\[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
Test:
Rosa's FloatVsDoubleBenchmark
Bits:
128 bits
Bits error versus x1
Bits error versus x2
Time: 35.4 s
Input Error: 0.5
Output Error: 0.3
Log: ⚲
Profile: 🕒
\((\left(\frac{3}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(3 \cdot x1\right) - (x2 * 2 + x1)_*\right) + \left(x1 + x1\right))_* + (\left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2\right))_*}{\frac{(x1 * x1 + 1)_*}{2}}\right) * \left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*}{\frac{(x1 * x1 + 1)_*}{x1}}\right) + \left(\left(\frac{x1 \cdot x1}{(x1 * x1 + 1)_*} \cdot (\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*\right) \cdot \left(4 - \frac{2}{(x1 * x1 + 1)_*}\right) - (\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*}{\frac{(x1 * x1 + 1)_*}{x1}}\right) * 6 + \left(\left(x1 \cdot x1\right) \cdot 6\right))_*\right))_*\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(3 \cdot x1\right)\right) + \left({x1}^3\right))_*\right))_*\)
  1. Started with
    \[x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)\]
    0.5
  2. Applied simplify to get
    \[\color{red}{x1 + \left(\left(\left(\left(\left(\left(\left(2 \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) \cdot \left(\frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 3\right) + \left(x1 \cdot x1\right) \cdot \left(4 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1} - 6\right)\right) \cdot \left(x1 \cdot x1 + 1\right) + \left(\left(3 \cdot x1\right) \cdot x1\right) \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 + 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right) + \left(x1 \cdot x1\right) \cdot x1\right) + x1\right) + 3 \cdot \frac{\left(\left(3 \cdot x1\right) \cdot x1 - 2 \cdot x2\right) - x1}{x1 \cdot x1 + 1}\right)} \leadsto \color{blue}{(\left((\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right) * \left({x1}^2\right) + \left(\frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right))_*\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)}\]
    0.5
  3. Using strategy rm
    0.5
  4. Applied fma-udef to get
    \[(\color{red}{\left((\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right) * \left({x1}^2\right) + \left(\frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right))_*\right)} * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right) \leadsto (\color{blue}{\left(\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right) \cdot {x1}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right)} * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)\]
    0.5
  5. Applied taylor to get
    \[(\left(\left(\frac{4}{(x1 * x1 + 1)_*} \cdot (\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_* - 6\right) \cdot {x1}^2 + \frac{\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2\right))_*}{(x1 * x1 + 1)_*} - \left(3 + \frac{x1}{(x1 * x1 + 1)_*}\right)\right) \cdot x1}{\frac{\frac{(x1 * x1 + 1)_*}{2}}{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}}\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right) \leadsto (\left(\left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2\right))_* \cdot \left((\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1\right)}{{\left((x1 * x1 + 1)_*\right)}^2} + 4 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{(x1 * x1 + 1)_*}\right) - \left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{{\left((x1 * x1 + 1)_*\right)}^2} + \left(6 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1}{(x1 * x1 + 1)_*} + 6 \cdot {x1}^2\right)\right)\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)\]
    3.0
  6. Taylor expanded around 0 to get
    \[(\color{red}{\left(\left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2\right))_* \cdot \left((\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1\right)}{{\left((x1 * x1 + 1)_*\right)}^2} + 4 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{(x1 * x1 + 1)_*}\right) - \left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{{\left((x1 * x1 + 1)_*\right)}^2} + \left(6 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1}{(x1 * x1 + 1)_*} + 6 \cdot {x1}^2\right)\right)\right)} * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right) \leadsto (\color{blue}{\left(\left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2\right))_* \cdot \left((\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1\right)}{{\left((x1 * x1 + 1)_*\right)}^2} + 4 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{(x1 * x1 + 1)_*}\right) - \left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{{\left((x1 * x1 + 1)_*\right)}^2} + \left(6 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1}{(x1 * x1 + 1)_*} + 6 \cdot {x1}^2\right)\right)\right)} * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)\]
    3.0
  7. Applied simplify to get
    \[\color{red}{(\left(\left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2\right))_* \cdot \left((\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1\right)}{{\left((x1 * x1 + 1)_*\right)}^2} + 4 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{(x1 * x1 + 1)_*}\right) - \left(2 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot {x1}^2}{{\left((x1 * x1 + 1)_*\right)}^2} + \left(6 \cdot \frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_* \cdot x1}{(x1 * x1 + 1)_*} + 6 \cdot {x1}^2\right)\right)\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(x1 \cdot 3\right) * x1 + \left(x2 \cdot 2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(x1 \cdot 3\right)\right) + \left({x1}^3\right))_*\right))_* + \left(\left(x1 + x1\right) + \frac{3}{(x1 * x1 + 1)_*} \cdot \left(x1 \cdot \left(x1 \cdot 3\right) - (x2 * 2 + x1)_*\right)\right)} \leadsto \color{blue}{(\left(\frac{3}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(3 \cdot x1\right) - (x2 * 2 + x1)_*\right) + \left(x1 + x1\right))_* + (\left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2\right))_*}{\frac{(x1 * x1 + 1)_*}{2}}\right) * \left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*}{\frac{(x1 * x1 + 1)_*}{x1}}\right) + \left(\left(\frac{x1 \cdot x1}{(x1 * x1 + 1)_*} \cdot (\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*\right) \cdot \left(4 - \frac{2}{(x1 * x1 + 1)_*}\right) - (\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*}{\frac{(x1 * x1 + 1)_*}{x1}}\right) * 6 + \left(\left(x1 \cdot x1\right) \cdot 6\right))_*\right))_*\right) * \left((x1 * x1 + 1)_*\right) + \left((\left(\frac{(\left(3 \cdot x1\right) * x1 + \left(2 \cdot x2 - x1\right))_*}{(x1 * x1 + 1)_*}\right) * \left(x1 \cdot \left(3 \cdot x1\right)\right) + \left({x1}^3\right))_*\right))_*}\]
    0.3
  8. Removed slow pow expressions

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))))))