Average Error: 10.3 → 5.8
Time: 53.5s
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 4.104469908521964 \cdot 10^{+152}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_* \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc} + 1\right)}}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 4.104469908521964e+152

    1. Initial program 6.5

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

    2. Initial simplification6.5

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

    4. Applied flip--6.5

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

    5. Applied associate-/l/6.5

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

    6. Simplified6.5

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

    8. Applied expm1-log1p-u6.5

      \[\leadsto \color{blue}{(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_* \cdot \left(1 + \frac{Om}{Omc} \cdot \frac{Om}{Omc}\right)}}\right))} - 1)^*}\]

    if 4.104469908521964e+152 < (/ t l)

    1. Initial program 33.9

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

    2. Initial simplification33.9

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

    4. Applied sqrt-div33.9

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

    5. Taylor expanded around -inf 1.4

      \[\leadsto \sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\color{blue}{\frac{t \cdot \sqrt{2}}{\ell}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 4.104469908521964 \cdot 10^{+152}:\\ \;\;\;\;(e^{\log_* (1 + \sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{4}}{(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1)_* \cdot \left(\frac{Om}{Omc} \cdot \frac{Om}{Omc} + 1\right)}}\right))} - 1)^*\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Runtime

Time bar (total: 53.5s)Debug logProfile

herbie shell --seed 2018249 +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)))))))