Average Error: 10.2 → 10.2
Time: 12.2s
Precision: binary64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{1}{\ell} \cdot t\right)}^{2}}}\right)\]
\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
↓
\sin^{-1} \left(\frac{\sqrt{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}{\sqrt{1 + 2 \cdot {\left(\frac{1}{\ell} \cdot t\right)}^{2}}}\right)
double code(double t, double l, double Om, double Omc) {
	return ((double) asin(((double) sqrt(((double) (((double) (1.0 - ((double) pow(((double) (Om / Omc)), 2.0)))) / ((double) (1.0 + ((double) (2.0 * ((double) pow(((double) (t / l)), 2.0))))))))))));
}
↓
double code(double t, double l, double Om, double Omc) {
	return ((double) asin(((double) (((double) sqrt(((double) (1.0 - ((double) pow(((double) (Om / Omc)), 2.0)))))) / ((double) sqrt(((double) (1.0 + ((double) (2.0 * ((double) pow(((double) (((double) (1.0 / l)) * t)), 2.0))))))))))));
}

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. Initial program 10.2

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

  2. Taylor expanded around -inf 51.2

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

  3. Simplified10.2

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

  5. Applied sqrt-div10.3

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

  6. Simplified10.2

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

  7. Final simplification10.2

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

Reproduce

herbie shell --seed 2020181 
(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)))))))