\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
Test:
Octave 3.8, jcobi/3
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 58.5 s
Input Error: 1.9
Output Error: 1.2
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\frac{\sqrt{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}}{\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)} \cdot \sqrt{\alpha + \left(2 + \beta\right)}}\right)}^2 & \text{when } \beta \le 6.898948f+25 \\ \frac{\left(1.0 + \alpha\right) + (\left(\frac{1}{\beta}\right) * \left(\frac{1}{\alpha}\right) + \left(\frac{1}{\beta}\right))_*}{\left(2 + \left(\alpha + \beta\right)\right) \cdot (5.0 * \left(\alpha + \beta\right) + 6.0)_*} & \text{otherwise} \end{cases}\)

    if beta < 6.898948f+25

    1. Started with
      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
      1.1
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}} \leadsto \color{blue}{\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}}\]
      1.8
    3. Using strategy rm
      1.8
    4. Applied add-sqr-sqrt to get
      \[\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \color{red}{\left(\alpha + \left(2 + \beta\right)\right)}} \leadsto \frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \color{blue}{{\left(\sqrt{\alpha + \left(2 + \beta\right)}\right)}^2}}\]
      1.2
    5. Applied add-sqr-sqrt to get
      \[\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\color{red}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right)} \cdot {\left(\sqrt{\alpha + \left(2 + \beta\right)}\right)}^2} \leadsto \frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\color{blue}{{\left(\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)}\right)}^2} \cdot {\left(\sqrt{\alpha + \left(2 + \beta\right)}\right)}^2}\]
      1.3
    6. Applied square-unprod to get
      \[\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\color{red}{{\left(\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)}\right)}^2 \cdot {\left(\sqrt{\alpha + \left(2 + \beta\right)}\right)}^2}} \leadsto \frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\color{blue}{{\left(\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)} \cdot \sqrt{\alpha + \left(2 + \beta\right)}\right)}^2}}\]
      1.3
    7. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}}{{\left(\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)} \cdot \sqrt{\alpha + \left(2 + \beta\right)}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}\right)}^2}}{{\left(\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)} \cdot \sqrt{\alpha + \left(2 + \beta\right)}\right)}^2}\]
      1.3
    8. Applied square-undiv to get
      \[\color{red}{\frac{{\left(\sqrt{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}\right)}^2}{{\left(\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)} \cdot \sqrt{\alpha + \left(2 + \beta\right)}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}}{\sqrt{\left(\alpha + 1.0\right) + \left(2 + \beta\right)} \cdot \sqrt{\alpha + \left(2 + \beta\right)}}\right)}^2}\]
      1.3

    if 6.898948f+25 < beta

    1. Started with
      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
      7.8
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}} \leadsto \color{blue}{\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}}\]
      7.8
    3. Applied taylor to get
      \[\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)} \leadsto \frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{5.0 \cdot \beta + \left(6.0 + 5.0 \cdot \alpha\right)}\]
      26.9
    4. Taylor expanded around 0 to get
      \[\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\color{red}{5.0 \cdot \beta + \left(6.0 + 5.0 \cdot \alpha\right)}} \leadsto \frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\color{blue}{5.0 \cdot \beta + \left(6.0 + 5.0 \cdot \alpha\right)}}\]
      26.9
    5. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{5.0 \cdot \beta + \left(6.0 + 5.0 \cdot \alpha\right)}} \leadsto \color{blue}{\frac{(\beta * \alpha + \beta)_* + \left(\alpha + 1.0\right)}{\left(2 + \left(\beta + \alpha\right)\right) \cdot (5.0 * \left(\beta + \alpha\right) + 6.0)_*}}\]
      26.9
    6. Applied taylor to get
      \[\frac{(\beta * \alpha + \beta)_* + \left(\alpha + 1.0\right)}{\left(2 + \left(\beta + \alpha\right)\right) \cdot (5.0 * \left(\beta + \alpha\right) + 6.0)_*} \leadsto \frac{1.0 + \left(\alpha + (\left(\frac{1}{\beta}\right) * \left(\frac{1}{\alpha}\right) + \left(\frac{1}{\beta}\right))_*\right)}{\left(2 + \left(\beta + \alpha\right)\right) \cdot (5.0 * \left(\beta + \alpha\right) + 6.0)_*}\]
      0.2
    7. Taylor expanded around inf to get
      \[\frac{\color{red}{1.0 + \left(\alpha + (\left(\frac{1}{\beta}\right) * \left(\frac{1}{\alpha}\right) + \left(\frac{1}{\beta}\right))_*\right)}}{\left(2 + \left(\beta + \alpha\right)\right) \cdot (5.0 * \left(\beta + \alpha\right) + 6.0)_*} \leadsto \frac{\color{blue}{1.0 + \left(\alpha + (\left(\frac{1}{\beta}\right) * \left(\frac{1}{\alpha}\right) + \left(\frac{1}{\beta}\right))_*\right)}}{\left(2 + \left(\beta + \alpha\right)\right) \cdot (5.0 * \left(\beta + \alpha\right) + 6.0)_*}\]
      0.2
    8. Applied simplify to get
      \[\frac{1.0 + \left(\alpha + (\left(\frac{1}{\beta}\right) * \left(\frac{1}{\alpha}\right) + \left(\frac{1}{\beta}\right))_*\right)}{\left(2 + \left(\beta + \alpha\right)\right) \cdot (5.0 * \left(\beta + \alpha\right) + 6.0)_*} \leadsto \frac{\left(1.0 + \alpha\right) + (\left(\frac{1}{\beta}\right) * \left(\frac{1}{\alpha}\right) + \left(\frac{1}{\beta}\right))_*}{\left(2 + \left(\alpha + \beta\right)\right) \cdot (5.0 * \left(\alpha + \beta\right) + 6.0)_*}\]
      0.2
    9. Applied final simplification
  1. Removed slow pow expressions

Original test:


(lambda ((alpha default) (beta default))
  #:name "Octave 3.8, jcobi/3"
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))