Average Error: 10.3 → 5.6
Time: 8.7s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 1.7524108246432387 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array}\]
\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 1.7524108246432387 \cdot 10^{+153}:\\
\;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{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) 1.7524108246432387e+153)
   (asin
    (/
     (sqrt (- 1.0 (pow (/ Om Omc) 2.0)))
     (sqrt (+ 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 ((double) asin(((double) sqrt((((double) (1.0 - ((double) pow((Om / Omc), 2.0)))) / ((double) (1.0 + ((double) (2.0 * ((double) pow((t / l), 2.0)))))))))));
}
↓
double code(double t, double l, double Om, double Omc) {
	double tmp;
	if (((t / l) <= 1.7524108246432387e+153)) {
		tmp = ((double) asin((((double) sqrt(((double) (1.0 - ((double) pow((Om / Omc), 2.0)))))) / ((double) sqrt(((double) (1.0 + ((double) (2.0 * ((double) pow((t / l), 2.0)))))))))));
	} else {
		tmp = ((double) asin((((double) sqrt(((double) (1.0 - ((double) pow((Om / Omc), 2.0)))))) / ((double) ((t / l) * ((double) 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) < 1.7524108246432387e153

    1. Initial program 6.3

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

    3. Applied sqrt-div_binary646.4

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

    if 1.7524108246432387e153 < (/.f64 t l)

    1. Initial program 34.5

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

    3. Applied sqrt-div_binary6434.5

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

    4. Taylor expanded around inf 1.2

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

    5. Simplified1.2

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq 1.7524108246432387 \cdot 10^{+153}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\frac{t}{\ell} \cdot \sqrt{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020219 
(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)))))))