Toniolo and Linder, Equation (2)

Average Error: 10.7 → 1.1

Time: 18.7s

Precision: binary64

Cost: 26888

Math

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

↓

\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+164}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{t_1}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell \cdot \sqrt{t_1}}{\sqrt{2}}}{t}\right)\\ \end{array} \]

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)))))))

↓

(FPCore (t l Om Omc)
 :precision binary64
 (let* ((t_1 (- 1.0 (pow (/ Om Omc) 2.0))))
   (if (<= (/ t l) -1e+164)
     (asin (/ (* (sqrt 0.5) (- l)) t))
     (if (<= (/ t l) 2e+121)
       (asin (sqrt (/ t_1 (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (/ (/ (* l (sqrt t_1)) (sqrt 2.0)) t))))))

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))))));
}

↓

double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+164) {
		tmp = asin(((sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e+121) {
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin((((l * sqrt(t_1)) / sqrt(2.0)) / t));
	}
	return tmp;
}

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

↓

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 - ((om / omc) ** 2.0d0)
    if ((t / l) <= (-1d+164)) then
        tmp = asin(((sqrt(0.5d0) * -l) / t))
    else if ((t / l) <= 2d+121) then
        tmp = asin(sqrt((t_1 / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin((((l * sqrt(t_1)) / sqrt(2.0d0)) / t))
    end if
    code = tmp
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))))));
}

↓

public static double code(double t, double l, double Om, double Omc) {
	double t_1 = 1.0 - Math.pow((Om / Omc), 2.0);
	double tmp;
	if ((t / l) <= -1e+164) {
		tmp = Math.asin(((Math.sqrt(0.5) * -l) / t));
	} else if ((t / l) <= 2e+121) {
		tmp = Math.asin(Math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin((((l * Math.sqrt(t_1)) / Math.sqrt(2.0)) / t));
	}
	return tmp;
}

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))))))

↓

def code(t, l, Om, Omc):
	t_1 = 1.0 - math.pow((Om / Omc), 2.0)
	tmp = 0
	if (t / l) <= -1e+164:
		tmp = math.asin(((math.sqrt(0.5) * -l) / t))
	elif (t / l) <= 2e+121:
		tmp = math.asin(math.sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin((((l * math.sqrt(t_1)) / math.sqrt(2.0)) / t))
	return tmp

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

↓

function code(t, l, Om, Omc)
	t_1 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	tmp = 0.0
	if (Float64(t / l) <= -1e+164)
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(-l)) / t));
	elseif (Float64(t / l) <= 2e+121)
		tmp = asin(sqrt(Float64(t_1 / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(Float64(l * sqrt(t_1)) / sqrt(2.0)) / t));
	end
	return tmp
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

↓

function tmp_2 = code(t, l, Om, Omc)
	t_1 = 1.0 - ((Om / Omc) ^ 2.0);
	tmp = 0.0;
	if ((t / l) <= -1e+164)
		tmp = asin(((sqrt(0.5) * -l) / t));
	elseif ((t / l) <= 2e+121)
		tmp = asin(sqrt((t_1 / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin((((l * sqrt(t_1)) / sqrt(2.0)) / t));
	end
	tmp_2 = tmp;
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]

↓

code[t_, l_, Om_, Omc_] := Block[{t$95$1 = N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1e+164], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * (-l)), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 2e+121], N[ArcSin[N[Sqrt[N[(t$95$1 / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[(l * N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -1e164

   1. Initial program 34.3

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

   2. Taylor expanded in Om around 0 34.3

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

   3. Simplified 34.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell \cdot \ell} \cdot t, 1\right)}}\right)} \]
      Proof
      (sqrt.f64 (/.f64 1 (fma.f64 2 (*.f64 (/.f64 t (*.f64 l l)) t) 1))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (fma.f64 2 (*.f64 (/.f64 t (Rewrite<= unpow2_binary64 (pow.f64 l 2))) t) 1))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (fma.f64 2 (Rewrite<= associate-/r/_binary64 (/.f64 t (/.f64 (pow.f64 l 2) t))) 1))): 6 points increase in error, 8 points decrease in error
      (sqrt.f64 (/.f64 1 (fma.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t t) (pow.f64 l 2))) 1))): 51 points increase in error, 5 points decrease in error
      (sqrt.f64 (/.f64 1 (fma.f64 2 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 t 2)) (pow.f64 l 2)) 1))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2))) 1)))): 0 points increase in error, 0 points decrease in error
      (sqrt.f64 (/.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 2 (/.f64 (pow.f64 t 2) (pow.f64 l 2))))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around -inf 0.6

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

   if -1e164 < (/.f64 t l) < 2.00000000000000007e121

   1. Initial program 1.4

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

   2. Applied egg-rr 1.4

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

   if 2.00000000000000007e121 < (/.f64 t l)

   1. Initial program 30.0

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

   2. Applied egg-rr 1.5

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

   3. Taylor expanded in t around inf 8.1

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

   4. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell}{\sqrt{2} \cdot t}\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (/.f64 l (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite=> unpow2_binary64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))) (/.f64 l (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc))))) (/.f64 l (*.f64 (sqrt.f64 2) t))): 30 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)))) (/.f64 l (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))))) (/.f64 l (*.f64 (sqrt.f64 2) t))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 l (*.f64 (sqrt.f64 2) t)) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.3

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+164}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell \cdot \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{2}}}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	26624

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

Alternative 2
Error	1.1
Cost	26624

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

Alternative 3
Error	1.1
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+164}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 4
Error	1.1
Cost	20872

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+164}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell \cdot \mathsf{fma}\left(Om \cdot \frac{Om}{Omc}, \frac{-0.5}{Omc}, 1\right)}{t}\right)\\ \end{array} \]

Alternative 5
Error	1.1
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+61}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 20000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell \cdot \mathsf{fma}\left(Om \cdot \frac{Om}{Omc}, \frac{-0.5}{Omc}, 1\right)}{t}\right)\\ \end{array} \]

Alternative 6
Error	1.0
Cost	20680

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+61}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \end{array} \]

Alternative 7
Error	1.1
Cost	14664

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+61}:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 20000:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \frac{t \cdot \frac{t}{\ell}}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 8
Error	1.8
Cost	13896

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 9
Error	13.3
Cost	13640

\[\begin{array}{l} t_1 := \sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	13.3
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+205}:\\ \;\;\;\;\sin^{-1} \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 11
Error	2.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{-\ell}{\frac{t}{\sqrt{0.5}}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{t}{\ell} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 12
Error	2.1
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -50:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5} \cdot \left(-\ell\right)}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{-8}:\\ \;\;\;\;\sin^{-1} \left(1 - \frac{t}{\ell} \cdot \frac{t}{\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell}{\sqrt{2} \cdot t}\right)\\ \end{array} \]

Alternative 13
Error	31.6
Cost	6464

\[\sin^{-1} 1 \]

Reproduce

herbie shell --seed 2022316 
(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)))))))