Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 98.9%

Time: 11.5s

Alternatives: 12

Speedup: 1.2×

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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := {\left(\frac{t\_m}{l\_m}\right)}^{2}\\ t_2 := \sqrt{\frac{t\_m}{l\_m}}\\ t_3 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{t\_3}{1 + 2 \cdot t\_1}}\right) \leq 10^{-151}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_3}{\mathsf{fma}\left(\frac{t\_m}{l\_m} \cdot t\_2, t\_2, t\_1 + 1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (pow (/ t_m l_m) 2.0))
        (t_2 (sqrt (/ t_m l_m)))
        (t_3 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (asin (sqrt (/ t_3 (+ 1.0 (* 2.0 t_1))))) 1e-151)
     (asin
      (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
     (asin (sqrt (/ t_3 (fma (* (/ t_m l_m) t_2) t_2 (+ t_1 1.0))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = pow((t_m / l_m), 2.0);
	double t_2 = sqrt((t_m / l_m));
	double t_3 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if (asin(sqrt((t_3 / (1.0 + (2.0 * t_1))))) <= 1e-151) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else {
		tmp = asin(sqrt((t_3 / fma(((t_m / l_m) * t_2), t_2, (t_1 + 1.0)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(t_m / l_m) ^ 2.0
	t_2 = sqrt(Float64(t_m / l_m))
	t_3 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (asin(sqrt(Float64(t_3 / Float64(1.0 + Float64(2.0 * t_1))))) <= 1e-151)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	else
		tmp = asin(sqrt(Float64(t_3 / fma(Float64(Float64(t_m / l_m) * t_2), t_2, Float64(t_1 + 1.0)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$m / l$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(t$95$3 / N[(1.0 + N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-151], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(t$95$3 / N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2 + N[(t$95$1 + 1.0), $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))))))) < 9.9999999999999994e-152

   1. Initial program 46.3%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if 9.9999999999999994e-152 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. sqr-pow N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. sqr-pow N/A

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

      12. associate-*r* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. unpow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. metadata-eval N/A

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

      19. unpow1/2 N/A

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

      20. lower-sqrt.f64 N/A

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

      21. +-commutative N/A

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

   4. Applied rewrites 53.4%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-143}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1
         (asin
          (sqrt
           (/
            (- 1.0 (pow (/ Om Omc) 2.0))
            (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))))
   (if (<= t_1 2e-143)
     (asin
      (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
     t_1)))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0))))));
	double tmp;
	if (t_1 <= 2e-143) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

l_m =     private
t_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_m, l_m, om, omc)
use fmin_fmax_functions
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: t_1
    real(8) :: tmp
    t_1 = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t_m / l_m) ** 2.0d0))))))
    if (t_1 <= 2d-143) then
        tmp = asin((((sqrt(0.5d0) * l_m) / t_m) * sqrt((1.0d0 - ((om / omc) * (om / omc))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((t_m / l_m), 2.0))))));
	double tmp;
	if (t_1 <= 2e-143) {
		tmp = Math.asin((((Math.sqrt(0.5) * l_m) / t_m) * Math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	t_1 = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((t_m / l_m), 2.0))))))
	tmp = 0
	if t_1 <= 2e-143:
		tmp = math.asin((((math.sqrt(0.5) * l_m) / t_m) * math.sqrt((1.0 - ((Om / Omc) * (Om / Omc))))))
	else:
		tmp = t_1
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))))))
	tmp = 0.0
	if (t_1 <= 2e-143)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	t_1 = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t_m / l_m) ^ 2.0))))));
	tmp = 0.0;
	if (t_1 <= 2e-143)
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - ((Om / Omc) * (Om / Omc))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 2e-143], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 1.9999999999999999e-143

   1. Initial program 47.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if 1.9999999999999999e-143 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.0%

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

Alternative 3: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{1 - t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(-0.5, t\_1, 1\right) \cdot \sqrt{{\left(\mathsf{fma}\left(\frac{2}{l\_m} \cdot \frac{t\_m}{l\_m}, t\_m, 1\right)\right)}^{-1}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (* (/ Om Omc) (/ Om Omc))))
   (if (<=
        (asin
         (sqrt
          (/
           (- 1.0 (pow (/ Om Omc) 2.0))
           (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
        1e-71)
     (asin (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- 1.0 t_1))))
     (asin
      (*
       (fma -0.5 t_1 1.0)
       (sqrt (pow (fma (* (/ 2.0 l_m) (/ t_m l_m)) t_m 1.0) -1.0)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = (Om / Omc) * (Om / Omc);
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 1e-71) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - t_1))));
	} else {
		tmp = asin((fma(-0.5, t_1, 1.0) * sqrt(pow(fma(((2.0 / l_m) * (t_m / l_m)), t_m, 1.0), -1.0))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(Float64(Om / Omc) * Float64(Om / Omc))
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 1e-71)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(1.0 - t_1))));
	else
		tmp = asin(Float64(fma(-0.5, t_1, 1.0) * sqrt((fma(Float64(Float64(2.0 / l_m) * Float64(t_m / l_m)), t_m, 1.0) ^ -1.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-71], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(-0.5 * t$95$1 + 1.0), $MachinePrecision] * N[Sqrt[N[Power[N[(N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m + 1.0), $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))))))) < 9.9999999999999992e-72

   1. Initial program 62.9%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if 9.9999999999999992e-72 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 97.7%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 84.7%

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

   7. Applied rewrites 97.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \leq 10^{-71}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(-0.5, \frac{Om}{Omc} \cdot \frac{Om}{Omc}, 1\right) \cdot \sqrt{{\left(\mathsf{fma}\left(\frac{2}{\ell} \cdot \frac{t}{\ell}, t, 1\right)\right)}^{-1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 1.570794:\\ \;\;\;\;\sin^{-1} \left(\sqrt{{\left(\mathsf{fma}\left(\frac{2}{l\_m \cdot l\_m} \cdot t\_m, t\_m, 1\right)\right)}^{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      1.570794)
   (asin (sqrt (pow (fma (* (/ 2.0 (* l_m l_m)) t_m) t_m 1.0) -1.0)))
   (asin (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 1.570794) {
		tmp = asin(sqrt(pow(fma(((2.0 / (l_m * l_m)) * t_m), t_m, 1.0), -1.0)));
	} else {
		tmp = asin(sqrt((1.0 - ((Om / Omc) * (Om / Omc)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 1.570794)
		tmp = asin(sqrt((fma(Float64(Float64(2.0 / Float64(l_m * l_m)) * t_m), t_m, 1.0) ^ -1.0)));
	else
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1.570794], N[ArcSin[N[Sqrt[N[Power[N[(N[(N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m + 1.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $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))))))) < 1.57079400000000002

   1. Initial program 68.9%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 56.5

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

   5. Applied rewrites 56.5%

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

   if 1.57079400000000002 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 97.7%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.7

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

   5. Applied rewrites 97.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \leq 1.570794:\\ \;\;\;\;\sin^{-1} \left(\sqrt{{\left(\mathsf{fma}\left(\frac{2}{\ell \cdot \ell} \cdot t, t, 1\right)\right)}^{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot l\_m}{t\_m} \cdot \sqrt{1 - t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(-0.5, t\_1, 1\right) \cdot \left(1 - \frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (* (/ Om Omc) (/ Om Omc))))
   (if (<=
        (asin
         (sqrt
          (/
           (- 1.0 (pow (/ Om Omc) 2.0))
           (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
        1e-5)
     (asin (* (/ (* (sqrt 0.5) l_m) t_m) (sqrt (- 1.0 t_1))))
     (asin (* (fma -0.5 t_1 1.0) (- 1.0 (* (/ t_m l_m) (/ t_m l_m))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = (Om / Omc) * (Om / Omc);
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 1e-5) {
		tmp = asin((((sqrt(0.5) * l_m) / t_m) * sqrt((1.0 - t_1))));
	} else {
		tmp = asin((fma(-0.5, t_1, 1.0) * (1.0 - ((t_m / l_m) * (t_m / l_m)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(Float64(Om / Omc) * Float64(Om / Omc))
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 1e-5)
		tmp = asin(Float64(Float64(Float64(sqrt(0.5) * l_m) / t_m) * sqrt(Float64(1.0 - t_1))));
	else
		tmp = asin(Float64(fma(-0.5, t_1, 1.0) * Float64(1.0 - Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-5], N[ArcSin[N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $MachinePrecision] * N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(-0.5 * t$95$1 + 1.0), $MachinePrecision] * N[(1.0 - N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $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))))))) < 1.00000000000000008e-5

   1. Initial program 68.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if 1.00000000000000008e-5 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 97.4%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 97.1%

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

Alternative 6: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 10^{-5}:\\ \;\;\;\;\sin^{-1} \left(\left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right) \cdot \sqrt{1 - t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\mathsf{fma}\left(-0.5, t\_1, 1\right) \cdot \left(1 - \frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (* (/ Om Omc) (/ Om Omc))))
   (if (<=
        (asin
         (sqrt
          (/
           (- 1.0 (pow (/ Om Omc) 2.0))
           (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
        1e-5)
     (asin (* (* l_m (/ (sqrt 0.5) t_m)) (sqrt (- 1.0 t_1))))
     (asin (* (fma -0.5 t_1 1.0) (- 1.0 (* (/ t_m l_m) (/ t_m l_m))))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = (Om / Omc) * (Om / Omc);
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0)))))) <= 1e-5) {
		tmp = asin(((l_m * (sqrt(0.5) / t_m)) * sqrt((1.0 - t_1))));
	} else {
		tmp = asin((fma(-0.5, t_1, 1.0) * (1.0 - ((t_m / l_m) * (t_m / l_m)))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = Float64(Float64(Om / Omc) * Float64(Om / Omc))
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0)))))) <= 1e-5)
		tmp = asin(Float64(Float64(l_m * Float64(sqrt(0.5) / t_m)) * sqrt(Float64(1.0 - t_1))));
	else
		tmp = asin(Float64(fma(-0.5, t_1, 1.0) * Float64(1.0 - Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1e-5], N[ArcSin[N[(N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(-0.5 * t$95$1 + 1.0), $MachinePrecision] * N[(1.0 - N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $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))))))) < 1.00000000000000008e-5

   1. Initial program 68.6%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.4%

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

   if 1.00000000000000008e-5 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 97.4%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 97.1%

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

Alternative 7: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(-0.5 * N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[ArcSin[N[(t$95$1 * N[(1.0 - N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $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 97.4%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 97.1%

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

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

   1. Initial program 68.6%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 64.4%

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

Alternative 8: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[ArcSin[N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(-0.5 * t$95$1 + 1.0), $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] * l$95$m), $MachinePrecision] / t$95$m), $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 97.4%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.3

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

   5. Applied rewrites 96.3%

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

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

   1. Initial program 68.6%

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

   3. Taylor expanded in Om around 0

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

      1. associate-*r* N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 0.0%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 64.4%

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

Alternative 9: 77.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;t\_m \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{{\left(\mathsf{fma}\left(\frac{2}{l\_m}, \frac{t\_m \cdot t\_m}{l\_m}, 1\right)\right)}^{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{{\left(\mathsf{fma}\left(\frac{2}{l\_m \cdot l\_m} \cdot t\_m, t\_m, 1\right)\right)}^{-1}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= t_m 5e+83)
   (asin (sqrt (pow (fma (/ 2.0 l_m) (/ (* t_m t_m) l_m) 1.0) -1.0)))
   (asin (sqrt (pow (fma (* (/ 2.0 (* l_m l_m)) t_m) t_m 1.0) -1.0)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if (t_m <= 5e+83) {
		tmp = asin(sqrt(pow(fma((2.0 / l_m), ((t_m * t_m) / l_m), 1.0), -1.0)));
	} else {
		tmp = asin(sqrt(pow(fma(((2.0 / (l_m * l_m)) * t_m), t_m, 1.0), -1.0)));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (t_m <= 5e+83)
		tmp = asin(sqrt((fma(Float64(2.0 / l_m), Float64(Float64(t_m * t_m) / l_m), 1.0) ^ -1.0)));
	else
		tmp = asin(sqrt((fma(Float64(Float64(2.0 / Float64(l_m * l_m)) * t_m), t_m, 1.0) ^ -1.0)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[t$95$m, 5e+83], N[ArcSin[N[Sqrt[N[Power[N[(N[(2.0 / l$95$m), $MachinePrecision] * N[(N[(t$95$m * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[Power[N[(N[(N[(2.0 / N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision] * t$95$m + 1.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.00000000000000029e83

   1. Initial program 84.6%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 56.1

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

   5. Applied rewrites 56.1%

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

   6. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. unpow2 N/A

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

      5. times-frac N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if 5.00000000000000029e83 < t

   1. Initial program 71.6%

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

   3. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-*r/ N/A

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

      4. associate-*l/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{{\left(\mathsf{fma}\left(\frac{2}{\ell}, \frac{t \cdot t}{\ell}, 1\right)\right)}^{-1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{{\left(\mathsf{fma}\left(\frac{2}{\ell \cdot \ell} \cdot t, t, 1\right)\right)}^{-1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

l_m =     private
t_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 47.7

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

5. Applied rewrites 47.7%

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

Alternative 11: 48.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

l_m =     private
t_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. times-frac N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 47.7

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

5. Applied rewrites 47.7%

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

7. Applied rewrites 46.2%

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

9. Applied rewrites 46.2%

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

Alternative 12: 48.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \sin^{-1} \left(\mathsf{fma}\left(\frac{-0.5}{Omc}, \frac{Om \cdot Om}{Omc}, 1\right)\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (fma (/ -0.5 Omc) (/ (* Om Om) Omc) 1.0)))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	return asin(fma((-0.5 / Omc), ((Om * Om) / Omc), 1.0));
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	return asin(fma(Float64(-0.5 / Omc), Float64(Float64(Om * Om) / Omc), 1.0))
end

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Taylor expanded in Om around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

5. Applied rewrites 72.3%

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

7. Applied rewrites 72.3%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 46.0%

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

Reproduce

herbie shell --seed 2024350 
(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)))))))