 34.1

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

9. Applied rewrites 34.1%

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

10. Taylor expanded in t around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lift-sqrt.f64 31.3

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

12. Applied rewrites 31.3%

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

Reproduce

herbie shell --seed 2025134 
(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)))))))