Average Error: 10.7 → 1.0
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\sin^{-1} \left(\sqrt{\frac{1 - {\left(\frac{Om}{Omc}\right)}^{2}}{1 + 2 \cdot {\left(\frac{t}{\ell}\right)}^{2}}}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -1.1143453734621622 \cdot 10^{+76} \lor \neg \left(\frac{t}{\ell} \le 6.867237757530834 \cdot 10^{+62}\right):\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 - 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 - \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)}}\right)\\ \end{array}\]

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) < -1.1143453734621622e+76 or 6.867237757530834e+62 < (/ t l)

    1. Initial program 25.0

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

    2. Initial simplification25.0

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

    4. Applied add-log-exp25.0

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

    6. Applied add-sqr-sqrt25.1

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

    7. Applied add-sqr-sqrt25.1

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

    8. Applied times-frac25.0

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

    9. Applied rem-sqrt-square25.0

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

    10. Simplified25.0

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

    11. Taylor expanded around 0 1.1

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

    if -1.1143453734621622e+76 < (/ t l) < 6.867237757530834e+62

    1. Initial program 1.0

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

    2. Initial simplification1.0

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

    4. Applied flip-+1.0

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

    5. Applied associate-/r/1.0

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t}{\ell} \le -1.1143453734621622 \cdot 10^{+76} \lor \neg \left(\frac{t}{\ell} \le 6.867237757530834 \cdot 10^{+62}\right):\\ \;\;\;\;\sin^{-1} \left(\left|\frac{\sqrt{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}}{\frac{\sqrt{2} \cdot t}{\ell}}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin^{-1} \left(\sqrt{\left(1 - 2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \frac{1 - \frac{Om}{Omc} \cdot \frac{Om}{Omc}}{1 - \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(2 \cdot \left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right)\right)}}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018250 
(FPCore (t l Om Omc)
  :name "Toniolo and Linder, Equation (2)"
  (asin (sqrt (/ (- 1 (pow (/ Om Omc) 2)) (+ 1 (* 2 (pow (/ t l) 2)))))))