Toniolo and Linder, Equation (2)

Percentage Accurate: 83.9% → 98.7%

Time: 15.9s

Alternatives: 6

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: 83.9% 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: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{\mathsf{fma}\left(\frac{t\_m}{l\_m}, \frac{t\_m}{l\_m} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{l\_m}{t\_m}\right)\\ \end{array} \end{array} \]

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) 2e+142)
   (asin
    (sqrt
     (/
      (- 1.0 (/ (* Om (/ Om Omc)) Omc))
      (fma (/ t_m l_m) (* (/ t_m l_m) 2.0) 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) <= 2e+142) {
		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(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) <= 2e+142)
		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(0.5) * Float64(l_m / 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], 2e+142], 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[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) < 2.0000000000000001e142

   1. Initial program 98.8%

      \[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
   2. Add Preprocessing
   3. 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 98.8

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

   4. Applied rewrites 98.8%

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\color{blue}{\mathsf{fma}\left(\frac{t}{\ell}, 2 \cdot \frac{t}{\ell}, 1\right)}}}\right) \]
   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 98.8

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

   6. Applied rewrites 98.8%

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

   if 2.0000000000000001e142 < (/.f64 t l)

   1. Initial program 43.3%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 68.0

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

   7. Applied rewrites 68.0%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 97.5%

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

   12. Applied rewrites 97.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om \cdot \frac{Om}{Omc}}{Omc}}{\mathsf{fma}\left(\frac{t}{\ell}, \frac{t}{\ell} \cdot 2, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t_m = \left|t\right| \\ \begin{array}{l} \mathbf{if}\;\frac{t\_m}{l\_m} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-\frac{Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(l\_m \cdot \frac{\sqrt{0.5}}{t\_m}\right)\\ \end{array} \end{array} \]

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) 0.0001)
   (asin (sqrt (fma (- (/ Om Omc)) (/ Om Omc) 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) <= 0.0001) {
		tmp = asin(sqrt(fma(-(Om / Omc), (Om / Omc), 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) <= 0.0001)
		tmp = asin(sqrt(fma(Float64(-Float64(Om / Omc)), Float64(Om / Omc), 1.0)));
	else
		tmp = asin(Float64(l_m * Float64(sqrt(0.5) / t_m)));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t_m = N[Abs[t], $MachinePrecision]
code[t$95$m_, l$95$m_, Om_, Omc_] := If[LessEqual[N[(t$95$m / l$95$m), $MachinePrecision], 0.0001], N[ArcSin[N[Sqrt[N[((-N[(Om / Omc), $MachinePrecision]) * N[(Om / Omc), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]], $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) < 1.00000000000000005e-4

   1. Initial program 98.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}\right) \]
   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 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 97.9%

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

   if 1.00000000000000005e-4 < (/.f64 t l)

   1. Initial program 69.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 44.7%

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

   5. Taylor expanded in l around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 60.8

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

   7. Applied rewrites 60.8%

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

   8. Taylor expanded in Om around 0

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

   10. Applied rewrites 96.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 0.0001:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\mathsf{fma}\left(-\frac{Om}{Omc}, \frac{Om}{Omc}, 1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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)))))))