Average Error: 10.3 → 0.9
Time: 27.3s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right) \]
↓
\[\begin{array}{l} t_1 := \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\\ \mathbf{if}\;\frac{t}{\ell} \leq -9.428800306587746 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(t_1 \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.175308461442342 \cdot 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{Omc}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := t \cdot \sqrt{2}\\ \sin^{-1} \left(\left|\frac{t_1}{\mathsf{fma}\left(0.5, \frac{\ell}{t_2}, \frac{t_2}{\ell}\right)}\right|\right) \end{array}\\ \end{array} \]
\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 := \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}\\
\mathbf{if}\;\frac{t}{\ell} \leq -9.428800306587746 \cdot 10^{+153}:\\
\;\;\;\;\sin^{-1} \left(t_1 \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 1.175308461442342 \cdot 10^{+146}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{Omc}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_2 := t \cdot \sqrt{2}\\
\sin^{-1} \left(\left|\frac{t_1}{\mathsf{fma}\left(0.5, \frac{\ell}{t_2}, \frac{t_2}{\ell}\right)}\right|\right)
\end{array}\\


\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 (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))))
   (if (<= (/ t l) -9.428800306587746e+153)
     (asin (* t_1 (- (/ (* l (sqrt 0.5)) t))))
     (if (<= (/ t l) 1.175308461442342e+146)
       (asin
        (sqrt
         (*
          (+ 1.0 (/ Om Omc))
          (/ (- 1.0 (/ Om Omc)) (fma 2.0 (pow (/ t l) 2.0) 1.0)))))
       (let* ((t_2 (* t (sqrt 2.0))))
         (asin (fabs (/ t_1 (fma 0.5 (/ l t_2) (/ t_2 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 = sqrt(1.0 - pow((Om / Omc), 2.0));
	double tmp;
	if ((t / l) <= -9.428800306587746e+153) {
		tmp = asin(t_1 * -((l * sqrt(0.5)) / t));
	} else if ((t / l) <= 1.175308461442342e+146) {
		tmp = asin(sqrt((1.0 + (Om / Omc)) * ((1.0 - (Om / Omc)) / fma(2.0, pow((t / l), 2.0), 1.0))));
	} else {
		double t_2 = t * sqrt(2.0);
		tmp = asin(fabs(t_1 / fma(0.5, (l / t_2), (t_2 / 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) < -9.428800306587746e153

    1. Initial program 36.0

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

    2. Simplified36.0

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

    3. Taylor expanded in t around -inf 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 -9.428800306587746e153 < (/.f64 t l) < 1.17530846144234201e146

    1. Initial program 0.8

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

    2. Simplified0.8

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

    3. Applied *-un-lft-identity_binary640.8

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

    4. Applied add-sqr-sqrt_binary6425.4

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

    5. Applied unpow-prod-down_binary6425.4

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

    6. Applied add-sqr-sqrt_binary6425.4

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

    7. Applied difference-of-squares_binary6425.4

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

    8. Applied times-frac_binary6425.4

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

    9. Simplified25.4

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

    10. Simplified0.9

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

    if 1.17530846144234201e146 < (/.f64 t l)

    1. Initial program 33.3

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

    2. Simplified33.3

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

    3. Applied add-sqr-sqrt_binary6433.3

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

    4. Applied add-sqr-sqrt_binary6433.3

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

    5. Applied times-frac_binary6433.3

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

    6. Applied rem-sqrt-square_binary6433.3

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

    7. Taylor expanded in t around inf 1.1

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

    8. Simplified1.1

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -9.428800306587746 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \left(-\frac{\ell \cdot \sqrt{0.5}}{t}\right)\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.175308461442342 \cdot 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{Omc}}{\mathsf{fma}\left(2, {\left(\frac{t}{\ell}\right)}^{2}, 1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\mathsf{fma}\left(0.5, \frac{\ell}{t \cdot \sqrt{2}}, \frac{t \cdot \sqrt{2}}{\ell}\right)}\right|\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021352 
(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)))))))