Toniolo and Linder, Equation (2)

Percentage Accurate: 84.2% → 98.5%

Time: 16.6s

Alternatives: 10

Speedup: 2.5×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% 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

real(8) function code(t, l, om, omc)
    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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t\_m}{l\_m}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \mathsf{fma}\left(-0.125, \frac{l\_m \cdot l\_m}{\left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \sqrt{0.5}}, \frac{\sqrt{0.5}}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 5e+147)
   (asin
    (sqrt
     (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (pow (/ t_m l_m) 2.0))))))
   (asin
    (*
     l_m
     (fma
      -0.125
      (/ (* l_m l_m) (* (* t_m (* t_m t_m)) (sqrt 0.5)))
      (/ (sqrt 0.5) t_m))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+147) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t_m / l_m), 2.0))))));
	} else {
		tmp = asin((l_m * fma(-0.125, ((l_m * l_m) / ((t_m * (t_m * t_m)) * sqrt(0.5))), (sqrt(0.5) / t_m))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+147)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * (Float64(t_m / l_m) ^ 2.0))))));
	else
		tmp = asin(Float64(l_m * fma(-0.125, Float64(Float64(l_m * l_m) / Float64(Float64(t_m * Float64(t_m * t_m)) * sqrt(0.5))), Float64(sqrt(0.5) / t_m))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+147], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(-0.125 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.0000000000000002e147

   1. Initial program 98.1%

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

   if 5.0000000000000002e147 < (/.f64 t l)

   1. Initial program 44.9%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 90.4%

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

   6. Taylor expanded in Om around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sqrt{\frac{1}{2}}}, \color{blue}{\frac{\sqrt{\frac{1}{2}}}{t}}\right)\right) \]

      13. lower-sqrt.f64 99.7

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sqrt{0.5}}, \frac{\color{blue}{\sqrt{0.5}}}{t}\right)\right) \]

   8. Simplified 99.7%

      \[\leadsto \sin^{-1} \left(\ell \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sqrt{0.5}}, \frac{\sqrt{0.5}}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} t_1 := \sqrt{\frac{t\_m}{l\_m}}\\ \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\left(\frac{t\_m}{l\_m} \cdot 2\right) \cdot t\_1, t\_1, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \mathsf{fma}\left(-0.125, \frac{l\_m \cdot l\_m}{\left(t\_m \cdot \left(t\_m \cdot t\_m\right)\right) \cdot \sqrt{0.5}}, \frac{\sqrt{0.5}}{t\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (let* ((t_1 (sqrt (/ t_m l_m))))
   (if (<= (/ t_m l_m) 5e+147)
     (asin
      (sqrt
       (/
        (- 1.0 (pow (/ Om Omc) 2.0))
        (fma (* (* (/ t_m l_m) 2.0) t_1) t_1 1.0))))
     (asin
      (*
       l_m
       (fma
        -0.125
        (/ (* l_m l_m) (* (* t_m (* t_m t_m)) (sqrt 0.5)))
        (/ (sqrt 0.5) t_m)))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double t_1 = sqrt((t_m / l_m));
	double tmp;
	if ((t_m / l_m) <= 5e+147) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma((((t_m / l_m) * 2.0) * t_1), t_1, 1.0))));
	} else {
		tmp = asin((l_m * fma(-0.125, ((l_m * l_m) / ((t_m * (t_m * t_m)) * sqrt(0.5))), (sqrt(0.5) / t_m))));
	}
	return tmp;
}

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	t_1 = sqrt(Float64(t_m / l_m))
	tmp = 0.0
	if (Float64(t_m / l_m) <= 5e+147)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(Float64(Float64(Float64(t_m / l_m) * 2.0) * t_1), t_1, 1.0))));
	else
		tmp = asin(Float64(l_m * fma(-0.125, Float64(Float64(l_m * l_m) / Float64(Float64(t_m * Float64(t_m * t_m)) * sqrt(0.5))), Float64(sqrt(0.5) / t_m))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := Block[{t$95$1 = N[Sqrt[N[(t$95$m / l$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+147], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(-0.125 * N[(N[(l$95$m * l$95$m), $MachinePrecision] / N[(N[(t$95$m * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5.0000000000000002e147

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. *-rgt-identity N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. unpow1 N/A

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

      12. sqr-pow N/A

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

      13. associate-*r* N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied egg-rr 98.6%

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

   if 5.0000000000000002e147 < (/.f64 t l)

   1. Initial program 47.3%

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

   3. Taylor expanded in l around 0

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

   5. Simplified 87.7%

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

   6. Taylor expanded in Om around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. cube-mult N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot \left(t \cdot t\right)\right)} \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{{t}^{2}}\right) \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\color{blue}{\left(t \cdot {t}^{2}\right)} \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \color{blue}{\left(t \cdot t\right)}\right) \cdot \sqrt{\frac{1}{2}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2}}}}, \frac{\sqrt{\frac{1}{2}}}{t}\right)\right) \]

      12. lower-/.f64 N/A

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(\frac{-1}{8}, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sqrt{\frac{1}{2}}}, \color{blue}{\frac{\sqrt{\frac{1}{2}}}{t}}\right)\right) \]

      13. lower-sqrt.f64 97.9

          \[\leadsto \sin^{-1} \left(\ell \cdot \mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sqrt{0.5}}, \frac{\color{blue}{\sqrt{0.5}}}{t}\right)\right) \]

   8. Simplified 97.9%

      \[\leadsto \sin^{-1} \left(\ell \cdot \color{blue}{\mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sqrt{0.5}}, \frac{\sqrt{0.5}}{t}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 5 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(\left(\frac{t}{\ell} \cdot 2\right) \cdot \sqrt{\frac{t}{\ell}}, \sqrt{\frac{t}{\ell}}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \mathsf{fma}\left(-0.125, \frac{\ell \cdot \ell}{\left(t \cdot \left(t \cdot t\right)\right) \cdot \sqrt{0.5}}, \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024218 
(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)))))))