Average Error: 10.2 → 0.8
Time: 10.0s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := \frac{\sqrt{0.5}}{t}\\ t_2 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \left(t_1 \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+66}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \left(\ell \cdot t_1\right)\right)\\ \end{array} \]
(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 (/ (sqrt 0.5) t)) (t_2 (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc))))))
   (if (<= (/ t l) -2e+144)
     (asin (* t_2 (* t_1 (- l))))
     (if (<= (/ t l) 1e+66)
       (asin
        (/
         (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
         (sqrt (fma 2.0 (pow (/ t l) 2.0) 1.0))))
       (asin (* t_2 (* l t_1)))))))
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 = sqrt(0.5) / t;
	double t_2 = sqrt((1.0 - ((Om / Omc) * (Om / Omc))));
	double tmp;
	if ((t / l) <= -2e+144) {
		tmp = asin((t_2 * (t_1 * -l)));
	} else if ((t / l) <= 1e+66) {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) / sqrt(fma(2.0, pow((t / l), 2.0), 1.0))));
	} else {
		tmp = asin((t_2 * (l * t_1)));
	}
	return tmp;
}

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(sqrt(0.5) / t)
	t_2 = sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc))))
	tmp = 0.0
	if (Float64(t / l) <= -2e+144)
		tmp = asin(Float64(t_2 * Float64(t_1 * Float64(-l))));
	elseif (Float64(t / l) <= 1e+66)
		tmp = asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) / sqrt(fma(2.0, (Float64(t / l) ^ 2.0), 1.0))));
	else
		tmp = asin(Float64(t_2 * Float64(l * t_1)));
	end
	return tmp
end

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[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2e+144], N[ArcSin[N[(t$95$2 * N[(t$95$1 * (-l)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1e+66], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$2 * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

\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{\sqrt{0.5}}{t}\\
t_2 := \sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\\
\mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+144}:\\
\;\;\;\;\sin^{-1} \left(t_2 \cdot \left(t_1 \cdot \left(-\ell\right)\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 10^{+66}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(t_2 \cdot \left(\ell \cdot t_1\right)\right)\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 t l) < -2.00000000000000005e144

    1. Initial program 31.8

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

    2. Simplified31.8

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

    3. Taylor expanded in t around -inf 8.0

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

    4. Simplified0.2

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

    if -2.00000000000000005e144 < (/.f64 t l) < 9.99999999999999945e65

    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. Simplified1.0

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

    3. Applied egg-rr1.0

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

    if 9.99999999999999945e65 < (/.f64 t l)

    1. Initial program 24.4

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

    2. Simplified24.4

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

    3. Taylor expanded in t around inf 8.0

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

    4. Simplified0.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 10^{+66}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\ell \cdot \frac{\sqrt{0.5}}{t}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(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)))))))