Toniolo and Linder, Equation (2)

Percentage Accurate: 83.8% → 99.0%

Time: 16.2s

Alternatives: 10

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.8% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} t_1 := 1 - \mathsf{ratio\_of\_squares}\left(Om, Omc\right)\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 4 \cdot 10^{-146}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5 \cdot t\_1}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t\_1}{\mathsf{ratio\_of\_squares}\left(t\_m, l\_m\right) \cdot 2 + 1}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (ratio-of-squares Om Omc))))
   (if (<=
        (asin
         (sqrt
          (/
           (- 1.0 (pow (/ Om Omc) 2.0))
           (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
        4e-146)
     (asin (/ (* l_m (sqrt (* 0.5 t_1))) t_m))
     (asin (sqrt (/ t_1 (+ (* (ratio-of-squares t_m l_m) 2.0) 1.0)))))))

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 4.0000000000000001e-146

   1. Initial program 51.3%

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

   3. Taylor expanded in t around 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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-ratio-of-squares.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-ratio-of-squares.f64 N/A

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

      11. lift--.f64 76.2

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

   8. Applied rewrites 76.2%

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

   if 4.0000000000000001e-146 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-asin.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-ratio-of-squares.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 5 \cdot 10^{-116}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5 \cdot \left(1 - \mathsf{ratio\_of\_squares}\left(Om, Omc\right)\right)}}{t\_m} \cdot l\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{ratio\_of\_squares}\left(t\_m, l\_m\right) \cdot 2 + 1}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      5e-116)
   (asin (* (/ (sqrt (* 0.5 (- 1.0 (ratio-of-squares Om Omc)))) t_m) l_m))
   (asin (sqrt (/ 1.0 (+ (* (ratio-of-squares t_m l_m) 2.0) 1.0))))))

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))))))) < 5.0000000000000003e-116

   1. Initial program 56.6%

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

   3. Taylor expanded in l around 0

      \[\leadsto \sin^{-1} \color{blue}{\left(\ell \cdot \left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sin^{-1} \left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \color{blue}{\ell}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\left(\frac{-1}{8} \cdot \left(\frac{{\ell}^{2}}{{t}^{3} \cdot \sqrt{\frac{1}{2}}} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) + \frac{\sqrt{\frac{1}{2}}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right) \cdot \color{blue}{\ell}\right) \]

   5. Applied rewrites 13.7%

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{\sqrt{0.5}}{t} \cdot \sqrt{1 - \mathsf{ratio\_of\_squares}\left(Om, Omc\right)} + \left(\frac{\mathsf{ratio\_of\_squares}\left(\ell, \left({t}^{1.5}\right)\right)}{\sqrt{0.5}} \cdot \sqrt{1 - \mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right) \cdot -0.125\right) \cdot \ell\right)} \]

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-unprod N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-ratio-of-squares.f64 N/A

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

      9. lift--.f64 71.7

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

   8. Applied rewrites 71.7%

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

   if 5.0000000000000003e-116 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-asin.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-ratio-of-squares.f64 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 97.6%

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

Alternative 3: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 4 \cdot 10^{-146}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{ratio\_of\_squares}\left(t\_m, l\_m\right) \cdot 2 + 1}}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      4e-146)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin (sqrt (/ 1.0 (+ (* (ratio-of-squares t_m l_m) 2.0) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 4.0000000000000001e-146

   1. Initial program 51.3%

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

   3. Taylor expanded in t around 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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-ratio-of-squares.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-ratio-of-squares.f64 N/A

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

      11. lift--.f64 76.2

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 76.2%

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

   if 4.0000000000000001e-146 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.3%

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

   3. Taylor expanded in t around 0

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

      1. lower-asin.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-ratio-of-squares.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 97.4%

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

Alternative 4: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 4 \cdot 10^{-146}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{ratio\_square\_sum}\left(1, \left(\mathsf{ratio\_of\_squares}\left(t\_m, l\_m\right) \cdot 2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      4e-146)
   (asin (/ (* l_m (sqrt 0.5)) t_m))
   (asin (sqrt (ratio-square-sum 1.0 (* (ratio-of-squares t_m l_m) 2.0))))))

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 4.0000000000000001e-146

   1. Initial program 51.3%

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

   3. Taylor expanded in t around 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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-ratio-of-squares.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in t around inf

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

      1. associate-*l/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. sqrt-unprod N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. lift-ratio-of-squares.f64 N/A

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

      11. lift--.f64 76.2

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in Om around 0

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

   11. Applied rewrites 76.2%

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

   if 4.0000000000000001e-146 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 98.3%

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

   3. Taylor expanded in Om around 0

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

      1. metadata-eval N/A

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

      2. lower-ratio-square-sum.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 97.4

         \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{ratio\_square\_sum}\left(1, \left(\mathsf{ratio\_of\_squares}\left(t, \ell\right) \cdot 2\right)\right)}\right) \]

   5. Applied rewrites 97.4%

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

Alternative 5: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right) \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{ratio\_of\_squares}\left(l\_m, t\_m\right) \cdot 0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{ratio\_square\_sum}\left(1, \left(\mathsf{ratio\_of\_squares}\left(t\_m, l\_m\right) \cdot 2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<=
      (asin
       (sqrt
        (/
         (- 1.0 (pow (/ Om Omc) 2.0))
         (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
      5e-11)
   (asin (sqrt (* (ratio-of-squares l_m t_m) 0.5)))
   (asin (sqrt (ratio-square-sum 1.0 (* (ratio-of-squares t_m l_m) 2.0))))))

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))))))) < 5.00000000000000018e-11

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-ratio-of-squares.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in Om around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-ratio-of-squares.f64 73.0

         \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{ratio\_of\_squares}\left(\ell, t\right) \cdot 0.5}\right) \]

   8. Applied rewrites 73.0%

      \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{ratio\_of\_squares}\left(\ell, t\right) \cdot 0.5}\right) \]

   if 5.00000000000000018e-11 < (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 97.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   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. metadata-eval N/A

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

      2. lower-ratio-square-sum.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 96.9

         \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{ratio\_square\_sum}\left(1, \left(\mathsf{ratio\_of\_squares}\left(t, \ell\right) \cdot 2\right)\right)}\right) \]

   5. Applied rewrites 96.9%

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

Alternative 6: 83.3% accurate, 0.7× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.7%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-ratio-of-squares.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in Om around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-ratio-of-squares.f64 73.0

         \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{ratio\_of\_squares}\left(\ell, t\right) \cdot 0.5}\right) \]

   8. Applied rewrites 73.0%

      \[\leadsto \sin^{-1} \left(\sqrt{\mathsf{ratio\_of\_squares}\left(\ell, t\right) \cdot 0.5}\right) \]

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

   1. Initial program 97.8%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-ratio-of-squares.f64 97.0

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

   5. Applied rewrites 97.0%

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

Alternative 7: 57.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.45 \cdot 10^{+240}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{-\mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.45e+240)
   (asin (sqrt (- 1.0 (ratio-of-squares Om Omc))))
   (asin (sqrt (- (ratio-of-squares Om Omc))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.44999999999999999e240

   1. Initial program 85.1%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-ratio-of-squares.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if 1.44999999999999999e240 < (/.f64 t l)

   1. Initial program 83.2%

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

   3. Taylor expanded in Om around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-ratio-of-squares.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in t around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-ratio-of-squares.f64 48.0

         \[\leadsto \sin^{-1} \left(\sqrt{-\mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right) \]

   8. Applied rewrites 48.0%

      \[\leadsto \sin^{-1} \left(\sqrt{-\mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 1.45 \cdot 10^{+240}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{-\mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right)\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 1.45e+240)
   (asin (sqrt 1.0))
   (asin (sqrt (- (ratio-of-squares Om Omc))))))

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.44999999999999999e240

   1. Initial program 85.1%

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. lower-ratio-of-squares.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 54.8%

      \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]

   if 1.44999999999999999e240 < (/.f64 t l)

   1. Initial program 83.2%

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

   3. Taylor expanded in Om around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-ratio-of-squares.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. unpow2 N/A

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

      14. lower-ratio-of-squares.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in t around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lift-ratio-of-squares.f64 48.0

         \[\leadsto \sin^{-1} \left(\sqrt{-\mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right) \]

   8. Applied rewrites 48.0%

      \[\leadsto \sin^{-1} \left(\sqrt{-\mathsf{ratio\_of\_squares}\left(Om, Omc\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ l_m = \left|\ell\right| \\ \sin^{-1} \left(\frac{1}{\mathsf{ratio\_of\_squares}\left(t\_m, l\_m\right) + 1}\right) \end{array} \]

FPCore

t_m = (fabs.f64 t)
l_m = (fabs.f64 l)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (asin (/ 1.0 (+ (ratio-of-squares t_m l_m) 1.0))))

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in t around 0

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

   1. lower-asin.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower--.f64 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. lower-ratio-of-squares.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. unpow2 N/A

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

   14. lower-ratio-of-squares.f64 85.0

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

5. Applied rewrites 85.0%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-ratio-of-squares.f64 N/A

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

   6. sqrt-div N/A

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

   7. lift-ratio-of-squares.f64 N/A

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

   8. pow2 N/A

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

   9. pow2 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

7. Applied rewrites 84.9%

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

8. Taylor expanded in Om around 0

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

10. Applied rewrites 84.2%

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

11. Taylor expanded in t around 0

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

    1. +-commutative N/A

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

    2. lower-+.f64 N/A

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

    3. pow2 N/A

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

    4. pow2 N/A

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

    5. lift-ratio-of-squares.f64 62.6

       \[\leadsto \sin^{-1} \left(\frac{1}{\mathsf{ratio\_of\_squares}\left(t, \ell\right) + 1}\right) \]

13. Applied rewrites 62.6%

    \[\leadsto \sin^{-1} \left(\frac{1}{\mathsf{ratio\_of\_squares}\left(t, \ell\right) + 1}\right) \]
14. Add Preprocessing

Alternative 10: 50.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in t around 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

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

   2. unpow2 N/A

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

   3. unpow2 N/A

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

   4. lower-ratio-of-squares.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
7. Step-by-step derivation

8. Applied rewrites 49.6%

   \[\leadsto \sin^{-1} \left(\sqrt{1}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025058 
(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)))))))