Average Error: 10.2 → 10.2
Time: 47.3s
Precision: 64
Internal Precision: 128
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*} \cdot \left(\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}} \cdot \sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right)}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

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. Simplified10.2

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

  4. Applied *-un-lft-identity10.2

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

  5. Applied add-cube-cbrt10.2

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

  6. Applied times-frac10.2

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

  7. Simplified10.2

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

  8. Final simplification10.2

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

Reproduce

herbie shell --seed 2019093 +o rules:numerics
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))