Toniolo and Linder, Equation (2)

Percentage Accurate: 84.0% → 98.9%

Time: 12.4s

Alternatives: 8

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: 84.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 5 \cdot 10^{+144}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t\_m}{l\_m} \cdot \frac{t\_m}{l\_m}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right)}}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

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+144)
   (asin
    (sqrt
     (/
      (- 1.0 (pow (/ Om Omc) 2.0))
      (+ 1.0 (* 2.0 (* (/ t_m l_m) (/ t_m l_m)))))))
   (asin
    (/
     (* l_m (sqrt (fma (/ (/ Om Omc) Omc) (- Om) 1.0)))
     (* t_m (sqrt 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 5e+144) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t_m / l_m) * (t_m / l_m)))))));
	} else {
		tmp = asin(((l_m * sqrt(fma(((Om / Omc) / Omc), -Om, 1.0))) / (t_m * sqrt(2.0))));
	}
	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+144)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t_m / l_m) * Float64(t_m / l_m)))))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0))) / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 5e+144], 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[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 4.9999999999999999e144

   1. Initial program 91.1%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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. unpow2 N/A

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

      3. lower-*.f64 91.1

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

   4. Applied egg-rr 91.1%

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

   if 4.9999999999999999e144 < (/.f64 t l)

   1. Initial program 38.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   5. Simplified 32.8%

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

   7. Applied egg-rr 90.7%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 99.5

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

   9. Applied egg-rr 99.5%

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

Alternative 2: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right)}}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

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) 2e+26)
   (asin (sqrt (/ 1.0 (fma 2.0 (/ (* t_m (/ t_m l_m)) l_m) 1.0))))
   (asin
    (/
     (* l_m (sqrt (fma (/ (/ Om Omc) Omc) (- Om) 1.0)))
     (* t_m (sqrt 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+26) {
		tmp = asin(sqrt((1.0 / fma(2.0, ((t_m * (t_m / l_m)) / l_m), 1.0))));
	} else {
		tmp = asin(((l_m * sqrt(fma(((Om / Omc) / Omc), -Om, 1.0))) / (t_m * sqrt(2.0))));
	}
	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) <= 2e+26)
		tmp = asin(sqrt(Float64(1.0 / fma(2.0, Float64(Float64(t_m * Float64(t_m / l_m)) / l_m), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(fma(Float64(Float64(Om / Omc) / Omc), Float64(-Om), 1.0))) / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+26], N[ArcSin[N[Sqrt[N[(1.0 / N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] * (-Om) + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e26

   1. Initial program 90.0%

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

   3. Taylor expanded in Om around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.7

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

   7. Applied egg-rr 87.7%

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

   if 2.0000000000000001e26 < (/.f64 t l)

   1. Initial program 62.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   5. Simplified 34.2%

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

   7. Applied egg-rr 93.7%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 99.4

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

   9. Applied egg-rr 99.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot \frac{t}{\ell}}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{\mathsf{fma}\left(\frac{\frac{Om}{Omc}}{Omc}, -Om, 1\right)}}{t \cdot \sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

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) 2e+26)
   (asin (sqrt (/ 1.0 (fma 2.0 (/ (* t_m (/ t_m l_m)) l_m) 1.0))))
   (asin (/ l_m (* t_m (sqrt 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 2e+26) {
		tmp = asin(sqrt((1.0 / fma(2.0, ((t_m * (t_m / l_m)) / l_m), 1.0))));
	} else {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	}
	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) <= 2e+26)
		tmp = asin(sqrt(Float64(1.0 / fma(2.0, Float64(Float64(t_m * Float64(t_m / l_m)) / l_m), 1.0))));
	else
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 2e+26], N[ArcSin[N[Sqrt[N[(1.0 / N[(2.0 * N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 2.0000000000000001e26

   1. Initial program 90.0%

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

   3. Taylor expanded in Om around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 87.7

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

   7. Applied egg-rr 87.7%

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

   if 2.0000000000000001e26 < (/.f64 t l)

   1. Initial program 62.6%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   5. Simplified 34.2%

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

   7. Applied egg-rr 93.7%

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

   8. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 97.9

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

   10. Simplified 97.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t \cdot \frac{t}{\ell}}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{t \cdot \sqrt{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{\frac{Om}{Omc}}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{\frac{0.5}{t\_m}}}{\sqrt{t\_m}}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.005)
   (asin (sqrt (fma (- Om) (/ (/ Om Omc) Omc) 1.0)))
   (asin (/ (* l_m (sqrt (/ 0.5 t_m))) (sqrt 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) <= 0.005) {
		tmp = asin(sqrt(fma(-Om, ((Om / Omc) / Omc), 1.0)));
	} else {
		tmp = asin(((l_m * sqrt((0.5 / t_m))) / sqrt(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) <= 0.005)
		tmp = asin(sqrt(fma(Float64(-Om), Float64(Float64(Om / Omc) / Omc), 1.0)));
	else
		tmp = asin(Float64(Float64(l_m * sqrt(Float64(0.5 / t_m))) / sqrt(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], 0.005], N[ArcSin[N[Sqrt[N[((-Om) * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[N[(0.5 / t$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.8

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

   5. Simplified 63.8%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lower-/.f64 66.6

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

   7. Applied egg-rr 66.6%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   5. Simplified 37.5%

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

   6. Taylor expanded in Om around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 96.7

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

   8. Simplified 96.7%

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

      1. lift-sqrt.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lift-sqrt.f64 N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow1 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. sqrt-prod N/A

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

      11. lift-*.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. sqrt-prod N/A

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

      14. lift-*.f64 N/A

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

      15. sqrt-div N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/r* N/A

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

      18. sqrt-div N/A

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

      19. lower-/.f64 N/A

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

   10. Applied egg-rr 46.6%

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

Alternative 5: 97.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-Om, \frac{\frac{Om}{Omc}}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

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) 0.005)
   (asin (sqrt (fma (- Om) (/ (/ Om Omc) Omc) 1.0)))
   (asin (/ l_m (* t_m (sqrt 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin(sqrt(fma(-Om, ((Om / Omc) / Omc), 1.0)));
	} else {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	}
	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) <= 0.005)
		tmp = asin(sqrt(fma(Float64(-Om), Float64(Float64(Om / Omc) / Omc), 1.0)));
	else
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[N[Sqrt[N[((-Om) * N[(N[(Om / Omc), $MachinePrecision] / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.8

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

   5. Simplified 63.8%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lower-/.f64 66.6

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

   7. Applied egg-rr 66.6%

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

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   5. Simplified 37.5%

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

   7. Applied egg-rr 92.9%

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

   8. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 96.7

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

   10. Simplified 96.7%

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

Alternative 6: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m}{t\_m \cdot \sqrt{2}}\right)\\ \end{array} \end{array} \]

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) 0.005) (asin 1.0) (asin (/ l_m (* t_m (sqrt 2.0))))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.005d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l_m / (t_m * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l_m / (t_m * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.005:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l_m / (t_m * math.sqrt(2.0))))
	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) <= 0.005)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l_m / Float64(t_m * sqrt(2.0))));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.005)
		tmp = asin(1.0);
	else
		tmp = asin((l_m / (t_m * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l$95$m / N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.8

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

   5. Simplified 63.8%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 66.3%

      \[\leadsto \sin^{-1} \color{blue}{1} \]

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   5. Simplified 37.5%

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

   7. Applied egg-rr 92.9%

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

   8. Taylor expanded in Om around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 96.7

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

   10. Simplified 96.7%

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

Alternative 7: 96.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.005:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.005) (asin 1.0) (asin (* l_m (/ (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) <= 0.005) {
		tmp = asin(1.0);
	} else {
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	}
	return tmp;
}

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    real(8) :: tmp
    if ((t_m / l_m) <= 0.005d0) then
        tmp = asin(1.0d0)
    else
        tmp = asin((l_m * (sqrt(0.5d0) / t_m)))
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
t_m = Math.abs(t);
public static double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.005) {
		tmp = Math.asin(1.0);
	} else {
		tmp = Math.asin((l_m * (Math.sqrt(0.5) / t_m)));
	}
	return tmp;
}

Python

l_m = math.fabs(l)
t_m = math.fabs(t)
def code(t_m, l_m, Om, Omc):
	tmp = 0
	if (t_m / l_m) <= 0.005:
		tmp = math.asin(1.0)
	else:
		tmp = math.asin((l_m * (math.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) <= 0.005)
		tmp = asin(1.0);
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

MATLAB

l_m = abs(l);
t_m = abs(t);
function tmp_2 = code(t_m, l_m, Om, Omc)
	tmp = 0.0;
	if ((t_m / l_m) <= 0.005)
		tmp = asin(1.0);
	else
		tmp = asin((l_m * (sqrt(0.5) / t_m)));
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.005], N[ArcSin[1.0], $MachinePrecision], N[ArcSin[N[(l$95$m * N[(N[Sqrt[0.5], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.0050000000000000001

   1. Initial program 89.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. distribute-lft-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 63.8

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

   5. Simplified 63.8%

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

   6. Taylor expanded in Om around 0

      \[\leadsto \sin^{-1} \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 66.3%

      \[\leadsto \sin^{-1} \color{blue}{1} \]

   if 0.0050000000000000001 < (/.f64 t l)

   1. Initial program 66.3%

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

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

   5. Simplified 37.5%

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

   6. Taylor expanded in Om around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 96.7

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

   8. Simplified 96.7%

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

Alternative 8: 49.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

l_m = abs(l)
t_m = abs(t)
real(8) function code(t_m, l_m, om, omc)
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om
    real(8), intent (in) :: omc
    code = asin(1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in t around 0

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

   1. lower-sqrt.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-/l* N/A

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

   6. distribute-lft-neg-in N/A

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

   7. mul-1-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 47.7

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

5. Simplified 47.7%

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

6. Taylor expanded in Om around 0

   \[\leadsto \sin^{-1} \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 49.6%

   \[\leadsto \sin^{-1} \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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)))))))