Average Error: 10.4 → 10.4
Time: 50.0s
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\sin^{-1} \left(\left|\sqrt{\frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_*}}\right|\right)\]

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. Initial simplification10.4

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

  4. Applied add-cube-cbrt10.5

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

  5. Applied associate-*r*10.5

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

  7. Applied add-sqr-sqrt10.5

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

  8. Applied rem-sqrt-square10.5

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

  9. Simplified10.4

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

  10. Final simplification10.4

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

Runtime

Time bar (total: 50.0s)Debug logProfile

herbie shell --seed 2018277 +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)))))))