Toniolo and Linder, Equation (2)

Average Error: 10.3 → 0.8

Time: 15.8s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} t_1 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot t_1\right)\right)\right)\\ \end{array} \]

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 (* (/ Om Omc) (/ Om Omc))))
   (if (<= (/ t l) -5e+79)
     (asin (* (* l (/ (- (sqrt 0.5)) t)) (sqrt (- 1.0 t_1))))
     (if (<= (/ t l) 5e+84)
       (asin
        (sqrt
         (/ (- 1.0 (pow (/ Om Omc) 2.0)) (+ 1.0 (* 2.0 (/ (/ t l) (/ l t)))))))
       (asin (* (/ (sqrt 0.5) t) (+ l (* -0.5 (* l t_1)))))))))

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 = (Om / Omc) * (Om / Omc);
	double tmp;
	if ((t / l) <= -5e+79) {
		tmp = asin(((l * (-sqrt(0.5) / t)) * sqrt((1.0 - t_1))));
	} else if ((t / l) <= 5e+84) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = asin(((sqrt(0.5) / t) * (l + (-0.5 * (l * t_1)))));
	}
	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 = (om / omc) * (om / omc)
    if ((t / l) <= (-5d+79)) then
        tmp = asin(((l * (-sqrt(0.5d0) / t)) * sqrt((1.0d0 - t_1))))
    else if ((t / l) <= 5d+84) then
        tmp = asin(sqrt(((1.0d0 - ((om / omc) ** 2.0d0)) / (1.0d0 + (2.0d0 * ((t / l) / (l / t)))))))
    else
        tmp = asin(((sqrt(0.5d0) / t) * (l + ((-0.5d0) * (l * t_1)))))
    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 = (Om / Omc) * (Om / Omc);
	double tmp;
	if ((t / l) <= -5e+79) {
		tmp = Math.asin(((l * (-Math.sqrt(0.5) / t)) * Math.sqrt((1.0 - t_1))));
	} else if ((t / l) <= 5e+84) {
		tmp = Math.asin(Math.sqrt(((1.0 - Math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	} else {
		tmp = Math.asin(((Math.sqrt(0.5) / t) * (l + (-0.5 * (l * t_1)))));
	}
	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 = (Om / Omc) * (Om / Omc)
	tmp = 0
	if (t / l) <= -5e+79:
		tmp = math.asin(((l * (-math.sqrt(0.5) / t)) * math.sqrt((1.0 - t_1))))
	elif (t / l) <= 5e+84:
		tmp = math.asin(math.sqrt(((1.0 - math.pow((Om / Omc), 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))))
	else:
		tmp = math.asin(((math.sqrt(0.5) / t) * (l + (-0.5 * (l * t_1)))))
	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(Float64(Om / Omc) * Float64(Om / Omc))
	tmp = 0.0
	if (Float64(t / l) <= -5e+79)
		tmp = asin(Float64(Float64(l * Float64(Float64(-sqrt(0.5)) / t)) * sqrt(Float64(1.0 - t_1))));
	elseif (Float64(t / l) <= 5e+84)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / Float64(1.0 + Float64(2.0 * Float64(Float64(t / l) / Float64(l / t)))))));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) / t) * Float64(l + Float64(-0.5 * Float64(l * t_1)))));
	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 = (Om / Omc) * (Om / Omc);
	tmp = 0.0;
	if ((t / l) <= -5e+79)
		tmp = asin(((l * (-sqrt(0.5) / t)) * sqrt((1.0 - t_1))));
	elseif ((t / l) <= 5e+84)
		tmp = asin(sqrt(((1.0 - ((Om / Omc) ^ 2.0)) / (1.0 + (2.0 * ((t / l) / (l / t)))))));
	else
		tmp = asin(((sqrt(0.5) / t) * (l + (-0.5 * (l * t_1)))));
	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[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5e+79], N[ArcSin[N[(N[(l * N[((-N[Sqrt[0.5], $MachinePrecision]) / t), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+84], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] * N[(l + N[(-0.5 * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 t l) < -5e79

   1. Initial program 25.6

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

   2. Taylor expanded in t around -inf 8.2

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]

   3. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 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 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 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 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 25 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 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 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 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)))): 20 points increase in error, 23 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t)))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 8.2

      \[\leadsto \sin^{-1} \color{blue}{\left(-1 \cdot \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\right)} \]

   5. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)} \]
      Proof
      (*.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) (neg.f64 l)) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l))) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (neg.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 27 points increase in error, 28 points decrease in error
      (*.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc)))))): 25 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2))))))): 0 points increase in error, 0 points decrease in error

   if -5e79 < (/.f64 t l) < 5.0000000000000001e84

   1. Initial program 1.0

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

   2. Applied egg-rr 1.0

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

   if 5.0000000000000001e84 < (/.f64 t l)

   1. Initial program 26.5

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

   2. Taylor expanded in t around inf 7.8

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

   3. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \ell\right)\right)} \]
      Proof
      (*.f64 (sqrt.f64 (-.f64 1 (pow.f64 (/.f64 Om Omc) 2))) (*.f64 (/.f64 (sqrt.f64 1/2) t) l)): 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 (/.f64 (sqrt.f64 1/2) t) l)): 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 (/.f64 (sqrt.f64 1/2) t) l)): 25 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 (/.f64 (sqrt.f64 1/2) t) l)): 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 (/.f64 (sqrt.f64 1/2) t) l)): 0 points increase in error, 0 points decrease in error
      (*.f64 (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 19 points increase in error, 24 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in t around 0 7.8

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)} \]

   5. Simplified 0.3

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\ell \cdot \frac{\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)} \]
      Proof
      (*.f64 (*.f64 l (/.f64 (sqrt.f64 1/2) t)) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l)) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (sqrt.f64 (-.f64 1 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc))))): 27 points increase in error, 28 points decrease in error
      (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc)))))): 25 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (sqrt.f64 (-.f64 1 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in Om around 0 8.9

      \[\leadsto \sin^{-1} \color{blue}{\left(-0.5 \cdot \frac{\sqrt{0.5} \cdot \left({Om}^{2} \cdot \ell\right)}{{Omc}^{2} \cdot t} + \frac{\sqrt{0.5} \cdot \ell}{t}\right)} \]

   7. Simplified 0.5

      \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + -0.5 \cdot \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \ell\right)\right)\right)} \]
      Proof
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (+.f64 l (*.f64 -1/2 (*.f64 (*.f64 (/.f64 Om Omc) (/.f64 Om Omc)) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (+.f64 l (*.f64 -1/2 (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 Om Om) (*.f64 Omc Omc))) l)))): 25 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (+.f64 l (*.f64 -1/2 (*.f64 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 Om 2)) (*.f64 Omc Omc)) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (+.f64 l (*.f64 -1/2 (*.f64 (/.f64 (pow.f64 Om 2) (Rewrite<= unpow2_binary64 (pow.f64 Omc 2))) l)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (+.f64 l (*.f64 -1/2 (Rewrite<= associate-/r/_binary64 (/.f64 (pow.f64 Om 2) (/.f64 (pow.f64 Omc 2) l)))))): 7 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (+.f64 l (*.f64 -1/2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 Om 2) l) (pow.f64 Omc 2)))))): 9 points increase in error, 7 points decrease in error
      (*.f64 (/.f64 (sqrt.f64 1/2) t) (+.f64 l (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (pow.f64 Om 2) l) (pow.f64 Omc 2)) -1/2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) l) (*.f64 (/.f64 (sqrt.f64 1/2) t) (*.f64 (/.f64 (*.f64 (pow.f64 Om 2) l) (pow.f64 Omc 2)) -1/2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t)) (*.f64 (/.f64 (sqrt.f64 1/2) t) (*.f64 (/.f64 (*.f64 (pow.f64 Om 2) l) (pow.f64 Omc 2)) -1/2))): 17 points increase in error, 22 points decrease in error
      (+.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (sqrt.f64 1/2) t) (/.f64 (*.f64 (pow.f64 Om 2) l) (pow.f64 Omc 2))) -1/2))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (*.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (sqrt.f64 1/2) (*.f64 (pow.f64 Om 2) l)) (*.f64 t (pow.f64 Omc 2)))) -1/2)): 8 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 (sqrt.f64 1/2) l) t) (Rewrite<= *-commutative_binary64 (*.f64 -1/2 (/.f64 (*.f64 (sqrt.f64 1/2) (*.f64 (pow.f64 Om 2) l)) (*.f64 t (pow.f64 Omc 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1/2 (/.f64 (*.f64 (sqrt.f64 1/2) (*.f64 (pow.f64 Om 2) l)) (*.f64 t (pow.f64 Omc 2)))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 -1/2 (*.f64 (sqrt.f64 1/2) (*.f64 (pow.f64 Om 2) l))) (*.f64 t (pow.f64 Omc 2)))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 -1/2 (*.f64 (sqrt.f64 1/2) (*.f64 (pow.f64 Om 2) l))) (Rewrite=> *-commutative_binary64 (*.f64 (pow.f64 Omc 2) t))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t)): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r/_binary64 (*.f64 -1/2 (/.f64 (*.f64 (sqrt.f64 1/2) (*.f64 (pow.f64 Om 2) l)) (*.f64 (pow.f64 Omc 2) t)))) (/.f64 (*.f64 (sqrt.f64 1/2) l) t)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+79}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+84}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot \frac{\frac{t}{\ell}}{\frac{\ell}{t}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	26624

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

Alternative 2
Error	1.1
Cost	26624

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

Alternative 3
Error	1.0
Cost	20484

\[\begin{array}{l} t_1 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+65}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - t_1} \cdot \frac{\frac{-\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+27}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{1}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot t_1\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	20484

\[\begin{array}{l} t_1 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1 \cdot 10^{+65}:\\ \;\;\;\;\sin^{-1} \left(\left(\ell \cdot \frac{-\sqrt{0.5}}{t}\right) \cdot \sqrt{1 - t_1}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 4 \cdot 10^{+27}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}{1 + 2 \cdot \left(\frac{1}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot t_1\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	6.0
Cost	20164

\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 10^{+74}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1}{\mathsf{fma}\left(2, \frac{t}{\ell} \cdot \frac{t}{\ell}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	12.1
Cost	14272

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

Alternative 7
Error	18.2
Cost	14152

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell \cdot \ell} \cdot \left(t \cdot t\right)}}\right)\\ \mathbf{if}\;\ell \leq -2.299422372727205 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.3623602360351988 \cdot 10^{-150}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{0.5}}{t} \cdot \left(\ell + -0.5 \cdot \left(\ell \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.0769546093343803 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 8
Error	12.1
Cost	14144

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

Alternative 9
Error	18.2
Cost	14028

\[\begin{array}{l} t_1 := \sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\ell \cdot \ell} \cdot \left(t \cdot t\right)}}\right)\\ \mathbf{if}\;\ell \leq -2.299422372727205 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.3623602360351988 \cdot 10^{-150}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{elif}\;\ell \leq 2.0769546093343803 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 10
Error	24.5
Cost	13640

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.402655225876965 \cdot 10^{-29}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.036383575514708 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{\frac{Om}{Omc}}{\frac{Omc}{Om}}}\right)\\ \end{array} \]

Alternative 11
Error	24.6
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.402655225876965 \cdot 10^{-29}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{elif}\;\ell \leq 1.036383575514708 \cdot 10^{+26}:\\ \;\;\;\;\sin^{-1} \left(\frac{\frac{\ell}{t}}{\sqrt{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} 1\\ \end{array} \]

Alternative 12
Error	31.5
Cost	6464

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

Reproduce

herbie shell --seed 2022297 
(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)))))))