Average Error: 10.2 → 0.8
Time: 15.7s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{0.5}}{t}\\ t_2 := {\left(\frac{Om}{Omc}\right)}^{2}\\ t_3 := \log \left(\sqrt{e^{t_2}}\right)\\ t_4 := \sqrt{1 - t_2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1.1468627857485825 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(t_4 \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.29234323632466 \cdot 10^{+21}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \left(t_3 + t_3\right)}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_4 \cdot t_1\right)\\ \end{array} \]
\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}\\
t_2 := {\left(\frac{Om}{Omc}\right)}^{2}\\
t_3 := \log \left(\sqrt{e^{t_2}}\right)\\
t_4 := \sqrt{1 - t_2}\\
\mathbf{if}\;\frac{t}{\ell} \leq -1.1468627857485825 \cdot 10^{+141}:\\
\;\;\;\;\sin^{-1} \left(t_4 \cdot \left(-t_1\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 3.29234323632466 \cdot 10^{+21}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \left(t_3 + t_3\right)}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(t_4 \cdot t_1\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))
        (t_2 (pow (/ Om Omc) 2.0))
        (t_3 (log (sqrt (exp t_2))))
        (t_4 (sqrt (- 1.0 t_2))))
   (if (<= (/ t l) -1.1468627857485825e+141)
     (asin (* t_4 (- t_1)))
     (if (<= (/ t l) 3.29234323632466e+21)
       (asin (sqrt (/ (- 1.0 (+ t_3 t_3)) (fma 2.0 (pow (/ t l) 2.0) 1.0))))
       (asin (* t_4 t_1))))))
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 t_2 = pow((Om / Omc), 2.0);
	double t_3 = log(sqrt(exp(t_2)));
	double t_4 = sqrt((1.0 - t_2));
	double tmp;
	if ((t / l) <= -1.1468627857485825e+141) {
		tmp = asin((t_4 * -t_1));
	} else if ((t / l) <= 3.29234323632466e+21) {
		tmp = asin(sqrt(((1.0 - (t_3 + t_3)) / fma(2.0, pow((t / l), 2.0), 1.0))));
	} else {
		tmp = asin((t_4 * t_1));
	}
	return tmp;
}

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) < -1.14686278574858246e141

    1. Initial program 33.7

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

    2. Simplified33.7

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

    3. Taylor expanded in t around -inf 8.0

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

    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 -1.14686278574858246e141 < (/.f64 t l) < 3.2923432363246601e21

    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 add-log-exp_binary641.0

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

    4. Applied add-sqr-sqrt_binary641.0

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

    5. Applied log-prod_binary641.0

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

    if 3.2923432363246601e21 < (/.f64 t l)

    1. Initial program 20.0

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

    2. Simplified20.0

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

    3. Taylor expanded in t around inf 7.4

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

    4. Simplified0.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.1468627857485825 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.29234323632466 \cdot 10^{+21}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - \left(\log \left(\sqrt{e^{{\left(\frac{Om}{Omc}\right)}^{2}}}\right) + \log \left(\sqrt{e^{{\left(\frac{Om}{Omc}\right)}^{2}}}\right)\right)}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(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)))))))