Average Error: 10.5 → 5.8
Time: 1.5m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.0133829843594143 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}} \cdot \left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \frac{\sqrt{\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus Omc

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ t l) < 5.0133829843594143e+141

    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

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

    4. Applied *-un-lft-identity6.5

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

    5. Applied add-cube-cbrt6.6

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

    6. Applied times-frac6.6

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

    7. Applied sqrt-prod6.6

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

    8. Simplified6.6

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

    if 5.0133829843594143e+141 < (/ t l)

    1. Initial program 33.0

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

    2. Initial simplification33.0

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

    4. Applied *-un-lft-identity33.0

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

    5. Applied add-cube-cbrt33.0

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

    6. Applied times-frac33.0

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

    7. Applied sqrt-prod33.0

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

    8. Simplified33.0

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

    10. Applied sqrt-div33.0

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

    11. Taylor expanded around -inf 1.3

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le 5.0133829843594143 \cdot 10^{+141}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\frac{\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot 2 + 1}} \cdot \left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\left|\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}\right| \cdot \frac{\sqrt{\sqrt[3]{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}}{\frac{t \cdot \sqrt{2}}{\ell}}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018234 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))