Average Error: 10.5 → 10.6
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}} \cdot \sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{\sqrt{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}}\right))} - 1)^*\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Initial program 10.5

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

  2. Initial simplification10.5

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

  4. Applied expm1-log1p-u10.5

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

  6. Applied add-sqr-sqrt10.6

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

  7. Applied *-un-lft-identity10.6

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

  8. Applied times-frac10.5

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

  9. Applied sqrt-prod10.6

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

  10. Final simplification10.6

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018242 +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)))))))