\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 20.8 s
Input Error: 6.7
Output Error: 0.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{\beta}{\left(\alpha + 2.0\right) + \beta} - \left(\frac{4.0}{\alpha \cdot \alpha} + \left(\left(1 - 1.0\right) - \frac{2.0}{\alpha}\right)\right)}{2.0} & \text{when } \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999996f0 \\ \frac{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}{2.0} & \text{otherwise} \end{cases}\)

    if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999996f0

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      28.1
    2. Using strategy rm
      28.1
    3. Applied div-sub to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]
      28.0
    4. Applied associate-+l- to get
      \[\frac{\color{red}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
      23.8
    5. Using strategy rm
      23.8
    6. Applied add-sqr-sqrt to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{red}{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}} - 1.0\right)}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{{\left(\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}^2} - 1.0\right)}{2.0}\]
      23.9
    7. Applied taylor to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left({\left(\sqrt{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right)}^2 - 1.0\right)}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\left(4.0 \cdot \frac{1}{{\alpha}^2} + 1\right) - 2.0 \cdot \frac{1}{\alpha}\right) - 1.0\right)}{2.0}\]
      23.6
    8. Taylor expanded around inf to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{red}{\left(\left(4.0 \cdot \frac{1}{{\alpha}^2} + 1\right) - 2.0 \cdot \frac{1}{\alpha}\right)} - 1.0\right)}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{\left(\left(4.0 \cdot \frac{1}{{\alpha}^2} + 1\right) - 2.0 \cdot \frac{1}{\alpha}\right)} - 1.0\right)}{2.0}\]
      23.6
    9. Applied simplify to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\left(\left(4.0 \cdot \frac{1}{{\alpha}^2} + 1\right) - 2.0 \cdot \frac{1}{\alpha}\right) - 1.0\right)}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + 2.0\right) + \beta} - \left(\frac{4.0}{\alpha \cdot \alpha} + \left(\left(1 - 1.0\right) - \frac{2.0}{\alpha}\right)\right)}{2.0}\]
      0.1
    10. Applied final simplification

    if -0.9999996f0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      0.6
    2. Using strategy rm
      0.6
    3. Applied div-sub to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]
      0.5
  1. Removed slow pow expressions

Original test:


(lambda ((alpha default) (beta default))
  #:name "Octave 3.8, jcobi/1"
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))