Average Error: 10.1 → 1.2
Time: 5.5min
Precision: binary64
Cost: 27714
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4.4325947615188905 \cdot 10^{+44}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.4406746718840184 \cdot 10^{+56}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot {\left(1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array}\]
\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}
\mathbf{if}\;\frac{t}{\ell} \leq -4.4325947615188905 \cdot 10^{+44}:\\
\;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 1.4406746718840184 \cdot 10^{+56}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot {\left(1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)\right)}^{-0.5}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\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
 (if (<= (/ t l) -4.4325947615188905e+44)
   (asin (- (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (sqrt 0.5) (/ t l)))))
   (if (<= (/ t l) 1.4406746718840184e+56)
     (asin
      (*
       (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
       (pow (+ 1.0 (* 2.0 (* t (/ (/ t l) l)))) -0.5)))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (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 tmp;
	if ((t / l) <= -4.4325947615188905e+44) {
		tmp = asin(-(sqrt(1.0 - pow((Om / Omc), 2.0)) * (sqrt(0.5) / (t / l))));
	} else if ((t / l) <= 1.4406746718840184e+56) {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * pow((1.0 + (2.0 * (t * ((t / l) / l)))), -0.5));
	} else {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * (sqrt(0.5) / (t / l)));
	}
	return tmp;
}

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error1.0
Cost27714
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2.3765008719296084 \cdot 10^{+114}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.0800233055072413 \cdot 10^{+146}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}} \cdot {\left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array}\]
Alternative 2
Error1.0
Cost27266
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -2.11346151031864 \cdot 10^{+114}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.4558158866244431 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array}\]
Alternative 3
Error5.6
Cost26817
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 1.4558158866244431 \cdot 10^{+20}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array}\]
Alternative 4
Error10.1
Cost20352
\[\sin^{-1} \left(\sqrt{\frac{1 - Om \cdot \frac{\frac{Om}{Omc}}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Alternative 5
Error10.7
Cost19840
\[\sin^{-1} \left(\sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Alternative 6
Error29.7
Cost13633
\[\begin{array}{l} \mathbf{if}\;t \leq 5.546388378083076 \cdot 10^{+115}:\\ \;\;\;\;\sin^{-1} \left(1 - {\left(\frac{Om}{Omc}\right)}^{2} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 7
Error29.9
Cost6785
\[\begin{array}{l} \mathbf{if}\;t \leq 4.167382628070661 \cdot 10^{+116}:\\ \;\;\;\;\sin^{-1} 1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 8
Error49.6
Cost706
\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.035219707846875 \cdot 10^{-104}:\\ \;\;\;\;1\\ \mathbf{elif}\;\ell \leq 2.1941543636572065 \cdot 10^{-53}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]
Alternative 9
Error56.0
Cost64
\[1\]

Error

Time

Derivation

  1. Split input into 3 regimes

  2. if (/.f64 t l) < -4.43259476151889046e44

    1. Initial program 22.5

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

    2. Taylor expanded around -inf 8.4

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

    3. Simplified1.1

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

    4. Simplified1.1

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

    if -4.43259476151889046e44 < (/.f64 t l) < 1.44067467188401839e56

    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. Using strategy rm

    3. Applied div-inv_binary64_4160.9

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

    4. Applied sqrt-prod_binary64_4350.9

      \[\leadsto \sin^{-1} \color{blue}{\left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \sqrt{\frac{1}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)}\]
    5. Using strategy rm

    6. Applied inv-pow_binary64_5040.9

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

    7. Applied sqrt-pow1_binary64_4370.9

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

    8. Simplified0.9

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

    10. Applied *-un-lft-identity_binary64_4190.9

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

    11. Applied add-sqr-sqrt_binary64_44132.7

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

    12. Applied times-frac_binary64_42532.7

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

    13. Applied unpow-prod-down_binary64_49832.8

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

    14. Simplified32.8

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

    15. Simplified1.3

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

    16. Simplified1.3

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

    if 1.44067467188401839e56 < (/.f64 t l)

    1. Initial program 22.9

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

    2. Taylor expanded around inf 7.2

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

    3. Simplified1.1

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

    4. Simplified1.1

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -4.4325947615188905 \cdot 10^{+44}:\\ \;\;\;\;\sin^{-1} \left(-\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 1.4406746718840184 \cdot 10^{+56}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot {\left(1 + 2 \cdot \left(t \cdot \frac{\frac{t}{\ell}}{\ell}\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\sqrt{0.5}}{\frac{t}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021065 
(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)))))))