Average Error: 10.2 → 1.0
Time: 12.6s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{6}}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array} \]
(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
 (if (<= (/ t l) -2e+154)
   (asin
    (* (sqrt (- 1.0 (* (/ Om Omc) (/ Om Omc)))) (* (/ (sqrt 0.5) t) (- l))))
   (if (<= (/ t l) 5e+40)
     (asin
      (sqrt
       (/
        (- 1.0 (cbrt (pow (/ Om Omc) 6.0)))
        (fma 2.0 (pow (/ t l) 2.0) 1.0))))
     (asin
      (*
       (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
       (/ 1.0 (* (/ t l) (sqrt 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 tmp;
	if ((t / l) <= -2e+154) {
		tmp = asin((sqrt((1.0 - ((Om / Omc) * (Om / Omc)))) * ((sqrt(0.5) / t) * -l)));
	} else if ((t / l) <= 5e+40) {
		tmp = asin(sqrt(((1.0 - cbrt(pow((Om / Omc), 6.0))) / fma(2.0, pow((t / l), 2.0), 1.0))));
	} else {
		tmp = asin((sqrt((1.0 - pow((Om / Omc), 2.0))) * (1.0 / ((t / l) * sqrt(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)
	tmp = 0.0
	if (Float64(t / l) <= -2e+154)
		tmp = asin(Float64(sqrt(Float64(1.0 - Float64(Float64(Om / Omc) * Float64(Om / Omc)))) * Float64(Float64(sqrt(0.5) / t) * Float64(-l))));
	elseif (Float64(t / l) <= 5e+40)
		tmp = asin(sqrt(Float64(Float64(1.0 - cbrt((Float64(Om / Omc) ^ 6.0))) / fma(2.0, (Float64(t / l) ^ 2.0), 1.0))));
	else
		tmp = asin(Float64(sqrt(Float64(1.0 - (Float64(Om / Omc) ^ 2.0))) * Float64(1.0 / Float64(Float64(t / l) * sqrt(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_] := If[LessEqual[N[(t / l), $MachinePrecision], -2e+154], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision] * (-l)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5e+40], N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[Power[N[(Om / Omc), $MachinePrecision], 6.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[ArcSin[N[(N[Sqrt[N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[(t / l), $MachinePrecision] * N[Sqrt[2.0], $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}
\mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\
\;\;\;\;\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 5 \cdot 10^{+40}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{6}}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\frac{t}{\ell} \cdot \sqrt{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) < -2.00000000000000007e154

    1. Initial program 35.2

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

    2. Simplified35.2

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

    3. Taylor expanded in t around -inf 8.2

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

    4. Simplified0.3

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

    if -2.00000000000000007e154 < (/.f64 t l) < 5.00000000000000003e40

    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

      \[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{\sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{6}}}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right) \]

    if 5.00000000000000003e40 < (/.f64 t l)

    1. Initial program 21.6

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

    2. Simplified21.6

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

    3. Applied egg-rr21.6

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

    4. Taylor expanded in t around inf 1.3

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

    5. Simplified1.3

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2 \cdot 10^{+154}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\frac{\sqrt{0.5}}{t} \cdot \left(-\ell\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5 \cdot 10^{+40}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \sqrt[3]{{\left(\frac{Om}{Omc}\right)}^{6}}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{1}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022156 
(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)))))))