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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `t`

Derivation

Initial program 10.7
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Initial simplification10.7
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right)\]
Using strategy rm
Applied clear-num10.8
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{\frac{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}}\right)\]
Final simplification10.8
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\frac{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}\right)\]

Runtime

herbie shell --seed 2018255 +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)))))))