Toniolo and Linder, Equation (2)

Percentage Accurate: 84.3% → 98.9%

Time: 14.0s

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: 84.3% 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, 1.8× speedup

Math

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

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) 1e+151)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 1.0))))
   (asin
    (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (* 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) <= 1e+151) {
		tmp = asin(sqrt(((1.0 - ((Om * (Om / Omc)) / Omc)) / fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((l_m * 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) <= 1e+151)
		tmp = asin(sqrt(Float64(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc)) / fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(l_m * 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], 1e+151], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 1.00000000000000002e151

   1. Initial program 89.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 89.3

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

   4. Applied rewrites 89.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 89.3

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

   6. Applied rewrites 89.3%

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

   if 1.00000000000000002e151 < (/.f64 t l)

   1. Initial program 53.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 93.4

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

   5. Applied rewrites 93.4%

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

   7. Applied rewrites 99.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+151}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{-\mathsf{fma}\left(\frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (pow (/ t_m l_m) 2.0) 2e+33)
   (asin (/ 1.0 (sqrt (- (fma (/ (* t_m (/ t_m l_m)) l_m) -2.0 -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 (pow((t_m / l_m), 2.0) <= 2e+33) {
		tmp = asin((1.0 / sqrt(-fma(((t_m * (t_m / l_m)) / l_m), -2.0, -1.0))));
	} else {
		tmp = asin(((l_m * 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) ^ 2.0) <= 2e+33)
		tmp = asin(Float64(1.0 / sqrt(Float64(-fma(Float64(Float64(t_m * Float64(t_m / l_m)) / l_m), -2.0, -1.0)))));
	else
		tmp = asin(Float64(Float64(l_m * 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[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision], 2e+33], N[ArcSin[N[(1.0 / N[Sqrt[(-N[(N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) #s(literal 2 binary64)) < 1.9999999999999999e33

   1. Initial program 99.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in Om around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 96.1%

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

   if 1.9999999999999999e33 < (pow.f64 (/.f64 t l) #s(literal 2 binary64))

   1. Initial program 69.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 60.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{t}{\ell}\right)}^{2} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{-\mathsf{fma}\left(\frac{t \cdot \frac{t}{\ell}}{\ell}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;{\left(\frac{t\_m}{l\_m}\right)}^{2} \leq 2 \cdot 10^{+33}:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (pow (/ t_m l_m) 2.0) 2e+33)
   (asin (/ 1.0 (sqrt (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 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 (pow((t_m / l_m), 2.0) <= 2e+33) {
		tmp = asin((1.0 / sqrt(fma((t_m / l_m), ((t_m / l_m) * 2.0), 1.0))));
	} else {
		tmp = asin(((l_m * 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) ^ 2.0) <= 2e+33)
		tmp = asin(Float64(1.0 / sqrt(fma(Float64(t_m / l_m), Float64(Float64(t_m / l_m) * 2.0), 1.0))));
	else
		tmp = asin(Float64(Float64(l_m * 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[Power[N[(t$95$m / l$95$m), $MachinePrecision], 2.0], $MachinePrecision], 2e+33], N[ArcSin[N[(1.0 / N[Sqrt[N[(N[(t$95$m / l$95$m), $MachinePrecision] * N[(N[(t$95$m / l$95$m), $MachinePrecision] * 2.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (/.f64 t l) #s(literal 2 binary64)) < 1.9999999999999999e33

   1. Initial program 99.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in Om around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 96.1%

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

   if 1.9999999999999999e33 < (pow.f64 (/.f64 t l) #s(literal 2 binary64))

   1. Initial program 69.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 60.1%

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

Alternative 4: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 500000000:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{-\mathsf{fma}\left(\frac{t\_m \cdot \frac{t\_m}{l\_m}}{l\_m}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 500000000.0)
   (asin (/ 1.0 (sqrt (- (fma (/ (* t_m (/ t_m l_m)) l_m) -2.0 -1.0)))))
   (asin
    (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (/ (* 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) <= 500000000.0) {
		tmp = asin((1.0 / sqrt(-fma(((t_m * (t_m / l_m)) / l_m), -2.0, -1.0))));
	} else {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((l_m * 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) <= 500000000.0)
		tmp = asin(Float64(1.0 / sqrt(Float64(-fma(Float64(Float64(t_m * Float64(t_m / l_m)) / l_m), -2.0, -1.0)))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(l_m * 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], 500000000.0], N[ArcSin[N[(1.0 / N[Sqrt[(-N[(N[(N[(t$95$m * N[(t$95$m / l$95$m), $MachinePrecision]), $MachinePrecision] / l$95$m), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 5e8

   1. Initial program 88.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. frac-2neg N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 66.0%

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

   5. Taylor expanded in Om around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 70.5

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

   7. Applied rewrites 70.5%

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

   9. Applied rewrites 83.1%

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

   if 5e8 < (/.f64 t l)

   1. Initial program 75.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 91.0

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

   5. Applied rewrites 91.0%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 500000000:\\ \;\;\;\;\sin^{-1} \left(\frac{1}{\sqrt{-\mathsf{fma}\left(\frac{t \cdot \frac{t}{\ell}}{\ell}, -2, -1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.2)
   (asin (sqrt (- 1.0 (/ (* Om (/ Om Omc)) Omc))))
   (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) <= 0.2) {
		tmp = asin(sqrt((1.0 - ((Om * (Om / Omc)) / Omc))));
	} 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) <= 0.2d0) then
        tmp = asin(sqrt((1.0d0 - ((om * (om / omc)) / omc))))
    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) <= 0.2) {
		tmp = Math.asin(Math.sqrt((1.0 - ((Om * (Om / Omc)) / Omc))));
	} 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) <= 0.2:
		tmp = math.asin(math.sqrt((1.0 - ((Om * (Om / Omc)) / Omc))))
	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) <= 0.2)
		tmp = asin(sqrt(Float64(1.0 - Float64(Float64(Om * Float64(Om / Omc)) / Omc))));
	else
		tmp = asin(Float64(Float64(l_m * 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) <= 0.2)
		tmp = asin(sqrt((1.0 - ((Om * (Om / Omc)) / Omc))));
	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], 0.2], N[ArcSin[N[Sqrt[N[(1.0 - N[(N[(Om * N[(Om / Omc), $MachinePrecision]), $MachinePrecision] / Omc), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.20000000000000001

   1. Initial program 87.7%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.7

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

   5. Applied rewrites 56.7%

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

   7. Applied rewrites 63.2%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 76.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 96.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{l\_m \cdot \sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.2) (asin (sqrt 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) <= 0.2) {
		tmp = asin(sqrt(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) <= 0.2d0) then
        tmp = asin(sqrt(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) <= 0.2) {
		tmp = Math.asin(Math.sqrt(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) <= 0.2:
		tmp = math.asin(math.sqrt(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) <= 0.2)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(Float64(Float64(l_m * 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) <= 0.2)
		tmp = asin(sqrt(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], 0.2], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(l$95$m * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t$95$m), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.20000000000000001

   1. Initial program 87.7%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 61.8%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 76.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 96.2%

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

Alternative 7: 96.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.2:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t_m = (fabs.f64 t)
(FPCore (t_m l_m Om Omc)
 :precision binary64
 (if (<= (/ t_m l_m) 0.2) (asin (sqrt 1.0)) (asin (* (sqrt 0.5) (/ l_m t_m)))))

C

l_m = fabs(l);
t_m = fabs(t);
double code(double t_m, double l_m, double Om, double Omc) {
	double tmp;
	if ((t_m / l_m) <= 0.2) {
		tmp = asin(sqrt(1.0));
	} else {
		tmp = asin((sqrt(0.5) * (l_m / 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) <= 0.2d0) then
        tmp = asin(sqrt(1.0d0))
    else
        tmp = asin((sqrt(0.5d0) * (l_m / 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) <= 0.2) {
		tmp = Math.asin(Math.sqrt(1.0));
	} else {
		tmp = Math.asin((Math.sqrt(0.5) * (l_m / 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) <= 0.2:
		tmp = math.asin(math.sqrt(1.0))
	else:
		tmp = math.asin((math.sqrt(0.5) * (l_m / t_m)))
	return tmp

Julia

l_m = abs(l)
t_m = abs(t)
function code(t_m, l_m, Om, Omc)
	tmp = 0.0
	if (Float64(t_m / l_m) <= 0.2)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin(Float64(sqrt(0.5) * Float64(l_m / 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) <= 0.2)
		tmp = asin(sqrt(1.0));
	else
		tmp = asin((sqrt(0.5) * (l_m / 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], 0.2], N[ArcSin[N[Sqrt[1.0], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[0.5], $MachinePrecision] * N[(l$95$m / t$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 t l) < 0.20000000000000001

   1. Initial program 87.7%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 61.8%

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

   if 0.20000000000000001 < (/.f64 t l)

   1. Initial program 76.5%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sqrt.f64 88.6

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in Om around 0

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

   8. Applied rewrites 96.2%

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

   10. Applied rewrites 96.2%

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

Alternative 8: 51.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

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(sqrt(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(Math.sqrt(1.0));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

3. Taylor expanded in t around 0

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

   1. lower--.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 44.2

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

5. Applied rewrites 44.2%

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

6. Taylor expanded in Om around 0

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

8. Applied rewrites 48.2%

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

Reproduce

herbie shell --seed 2024226 
(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)))))))