Average Error: 10.6 → 1.9
Time: 24.6s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -8.210628599653104 \cdot 10^{+185}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{\frac{1}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot 0.125\right) \cdot \left(\frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} - {\left(\frac{Om}{Omc}\right)}^{2} \cdot \frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}}\right) - \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.0531447535757206 \cdot 10^{+112}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]
\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 -8.210628599653104 \cdot 10^{+185}:\\
\;\;\;\;\sin^{-1} \left(\left(\sqrt{\frac{1}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot 0.125\right) \cdot \left(\frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} - {\left(\frac{Om}{Omc}\right)}^{2} \cdot \frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}}\right) - \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\

\mathbf{elif}\;\frac{t}{\ell} \leq 3.0531447535757206 \cdot 10^{+112}:\\
\;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{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{\ell \cdot \sqrt{0.5}}{t}\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) -8.210628599653104e+185)
   (asin
    (-
     (*
      (* (sqrt (/ 1.0 (- 1.0 (pow (/ Om Omc) 2.0)))) 0.125)
      (-
       (/ (pow l 3.0) (* (sqrt 0.5) (pow t 3.0)))
       (* (pow (/ Om Omc) 2.0) (/ (pow l 3.0) (* (sqrt 0.5) (pow t 3.0))))))
     (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (* l (sqrt 0.5)) t))))
   (if (<= (/ t l) 3.0531447535757206e+112)
     (asin
      (sqrt
       (*
        (+ 1.0 (/ Om Omc))
        (/ (- 1.0 (/ Om Omc)) (+ 1.0 (* 2.0 (pow (/ t l) 2.0)))))))
     (asin (* (sqrt (- 1.0 (pow (/ Om Omc) 2.0))) (/ (* l (sqrt 0.5)) t))))))
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) <= -8.210628599653104e+185) {
		tmp = asin(((sqrt(1.0 / (1.0 - pow((Om / Omc), 2.0))) * 0.125) * ((pow(l, 3.0) / (sqrt(0.5) * pow(t, 3.0))) - (pow((Om / Omc), 2.0) * (pow(l, 3.0) / (sqrt(0.5) * pow(t, 3.0)))))) - (sqrt(1.0 - pow((Om / Omc), 2.0)) * ((l * sqrt(0.5)) / t)));
	} else if ((t / l) <= 3.0531447535757206e+112) {
		tmp = asin(sqrt((1.0 + (Om / Omc)) * ((1.0 - (Om / Omc)) / (1.0 + (2.0 * pow((t / l), 2.0))))));
	} else {
		tmp = asin(sqrt(1.0 - pow((Om / Omc), 2.0)) * ((l * sqrt(0.5)) / t));
	}
	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 3 regimes

  2. if (/.f64 t l) < -8.21062859965310352e185

    1. Initial program 29.7

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

    2. Taylor expanded around -inf 9.5

      \[\leadsto \sin^{-1} \color{blue}{\left(0.125 \cdot \left(\frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} \cdot \sqrt{\frac{1}{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}\right) - \left(\frac{\sqrt{0.5} \cdot \ell}{t} \cdot \sqrt{1 - \frac{{Om}^{2}}{{Omc}^{2}}} + 0.125 \cdot \left(\frac{{Om}^{2} \cdot {\ell}^{3}}{\sqrt{0.5} \cdot \left({t}^{3} \cdot {Omc}^{2}\right)} \cdot \sqrt{\frac{1}{1 - \frac{{Om}^{2}}{{Omc}^{2}}}}\right)\right)\right)}\]

    3. Simplified0.6

      \[\leadsto \sin^{-1} \color{blue}{\left(\left(\sqrt{\frac{1}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot 0.125\right) \cdot \left(\frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} - \frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} \cdot {\left(\frac{Om}{Omc}\right)}^{2}\right) - \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)}\]

    if -8.21062859965310352e185 < (/.f64 t l) < 3.05314475357572065e112

    1. Initial program 2.4

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

    3. Applied *-un-lft-identity_binary64_782.4

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

    4. Applied sqr-pow_binary64_502.4

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

    5. Applied *-un-lft-identity_binary64_782.4

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

    6. Applied difference-of-squares_binary64_472.5

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

    7. Applied times-frac_binary64_842.5

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

    if 3.05314475357572065e112 < (/.f64 t l)

    1. Initial program 30.2

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

    2. Taylor expanded around inf 7.5

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

    3. Simplified0.3

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

  4. Final simplification1.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \leq -8.210628599653104 \cdot 10^{+185}:\\ \;\;\;\;\sin^{-1} \left(\left(\sqrt{\frac{1}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}} \cdot 0.125\right) \cdot \left(\frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}} - {\left(\frac{Om}{Omc}\right)}^{2} \cdot \frac{{\ell}^{3}}{\sqrt{0.5} \cdot {t}^{3}}\right) - \sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \mathbf{elif}\;\frac{t}{\ell} \leq 3.0531447535757206 \cdot 10^{+112}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 + \frac{Om}{Omc}\right) \cdot \frac{1 - \frac{Om}{Omc}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}} \cdot \frac{\ell \cdot \sqrt{0.5}}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021084 
(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)))))))