Average Error: 10.5 → 5.7
Time: 11.9s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 3.042991592892838 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \sqrt{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array}\]
\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 3.042991592892838 \cdot 10^{+57}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \sqrt{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\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) 3.042991592892838e+57)
   (asin
    (*
     (sqrt (sqrt (- 1.0 (pow (/ Om Omc) 2.0))))
     (sqrt
      (/
       (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
       (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))
   (asin (/ (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (* (/ t l) (sqrt 2.0))))))
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) <= 3.042991592892838e+57) {
		tmp = asin(sqrt(sqrt(1.0 - pow((Om / Omc), 2.0))) * sqrt(sqrt(1.0 - pow((Om / Omc), 2.0)) / (1.0 + (2.0 * pow((t / l), 2.0)))));
	} else {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) / ((t / l) * sqrt(2.0)));
	}
	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

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 t l) < 3.04299159289283812e57

    1. Initial program 6.9

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

    3. Applied *-un-lft-identity_binary64_4236.9

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

    4. Applied add-sqr-sqrt_binary64_4456.9

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

    5. Applied times-frac_binary64_4296.9

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

    6. Applied sqrt-prod_binary64_4396.9

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

    7. Simplified6.9

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

    if 3.04299159289283812e57 < (/.f64 t l)

    1. Initial program 24.7

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

    3. Applied sqrt-div_binary64_44024.7

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

    4. Taylor expanded around inf 1.0

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

    5. Simplified1.0

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 3.042991592892838 \cdot 10^{+57}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot \sqrt{\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020289 
(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)))))))