Toniolo and Linder, Equation (2)

Percentage Accurate: 83.5% → 98.8%

Time: 9.1s

Alternatives: 6

Speedup: 0.7×

Specification

Math

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

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: 83.5% accurate, 1.0× speedup

Math

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

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: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{\left|\ell\right|}\\ t_2 := \sqrt{\frac{\sqrt{0.5 \cdot \left(1 - \frac{\frac{Om}{Omc}}{Omc} \cdot Om\right)}}{\left|t\right|}}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 0:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \left(t\_2 \cdot t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\mathsf{fma}\left(\frac{t + t}{\left|\ell\right|}, t\_1, 1\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (let* ((t_1 (/ t (fabs l)))
       (t_2
        (sqrt
         (/
          (sqrt (* 0.5 (- 1.0 (* (/ (/ Om Omc) Omc) Om))))
          (fabs t)))))
  (if (<=
       (asin
        (sqrt
         (/
          (- 1.0 (pow (/ Om Omc) 2.0))
          (+ 1.0 (* 2.0 (pow t_1 2.0))))))
       0.0)
    (asin (* (fabs l) (* t_2 t_2)))
    (asin
     (sqrt
      (/
       (- 1.0 (/ (* (/ Om Omc) Om) Omc))
       (fma (/ (+ t t) (fabs l)) t_1 1.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = t / fabs(l);
	double t_2 = sqrt((sqrt((0.5 * (1.0 - (((Om / Omc) / Omc) * Om)))) / fabs(t)));
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow(t_1, 2.0)))))) <= 0.0) {
		tmp = asin((fabs(l) * (t_2 * t_2)));
	} else {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / fma(((t + t) / fabs(l)), t_1, 1.0))));
	}
	return tmp;
}

Julia

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

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(t / N[Abs[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Sqrt[N[(0.5 * N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t], $MachinePrecision]), $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[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.0], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t + t), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.5%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 20.8%

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

   4. Applied rewrites 20.8%

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

   6. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 28.9%

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

   8. Applied rewrites 28.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 30.7%

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

   10. Applied rewrites 30.7%

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

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

   1. Initial program 83.5%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 83.5%

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

   3. Applied rewrites 83.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.5%

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

   5. Applied rewrites 83.5%

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

Alternative 2: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left|t\right|}{\left|\ell\right|}\\ \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {t\_1}^{2}}}\right) \leq 5 \cdot 10^{-98}:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \frac{\sqrt{\frac{0.5}{\left|t\right|}}}{\sqrt{\left|t\right|}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{\mathsf{fma}\left(\frac{\left|t\right| + \left|t\right|}{\left|\ell\right|}, t\_1, 1\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (let* ((t_1 (/ (fabs t) (fabs l))))
  (if (<=
       (asin
        (sqrt
         (/
          (- 1.0 (pow (/ Om Omc) 2.0))
          (+ 1.0 (* 2.0 (pow t_1 2.0))))))
       5e-98)
    (asin (* (fabs l) (/ (sqrt (/ 0.5 (fabs t))) (sqrt (fabs t)))))
    (asin
     (sqrt
      (/
       (- 1.0 (/ (* (/ Om Omc) Om) Omc))
       (fma (/ (+ (fabs t) (fabs t)) (fabs l)) t_1 1.0)))))))

C

double code(double t, double l, double Om, double Omc) {
	double t_1 = fabs(t) / fabs(l);
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow(t_1, 2.0)))))) <= 5e-98) {
		tmp = asin((fabs(l) * (sqrt((0.5 / fabs(t))) / sqrt(fabs(t)))));
	} else {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / fma(((fabs(t) + fabs(t)) / fabs(l)), t_1, 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	t_1 = Float64(abs(t) / abs(l))
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (t_1 ^ 2.0)))))) <= 5e-98)
		tmp = asin(Float64(abs(l) * Float64(sqrt(Float64(0.5 / abs(t))) / sqrt(abs(t)))));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / fma(Float64(Float64(abs(t) + abs(t)) / abs(l)), t_1, 1.0))));
	end
	return tmp
end

Wolfram

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(N[Abs[t], $MachinePrecision] / N[Abs[l], $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[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 5e-98], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[(N[Sqrt[N[(0.5 / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Abs[t], $MachinePrecision] + N[Abs[t], $MachinePrecision]), $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision] * t$95$1 + 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))))))) < 5.0000000000000002e-98

   1. Initial program 83.5%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 20.8%

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

   4. Applied rewrites 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 23.8%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. sqrt-div N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lower-unsound-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-unsound-sqrt.f64 15.5%

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

   11. Applied rewrites 15.5%

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

   if 5.0000000000000002e-98 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 83.5%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 83.5%

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

   3. Applied rewrites 83.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.5%

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

   5. Applied rewrites 83.5%

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

Alternative 3: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{\left|t\right|}{\left|\ell\right|}\right)}^{2}}}\right) \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \frac{\sqrt{\frac{0.5}{\left|t\right|}}}{\sqrt{\left|t\right|}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc} \cdot Om}{Omc}}{1}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (if (<=
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (pow (/ (fabs t) (fabs l)) 2.0))))))
     0.4)
  (asin (* (fabs l) (/ (sqrt (/ 0.5 (fabs t))) (sqrt (fabs t)))))
  (asin (sqrt (/ (- 1.0 (/ (* (/ Om Omc) Om) Omc)) 1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((fabs(t) / fabs(l)), 2.0)))))) <= 0.4) {
		tmp = asin((fabs(l) * (sqrt((0.5 / fabs(t))) / sqrt(fabs(t)))));
	} else {
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((abs(t) / abs(l)) ** 2.0d0)))))) <= 0.4d0) then
        tmp = asin((abs(l) * (sqrt((0.5d0 / abs(t))) / sqrt(abs(t)))))
    else
        tmp = asin(sqrt(((1.0d0 - (((om / omc) * om) / omc)) / 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * Math.pow((Math.abs(t) / Math.abs(l)), 2.0)))))) <= 0.4) {
		tmp = Math.asin((Math.abs(l) * (Math.sqrt((0.5 / Math.abs(t))) / Math.sqrt(Math.abs(t)))));
	} else {
		tmp = Math.asin(Math.sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	}
	return tmp;
}

Python

def code(t, l, Om, Omc):
	tmp = 0
	if math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * math.pow((math.fabs(t) / math.fabs(l)), 2.0)))))) <= 0.4:
		tmp = math.asin((math.fabs(l) * (math.sqrt((0.5 / math.fabs(t))) / math.sqrt(math.fabs(t)))))
	else:
		tmp = math.asin(math.sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)))
	return tmp

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(abs(t) / abs(l)) ^ 2.0)))))) <= 0.4)
		tmp = asin(Float64(abs(l) * Float64(sqrt(Float64(0.5 / abs(t))) / sqrt(abs(t)))));
	else
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(Om / Omc) * Om) / Omc)) / 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, Om, Omc)
	tmp = 0.0;
	if (asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((abs(t) / abs(l)) ^ 2.0)))))) <= 0.4)
		tmp = asin((abs(l) * (sqrt((0.5 / abs(t))) / sqrt(abs(t)))));
	else
		tmp = asin(sqrt(((1.0 - (((Om / Omc) * Om) / Omc)) / 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, 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[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.4], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[(N[Sqrt[N[(0.5 / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(N[(Om / Omc), $MachinePrecision] * Om), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.40000000000000002

   1. Initial program 83.5%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 20.8%

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

   4. Applied rewrites 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 23.8%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. sqrt-div N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lower-unsound-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-unsound-sqrt.f64 15.5%

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

   11. Applied rewrites 15.5%

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

   if 0.40000000000000002 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 83.5%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 51.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 51.2%

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

   6. Applied rewrites 51.2%

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

Alternative 4: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{\left|t\right|}{\left|\ell\right|}\right)}^{2}}}\right) \leq 0.4:\\ \;\;\;\;\sin^{-1} \left(\left|\ell\right| \cdot \frac{\sqrt{\frac{0.5}{\left|t\right|}}}{\sqrt{\left|t\right|}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\mathsf{fma}\left(\frac{Om}{Omc \cdot Omc}, Om, -1\right)}{-1}}\right)\\ \end{array} \]

FPCore

(FPCore (t l Om Omc)
  :precision binary64
  (if (<=
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (+ 1.0 (* 2.0 (pow (/ (fabs t) (fabs l)) 2.0))))))
     0.4)
  (asin (* (fabs l) (/ (sqrt (/ 0.5 (fabs t))) (sqrt (fabs t)))))
  (asin (sqrt (/ (fma (/ Om (* Omc Omc)) Om -1.0) -1.0)))))

C

double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((fabs(t) / fabs(l)), 2.0)))))) <= 0.4) {
		tmp = asin((fabs(l) * (sqrt((0.5 / fabs(t))) / sqrt(fabs(t)))));
	} else {
		tmp = asin(sqrt((fma((Om / (Omc * Omc)), Om, -1.0) / -1.0)));
	}
	return tmp;
}

Julia

function code(t, l, Om, Omc)
	tmp = 0.0
	if (asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(abs(t) / abs(l)) ^ 2.0)))))) <= 0.4)
		tmp = asin(Float64(abs(l) * Float64(sqrt(Float64(0.5 / abs(t))) / sqrt(abs(t)))));
	else
		tmp = asin(sqrt(Float64(fma(Float64(Om / Float64(Omc * Omc)), Om, -1.0) / -1.0)));
	end
	return tmp
end

Wolfram

code[t_, l_, 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[(N[Abs[t], $MachinePrecision] / N[Abs[l], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.4], N[ArcSin[N[(N[Abs[l], $MachinePrecision] * N[(N[Sqrt[N[(0.5 / N[Abs[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[Sqrt[N[(N[(N[(Om / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision] * Om + -1.0), $MachinePrecision] / -1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64))))))) < 0.40000000000000002

   1. Initial program 83.5%

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

   2. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 20.8%

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

   4. Applied rewrites 20.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 22.5%

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

   7. Taylor expanded in Om around 0

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

   9. Applied rewrites 23.8%

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

       1. lift-sqrt.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. sqrt-div N/A

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

       6. lower-unsound-/.f64 N/A

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

       7. lower-unsound-sqrt.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-unsound-sqrt.f64 15.5%

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

   11. Applied rewrites 15.5%

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

   if 0.40000000000000002 < (asin.f64 (sqrt.f64 (/.f64 (-.f64 #s(literal 1 binary64) (pow.f64 (/.f64 Om Omc) #s(literal 2 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 2 binary64) (pow.f64 (/.f64 t l) #s(literal 2 binary64)))))))

   1. Initial program 83.5%

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

      1. remove-double-neg N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. remove-double-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. associate-*l* N/A

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

      13. *-commutative N/A

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

      14. count-2-rev N/A

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

      15. *-commutative N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. div-add-rev N/A

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

      21. lower-/.f64 N/A

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

      22. lower-+.f64 83.5%

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

   3. Applied rewrites 83.5%

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 48.1%

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

Alternative 5: 48.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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((abs(l) * (sqrt((0.5d0 / abs(t))) / sqrt(abs(t)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.5%

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

2. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 20.8%

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

4. Applied rewrites 20.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 22.5%

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

7. Taylor expanded in Om around 0

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

9. Applied rewrites 23.8%

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

    1. lift-sqrt.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-/r* N/A

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

    5. sqrt-div N/A

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

    6. lower-unsound-/.f64 N/A

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

    7. lower-unsound-sqrt.f64 N/A

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

    8. lower-/.f64 N/A

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

    9. lower-unsound-sqrt.f64 15.5%

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

11. Applied rewrites 15.5%

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

Alternative 6: 48.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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(0.5d0) / abs(t)) * abs(l)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.5%

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

2. Taylor expanded in l around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 20.8%

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

4. Applied rewrites 20.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

6. Applied rewrites 22.5%

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

7. Taylor expanded in Om around 0

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

9. Applied rewrites 23.8%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-*.f64 23.8%

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

    4. lift-sqrt.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. sqrt-div N/A

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

    7. lower-unsound-sqrt.f64 N/A

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

    8. lower-sqrt.f64 N/A

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

    9. lift-*.f64 N/A

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

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

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

    11. lower-unsound-/.f64 N/A

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

    12. lower-unsound-sqrt.f64 N/A

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

    13. lower-fabs.f64 30.6%

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

11. Applied rewrites 30.6%

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

Reproduce

herbie shell --seed 2025210 
(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)))))))