Average Error: 10.3 → 0.8
Time: 10.9s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := \mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)\\ t_2 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\ \mathbf{if}\;\frac{t}{\ell} \leq -6.246144260202978 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - t_2} \cdot \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.3013591093358795 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{t_1}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t} + \left(t_2 \cdot \sqrt{\frac{1}{t_1}}\right) \cdot -0.5\right)\\ \end{array} \]
(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 (fma 2.0 (pow (/ t l) 2.0) 1.0)) (t_2 (* (/ Om Omc) (/ Om Omc))))
   (if (<= (/ t l) -6.246144260202978e+156)
     (asin (* (sqrt (- 1.0 t_2)) (* l (- (/ (sqrt 0.5) t)))))
     (if (<= (/ t l) 3.3013591093358795e+134)
       (expm1 (log1p (asin (sqrt (/ (- 1.0 (pow (/ Om Omc) 2.0)) t_1)))))
       (asin
        (+ (* (sqrt 0.5) (/ l t)) (* (* t_2 (sqrt (/ 1.0 t_1))) -0.5)))))))
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 = fma(2.0, pow((t / l), 2.0), 1.0);
	double t_2 = (Om / Omc) * (Om / Omc);
	double tmp;
	if ((t / l) <= -6.246144260202978e+156) {
		tmp = asin((sqrt((1.0 - t_2)) * (l * -(sqrt(0.5) / t))));
	} else if ((t / l) <= 3.3013591093358795e+134) {
		tmp = expm1(log1p(asin(sqrt(((1.0 - pow((Om / Omc), 2.0)) / t_1)))));
	} else {
		tmp = asin(((sqrt(0.5) * (l / t)) + ((t_2 * sqrt((1.0 / t_1))) * -0.5)));
	}
	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 = fma(2.0, (Float64(t / l) ^ 2.0), 1.0)
	t_2 = Float64(Float64(Om / Omc) * Float64(Om / Omc))
	tmp = 0.0
	if (Float64(t / l) <= -6.246144260202978e+156)
		tmp = asin(Float64(sqrt(Float64(1.0 - t_2)) * Float64(l * Float64(-Float64(sqrt(0.5) / t)))));
	elseif (Float64(t / l) <= 3.3013591093358795e+134)
		tmp = expm1(log1p(asin(sqrt(Float64(Float64(1.0 - (Float64(Om / Omc) ^ 2.0)) / t_1)))));
	else
		tmp = asin(Float64(Float64(sqrt(0.5) * Float64(l / t)) + Float64(Float64(t_2 * sqrt(Float64(1.0 / t_1))) * -0.5)));
	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[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(Om / Omc), $MachinePrecision] * N[(Om / Omc), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -6.246144260202978e+156], N[ArcSin[N[(N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision] * N[(l * (-N[(N[Sqrt[0.5], $MachinePrecision] / t), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 3.3013591093358795e+134], N[(Exp[N[Log[1 + N[ArcSin[N[Sqrt[N[(N[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], N[ArcSin[N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$2 * N[Sqrt[N[(1.0 / t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -0.5), $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 := \mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)\\
t_2 := \frac{Om}{Omc} \cdot \frac{Om}{Omc}\\
\mathbf{if}\;\frac{t}{\ell} \leq -6.246144260202978 \cdot 10^{+156}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{1 - t_2} \cdot \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 3.3013591093358795 \cdot 10^{+134}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{t_1}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t} + \left(t_2 \cdot \sqrt{\frac{1}{t_1}}\right) \cdot -0.5\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) < -6.2461442602029777e156

    1. Initial program 33.8

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

    2. Simplified33.8

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

    3. Taylor expanded in t around -inf 7.6

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

    4. Simplified0.2

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

    if -6.2461442602029777e156 < (/.f64 t l) < 3.30135910933587949e134

    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

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)} \]

    if 3.30135910933587949e134 < (/.f64 t l)

    1. Initial program 32.4

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

    2. Simplified32.4

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

    3. Taylor expanded in Om around 0 38.7

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

    4. Simplified32.4

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

    5. Taylor expanded in t around inf 0.7

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

    6. Simplified0.7

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -6.246144260202978 \cdot 10^{+156}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \left(\ell \cdot \left(-\frac{\sqrt{0.5}}{t}\right)\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.3013591093358795 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{0.5} \cdot \frac{\ell}{t} + \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right) \cdot -0.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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)))))))