\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
Test:
Rosa's FloatVsDoubleBenchmark
Bits:
128 bits
Bits error versus x1
Bits error versus x2
Time: 30.0 s
Input Error: 0.6
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\left((9 * \left(\frac{x1 \cdot x1}{(x1 * x1 + 1)_*}\right) + \left(2 \cdot x1\right))_* - \frac{(x2 * 2 + x1)_*}{\frac{(x1 * x1 + 1)_*}{3}}\right) + (\left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{\frac{(x1 * x1 + 1)_*}{4}} - 6\right) * \left(x1 \cdot x1\right) + \left(\frac{\frac{(\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(\frac{x1}{(x1 * x1 + 1)_*} + 3\right)}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{x1 \cdot (\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right))_*\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(\left(3 \cdot x1\right) \cdot x1\right) + \left({x1}^3\right))_*\right))_*\)
  1. Started with
    \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
    0.6
  2. Applied simplify to get
    \[\color{red}{x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)} \leadsto \color{blue}{(\left((\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right) * \left({x1}^2\right) + \left(\frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right))_*\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)}\]
    0.3
  3. Using strategy rm
    0.3
  4. Applied fma-udef to get
    \[(\color{red}{\left((\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right) * \left({x1}^2\right) + \left(\frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right))_*\right)} * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right) \leadsto (\color{blue}{\left(\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right) \cdot {x1}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right)} * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)\]
    0.3
  5. Using strategy rm
    0.3
  6. Applied add-sqr-sqrt to get
    \[(\left(\color{red}{\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right)} \cdot {x1}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right) \leadsto (\left(\color{blue}{{\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6}\right)}^2} \cdot {x1}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)\]
    20.6
  7. Applied square-unprod to get
    \[(\left(\color{red}{{\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6}\right)}^2 \cdot {x1}^2} + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right) \leadsto (\left(\color{blue}{{\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6} \cdot x1\right)}^2} + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)\]
    20.6
  8. Applied taylor to get
    \[(\left({\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6} \cdot x1\right)}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right) \leadsto (\left({\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6} \cdot x1\right)}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(9 \cdot \frac{{x1}^2}{(x1 * x1 + 1)_*} + 2 \cdot x1\right) - 3 \cdot \frac{(x2 * 2 + x1)_*}{(x1 * x1 + 1)_*}\right)\]
    20.6
  9. Taylor expanded around 0 to get
    \[(\left({\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6} \cdot x1\right)}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \color{red}{\left(\left(9 \cdot \frac{{x1}^2}{(x1 * x1 + 1)_*} + 2 \cdot x1\right) - 3 \cdot \frac{(x2 * 2 + x1)_*}{(x1 * x1 + 1)_*}\right)} \leadsto (\left({\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6} \cdot x1\right)}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \color{blue}{\left(\left(9 \cdot \frac{{x1}^2}{(x1 * x1 + 1)_*} + 2 \cdot x1\right) - 3 \cdot \frac{(x2 * 2 + x1)_*}{(x1 * x1 + 1)_*}\right)}\]
    20.6
  10. Applied simplify to get
    \[(\left({\left(\sqrt{\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6} \cdot x1\right)}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(9 \cdot \frac{{x1}^2}{(x1 * x1 + 1)_*} + 2 \cdot x1\right) - 3 \cdot \frac{(x2 * 2 + x1)_*}{(x1 * x1 + 1)_*}\right) \leadsto \left((9 * \left(\frac{x1 \cdot x1}{(x1 * x1 + 1)_*}\right) + \left(2 \cdot x1\right))_* - \frac{(x2 * 2 + x1)_*}{\frac{(x1 * x1 + 1)_*}{3}}\right) + (\left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{\frac{(x1 * x1 + 1)_*}{4}} - 6\right) * \left(x1 \cdot x1\right) + \left(\frac{\frac{(\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(\frac{x1}{(x1 * x1 + 1)_*} + 3\right)}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{x1 \cdot (\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right))_*\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(\left(3 \cdot x1\right) \cdot x1\right) + \left({x1}^3\right))_*\right))_*\]
    0.4
  11. Applied final simplification

Original test:


(lambda ((x1 default) (x2 default))
  #:name "Rosa's FloatVsDoubleBenchmark"
  (+ x1 (+ (+ (+ (+ (* (+ (* (* (* 2 x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) (- (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)) 3)) (* (* x1 x1) (- (* 4 (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))) 6))) (+ (* x1 x1) 1)) (* (* (* 3 x1) x1) (/ (- (+ (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1)))) (* (* x1 x1) x1)) x1) (* 3 (/ (- (- (* (* 3 x1) x1) (* 2 x2)) x1) (+ (* x1 x1) 1))))))