Average Error: 10.5 → 1.2
Time: 8.3s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{t_1}{\frac{\sqrt{2}}{\ell} \cdot \left(-t\right)}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\frac{t_1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{t_1}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array} \]
(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 (- 1.0 (pow (/ Om Omc) 2.0)))))
   (if (<= (/ t l) -5e+157)
     (asin (/ t_1 (* (/ (sqrt 2.0) l) (- t))))
     (if (<= (/ t l) 5e+149)
       (asin (/ t_1 (sqrt (fma 2.0 (pow (/ t l) 2.0) 1.0))))
       (asin (/ t_1 (/ (* t (sqrt 2.0)) l)))))))
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((1.0 - pow((Om / Omc), 2.0)));
	double tmp;
	if ((t / l) <= -5e+157) {
		tmp = asin((t_1 / ((sqrt(2.0) / l) * -t)));
	} else if ((t / l) <= 5e+149) {
		tmp = asin((t_1 / sqrt(fma(2.0, pow((t / l), 2.0), 1.0))));
	} else {
		tmp = asin((t_1 / ((t * sqrt(2.0)) / l)));
	}
	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 = sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)))
	tmp = 0.0
	if (Float64(t / l) <= -5e+157)
		tmp = asin(Float64(t_1 / Float64(Float64(sqrt(2.0) / l) * Float64(-t))));
	elseif (Float64(t / l) <= 5e+149)
		tmp = asin(Float64(t_1 / sqrt(fma(2.0, (Float64(t / l) ^ 2.0), 1.0))));
	else
		tmp = asin(Float64(t_1 / Float64(Float64(t * sqrt(2.0)) / l)));
	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[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -5e+157], N[ArcSin[N[(t$95$1 / N[(N[(N[Sqrt[2.0], $MachinePrecision] / l), $MachinePrecision] * (-t)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+149], N[ArcSin[N[(t$95$1 / 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$1 / N[(N[(t * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / l), $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 := \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\\
\mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+157}:\\
\;\;\;\;\sin^{-1} \left(\frac{t_1}{\frac{\sqrt{2}}{\ell} \cdot \left(-t\right)}\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+149}:\\
\;\;\;\;\sin^{-1} \left(\frac{t_1}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\

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


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 t l) < -4.99999999999999976e157

    1. Initial program 33.6

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

    2. Simplified33.6

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

    3. Applied egg-rr33.6

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

    4. Taylor expanded in t around -inf 1.4

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

    5. Simplified1.4

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

    if -4.99999999999999976e157 < (/.f64 t l) < 4.9999999999999999e149

    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-rr1.1

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

    if 4.9999999999999999e149 < (/.f64 t l)

    1. Initial program 35.4

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

    2. Simplified35.4

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

    3. Applied egg-rr35.4

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

    4. Taylor expanded in t around inf 1.6

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -5 \cdot 10^{+157}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{\sqrt{2}}{\ell} \cdot \left(-t\right)}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022202 
(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)))))))