Average Error: 9.8 → 0.8
Time: 12.6s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := \frac{\ell \cdot \sqrt{0.5}}{t}\\ t_2 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\ t_3 := \sqrt{t_2}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1.7351943106705547 \cdot 10^{+68}:\\ \;\;\;\;\sin^{-1} \left(t_3 \cdot \left(-t_1\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 494140.4287704275:\\ \;\;\;\;\begin{array}{l} t_4 := \mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)\\ t_5 := \sqrt{\sqrt{\sin^{-1} \left(\frac{t_3}{\sqrt{t_4}}\right)}}\\ \sqrt{\sin^{-1} \left(\sqrt{\frac{t_2}{t_4}}\right)} \cdot \left(t_5 \cdot t_5\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(t_3 \cdot t_1\right)\\ \end{array} \]
\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 := 1 - {\left(\frac{Om}{Omc}\right)}^{2}\\
t_3 := \sqrt{t_2}\\
\mathbf{if}\;\frac{t}{\ell} \leq -1.7351943106705547 \cdot 10^{+68}:\\
\;\;\;\;\sin^{-1} \left(t_3 \cdot \left(-t_1\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 494140.4287704275:\\
\;\;\;\;\begin{array}{l}
t_4 := \mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)\\
t_5 := \sqrt{\sqrt{\sin^{-1} \left(\frac{t_3}{\sqrt{t_4}}\right)}}\\
\sqrt{\sin^{-1} \left(\sqrt{\frac{t_2}{t_4}}\right)} \cdot \left(t_5 \cdot t_5\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(t_3 \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 (- 1.0 (pow (/ Om Omc) 2.0)))
        (t_3 (sqrt t_2)))
   (if (<= (/ t l) -1.7351943106705547e+68)
     (asin (* t_3 (- t_1)))
     (if (<= (/ t l) 494140.4287704275)
       (let* ((t_4 (fma 2.0 (pow (/ t l) 2.0) 1.0))
              (t_5 (sqrt (sqrt (asin (/ t_3 (sqrt t_4)))))))
         (* (sqrt (asin (sqrt (/ t_2 t_4)))) (* t_5 t_5)))
       (asin (* t_3 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 = 1.0 - pow((Om / Omc), 2.0);
	double t_3 = sqrt(t_2);
	double tmp;
	if ((t / l) <= -1.7351943106705547e+68) {
		tmp = asin(t_3 * -t_1);
	} else if ((t / l) <= 494140.4287704275) {
		double t_4 = fma(2.0, pow((t / l), 2.0), 1.0);
		double t_5 = sqrt(sqrt(asin(t_3 / sqrt(t_4))));
		tmp = sqrt(asin(sqrt(t_2 / t_4))) * (t_5 * t_5);
	} else {
		tmp = asin(t_3 * 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) < -1.7351943106705547e68

    1. Initial program 23.8

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

    2. Simplified23.8

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

    3. Taylor expanded in t around -inf 7.2

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

    4. Simplified0.3

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

    if -1.7351943106705547e68 < (/.f64 t l) < 494140.42877042748

    1. Initial program 1.1

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

    2. Simplified1.1

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

    3. Applied add-sqr-sqrt_binary641.9

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

    4. Applied sqrt-div_binary641.9

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

    5. Applied add-sqr-sqrt_binary641.1

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

    if 494140.42877042748 < (/.f64 t l)

    1. Initial program 18.6

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

    2. Simplified18.6

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

    3. Taylor expanded in t around inf 8.2

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

    4. Simplified0.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.7351943106705547 \cdot 10^{+68}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 494140.4287704275:\\ \;\;\;\;\sqrt{\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)} \cdot \left(\sqrt{\sqrt{\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)}} \cdot \sqrt{\sqrt{\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022094 
(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)))))))