Average Error: 10.4 → 10.4
Time: 7.3m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\sqrt{\frac{{1}^{3} - {\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}^{3}}{\left(1 + \frac{2}{\frac{\ell}{t} \cdot \frac{\ell}{t}}\right) \cdot \left(\left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc}\right) + \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc} + 1\right)\right)}}\right)\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 10.4

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

  2. Taylor expanded around inf 49.4

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

  3. Applied simplify10.4

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

  5. Applied flip3--10.4

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

  6. Applied associate-/l/10.4

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

  7. Applied simplify10.4

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

Runtime

Time bar (total: 7.3m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))