Average Error: 10.1 → 10.3
Time: 8.8s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\sqrt{1} + {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{1}} \cdot \sqrt{\frac{\sqrt{1} - {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
\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(\sqrt{\frac{\sqrt{1} + {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{1}} \cdot \sqrt{\frac{\sqrt{1} - {\left(\frac{Om}{Omc}\right)}^{\left(\frac{2}{2}\right)}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)
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) {
	return asin((sqrt(((sqrt(1.0) + pow((Om / Omc), (2.0 / 2.0))) / 1.0)) * sqrt(((sqrt(1.0) - pow((Om / Omc), (2.0 / 2.0))) / (1.0 + (2.0 * pow((t / l), 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 10.1

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

  3. Applied *-un-lft-identity10.1

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

  4. Applied sqr-pow10.1

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

  5. Applied add-sqr-sqrt10.1

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

  6. Applied difference-of-squares10.2

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

  7. Applied times-frac10.2

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

  8. Applied sqrt-prod10.3

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

  9. Final simplification10.3

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

Reproduce

herbie shell --seed 2020058 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  :precision binary64
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))