Average Error: 10.5 → 10.5
Time: 13.7s
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\mathsf{hypot}\left({\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{\frac{3}{2}}, {1}^{\frac{3}{4}}\right) \cdot \frac{\sqrt{{1}^{\frac{3}{2}} - {\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{3}}}{\sqrt{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\]
\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(\mathsf{hypot}\left({\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{\frac{3}{2}}, {1}^{\frac{3}{4}}\right) \cdot \frac{\sqrt{{1}^{\frac{3}{2}} - {\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{3}}}{\sqrt{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\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((hypot(pow(sqrt(pow((Om / Omc), 2.0)), 1.5), pow(1.0, 0.75)) * (sqrt((pow(1.0, 1.5) - pow(sqrt(pow((Om / Omc), 2.0)), 3.0))) / sqrt((((pow((Om / Omc), 2.0) * (pow((Om / Omc), 2.0) + 1.0)) + (1.0 * 1.0)) * (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 10.5

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

  3. Applied flip3--10.5

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

  4. Applied associate-/l/10.5

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

  5. Simplified10.5

    \[\leadsto \sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}{\color{blue}{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}}\right)\]
  6. Using strategy rm

  7. Applied sqrt-div10.5

    \[\leadsto \sin^{-1} \color{blue}{\left(\frac{\sqrt{{1}^{3} - {\left({\left(\frac{Om}{Omc}\right)}^{2}\right)}^{3}}}{\sqrt{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)}\]
  8. Using strategy rm

  9. Applied *-un-lft-identity10.5

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

  10. Applied add-sqr-sqrt10.5

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

  11. Applied unpow-prod-down10.5

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

  12. Applied sqr-pow10.5

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

  13. Applied difference-of-squares10.5

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

  14. Applied sqrt-prod10.5

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

  15. Applied times-frac10.5

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

  16. Simplified10.5

    \[\leadsto \sin^{-1} \left(\color{blue}{\mathsf{hypot}\left({\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{\frac{3}{2}}, {1}^{\frac{3}{4}}\right)} \cdot \frac{\sqrt{{1}^{\left(\frac{3}{2}\right)} - {\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{3}}}{\sqrt{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\]

  17. Simplified10.5

    \[\leadsto \sin^{-1} \left(\mathsf{hypot}\left({\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{\frac{3}{2}}, {1}^{\frac{3}{4}}\right) \cdot \color{blue}{\frac{\sqrt{{1}^{\frac{3}{2}} - {\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{3}}}{\sqrt{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}}\right)\]

  18. Final simplification10.5

    \[\leadsto \sin^{-1} \left(\mathsf{hypot}\left({\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{\frac{3}{2}}, {1}^{\frac{3}{4}}\right) \cdot \frac{\sqrt{{1}^{\frac{3}{2}} - {\left(\sqrt{{\left(\frac{Om}{Omc}\right)}^{2}}\right)}^{3}}}{\sqrt{\left({\left(\frac{Om}{Omc}\right)}^{2} \cdot \left({\left(\frac{Om}{Omc}\right)}^{2} + 1\right) + 1 \cdot 1\right) \cdot \left(1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}\right)}}\right)\]

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(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)))))))