Average Error: 10.1 → 0.9
Time: 10.6s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := {\left(\frac{Om}{Omc}\right)}^{2}\\ t_2 := \sqrt{1 - t_1}\\ \mathbf{if}\;\frac{t}{\ell} \leq -1.0330560465423799 \cdot 10^{+45}:\\ \;\;\;\;\sin^{-1} \left(t_2 \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 6.337213396890452 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{t_1}\right)}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{t_2}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\right)\right)\\ \end{array} \]
\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 := {\left(\frac{Om}{Omc}\right)}^{2}\\
t_2 := \sqrt{1 - t_1}\\
\mathbf{if}\;\frac{t}{\ell} \leq -1.0330560465423799 \cdot 10^{+45}:\\
\;\;\;\;\sin^{-1} \left(t_2 \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 6.337213396890452 \cdot 10^{+148}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{t_1}\right)}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{t_2}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\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 (pow (/ Om Omc) 2.0)) (t_2 (sqrt (- 1.0 t_1))))
   (if (<= (/ t l) -1.0330560465423799e+45)
     (asin (* t_2 (- (/ (* l (sqrt 0.5)) t))))
     (if (<= (/ t l) 6.337213396890452e+148)
       (expm1
        (log1p
         (asin
          (sqrt (/ (- 1.0 (log (exp t_1))) (fma 2.0 (pow (/ t l) 2.0) 1.0))))))
       (expm1 (log1p (asin (/ t_2 (/ (* t (sqrt 2.0)) 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 = pow((Om / Omc), 2.0);
	double t_2 = sqrt(1.0 - t_1);
	double tmp;
	if ((t / l) <= -1.0330560465423799e+45) {
		tmp = asin(t_2 * -((l * sqrt(0.5)) / t));
	} else if ((t / l) <= 6.337213396890452e+148) {
		tmp = expm1(log1p(asin(sqrt((1.0 - log(exp(t_1))) / fma(2.0, pow((t / l), 2.0), 1.0)))));
	} else {
		tmp = expm1(log1p(asin(t_2 / ((t * sqrt(2.0)) / l))));
	}
	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) < -1.0330560465423799e45

    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. 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. Simplified0.3

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

    if -1.0330560465423799e45 < (/.f64 t l) < 6.33721339689045184e148

    1. Initial program 0.9

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

    2. Simplified0.9

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

    3. Applied add-log-exp_binary640.9

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

    4. Applied expm1-log1p-u_binary640.9

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

    if 6.33721339689045184e148 < (/.f64 t l)

    1. Initial program 34.8

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

    2. Simplified34.8

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

    3. Applied add-log-exp_binary6434.8

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

    4. Applied expm1-log1p-u_binary6434.8

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

    5. Applied sqrt-div_binary6434.8

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

    6. Simplified34.8

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

    7. Taylor expanded in t around inf 1.3

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}}\right)\right)\right) \]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -1.0330560465423799 \cdot 10^{+45}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 6.337213396890452 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\sqrt{\frac{1 - \log \left(e^{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\mathsf{log1p}\left(\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021332 
(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)))))))