Average Error: 10.2 → 3.1
Time: 16.6s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{0.5}}{t}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1.796225185050764 \cdot 10^{+128}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot t_1\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.4981170521793895 \cdot 10^{+106}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, \frac{\frac{t}{\ell}}{\frac{\ell}{t}}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\\ \end{array} \]
(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 (/ (* l (sqrt 0.5)) t)))
   (if (<= (/ t l) -1.796225185050764e+128)
     (asin (- (* (sqrt (- 1.0 (/ (* Om Om) (* Omc Omc)))) t_1)))
     (if (<= (/ t l) 1.4981170521793895e+106)
       (asin
        (sqrt
         (/ (- 1.0 (pow (/ Om Omc) 2.0)) (fma 2.0 (/ (/ t l) (/ l t)) 1.0))))
       (asin (* t_1 (sqrt (- 1.0 (/ (pow Om 2.0) (pow Omc 2.0))))))))))
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 = (l * sqrt(0.5)) / t;
	double tmp;
	if ((t / l) <= -1.796225185050764e+128) {
		tmp = asin(-(sqrt((1.0 - ((Om * Om) / (Omc * Omc)))) * t_1));
	} else if ((t / l) <= 1.4981170521793895e+106) {
		tmp = asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / fma(2.0, ((t / l) / (l / t)), 1.0))));
	} else {
		tmp = asin((t_1 * sqrt((1.0 - (pow(Om, 2.0) / pow(Omc, 2.0))))));
	}
	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(Float64(l * sqrt(0.5)) / t)
	tmp = 0.0
	if (Float64(t / l) <= -1.796225185050764e+128)
		tmp = asin(Float64(-Float64(sqrt(Float64(1.0 - Float64(Float64(Om * Om) / Float64(Omc * Omc)))) * t_1)));
	elseif (Float64(t / l) <= 1.4981170521793895e+106)
		tmp = asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / fma(2.0, Float64(Float64(t / l) / Float64(l / t)), 1.0))));
	else
		tmp = asin(Float64(t_1 * sqrt(Float64(1.0 - Float64((Om ^ 2.0) / (Omc ^ 2.0))))));
	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[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -1.796225185050764e+128], N[ArcSin[(-N[(N[Sqrt[N[(1.0 - N[(N[(Om * Om), $MachinePrecision] / N[(Omc * Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision])], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 1.4981170521793895e+106], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[(N[(t / l), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(t$95$1 * N[Sqrt[N[(1.0 - N[(N[Power[Om, 2.0], $MachinePrecision] / N[Power[Omc, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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{\ell \cdot \sqrt{0.5}}{t}\\
\mathbf{if}\;\frac{t}{\ell} \leq -1.796225185050764 \cdot 10^{+128}:\\
\;\;\;\;\sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot t_1\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 1.4981170521793895 \cdot 10^{+106}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, \frac{\frac{t}{\ell}}{\frac{\ell}{t}}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(t_1 \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\\


\end{array}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 t l) < -1.79622518505076397e128

    1. Initial program 31.4

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

    2. Simplified31.4

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

    3. Taylor expanded in t around -inf 8.4

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

    4. Simplified8.4

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

    if -1.79622518505076397e128 < (/.f64 t l) < 1.4981170521793895e106

    1. Initial program 0.9

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

    2. Simplified0.9

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

    3. Applied egg-rr0.9

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

    if 1.4981170521793895e106 < (/.f64 t l)

    1. Initial program 27.3

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

    2. Simplified27.3

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

    3. Taylor expanded in t around inf 7.2

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

  4. Final simplification3.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.796225185050764 \cdot 10^{+128}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - \frac{Om \cdot Om}{Omc \cdot Omc}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.4981170521793895 \cdot 10^{+106}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, \frac{\frac{t}{\ell}}{\frac{\ell}{t}}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\ell \cdot \sqrt{0.5}}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(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)))))))