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

↓

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

Error

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

Derivation

Initial program 10.5
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Taylor expanded around inf 49.4
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1 - \color{blue}{e^{2 \cdot \left(\log \left(\frac{1}{Omc}\right) - \log \left(\frac{1}{Om}\right)\right)}}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Applied simplify10.5
\[\leadsto \color{blue}{\sin^{-1} \left(\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\frac{\frac{2}{\frac{\ell}{t}}}{\frac{\ell}{t}} + 1}}\right)}\]

Runtime

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))