Average Error: 10.4 → 0.9
Time: 26.3s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ t_2 := \sqrt{t_1}\\ \mathbf{if}\;\frac{t}{\ell} \leq -2.696253691509278 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5.1099519620953986 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{t_1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|t_2 \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right|\right)\\ \end{array} \]
(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 (- 1.0 (pow (/ Om Omc) 2.0))) (t_2 (sqrt t_1)))
   (if (<= (/ t l) -2.696253691509278e+147)
     (asin (* t_2 (- (/ (* l (sqrt 0.5)) t))))
     (if (<= (/ t l) 5.1099519620953986e+107)
       (expm1 (log1p (asin (sqrt (/ t_1 (fma 2.0 (pow (/ t l) 2.0) 1.0))))))
       (asin (fabs (* t_2 (/ (sqrt 0.5) (/ t 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 = 1.0 - pow((Om / Omc), 2.0);
	double t_2 = sqrt(t_1);
	double tmp;
	if ((t / l) <= -2.696253691509278e+147) {
		tmp = asin((t_2 * -((l * sqrt(0.5)) / t)));
	} else if ((t / l) <= 5.1099519620953986e+107) {
		tmp = expm1(log1p(asin(sqrt((t_1 / fma(2.0, pow((t / l), 2.0), 1.0))))));
	} else {
		tmp = asin(fabs((t_2 * (sqrt(0.5) / (t / 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 = Float64(1.0 - (Float64(Om / Omc) ^ 2.0))
	t_2 = sqrt(t_1)
	tmp = 0.0
	if (Float64(t / l) <= -2.696253691509278e+147)
		tmp = asin(Float64(t_2 * Float64(-Float64(Float64(l * sqrt(0.5)) / t))));
	elseif (Float64(t / l) <= 5.1099519620953986e+107)
		tmp = expm1(log1p(asin(sqrt(Float64(t_1 / fma(2.0, (Float64(t / l) ^ 2.0), 1.0))))));
	else
		tmp = asin(abs(Float64(t_2 * Float64(sqrt(0.5) / Float64(t / 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[(1.0 - N[Power[N[(Om / Omc), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, If[LessEqual[N[(t / l), $MachinePrecision], -2.696253691509278e+147], N[ArcSin[N[(t$95$2 * (-N[(N[(l * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(t / l), $MachinePrecision], 5.1099519620953986e+107], N[(Exp[N[Log[1 + N[ArcSin[N[Sqrt[N[(t$95$1 / N[(2.0 * N[Power[N[(t / l), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision], N[ArcSin[N[Abs[N[(t$95$2 * N[(N[Sqrt[0.5], $MachinePrecision] / N[(t / l), $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 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\
t_2 := \sqrt{t_1}\\
\mathbf{if}\;\frac{t}{\ell} \leq -2.696253691509278 \cdot 10^{+147}:\\
\;\;\;\;\sin^{-1} \left(t_2 \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 5.1099519620953986 \cdot 10^{+107}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{t_1}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\left|t_2 \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right|\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) < -2.6962536915092781e147

    1. Initial program 32.8

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

    2. Simplified32.8

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

    3. Taylor expanded in t around -inf 8.5

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

    4. Simplified0.3

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

    if -2.6962536915092781e147 < (/.f64 t l) < 5.10995196209539865e107

    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 expm1-log1p-u_binary641.0

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

    if 5.10995196209539865e107 < (/.f64 t l)

    1. Initial program 28.9

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

    2. Simplified28.9

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

    3. Applied add-sqr-sqrt_binary6428.9

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

    4. Applied add-sqr-sqrt_binary6428.9

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

    5. Applied times-frac_binary6428.9

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

    6. Applied rem-sqrt-square_binary6428.9

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

    7. Taylor expanded in t around inf 8.1

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

    8. Simplified1.2

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2.696253691509278 \cdot 10^{+147}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 5.1099519620953986 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right|\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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)))))))