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

↓

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

Error

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

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Out

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Derivation

Initial program 10.3
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
Using strategy rm
Applied clear-num10.3
\[\leadsto \sin^{-1} \left(\sqrt{\color{blue}{\frac{1}{\frac{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}}\right)\]
Final simplification10.3
\[\leadsto \sin^{-1} \left(\sqrt{\frac{1}{\frac{1 + {\left(\frac{t}{\ell}\right)}^{2} \cdot 2}{1 - {\left(\frac{Om}{Omc}\right)}^{2}}}}\right)\]

Runtime

herbie shell --seed 2018217 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))