Average Error: 9.8 → 9.8
Time: 1.3m
Precision: 64
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{1}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}} - \frac{\frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 9.8

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

  2. Simplified9.8

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

  3. Taylor expanded around inf 25.8

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

  4. Simplified9.8

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

  6. Applied div-sub9.8

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

  7. Final simplification9.8

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

Reproduce

herbie shell --seed 2019100 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))