\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 22.8 s
Input Error: 6.8
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{\frac{\beta}{2.0}}{\left(\beta + \alpha\right) + 2.0} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right) + \frac{\frac{2.0}{\alpha} + \frac{8.0}{{\alpha}^3}}{2.0} & \text{when } \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9960165f0 \\ \frac{e^{\log \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0\right)}}{2.0} & \text{otherwise} \end{cases}\)

    if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9960165f0

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      26.4
    2. Using strategy rm
      26.4
    3. Applied div-sub to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]
      26.4
    4. Applied associate-+l- to get
      \[\frac{\color{red}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
      22.9
    5. Using strategy rm
      22.9
    6. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0} \leadsto \frac{\color{blue}{{\left(\sqrt{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
      24.2
    7. Applied taylor to get
      \[\frac{{\left(\sqrt{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^2 - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0} \leadsto \frac{{\left(\sqrt{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^2 - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\]
      8.0
    8. Taylor expanded around inf to get
      \[\frac{{\left(\sqrt{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^2 - \color{red}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0} \leadsto \frac{{\left(\sqrt{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^2 - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0}\]
      8.0
    9. Applied simplify to get
      \[\frac{{\left(\sqrt{\frac{\beta}{\left(\alpha + \beta\right) + 2.0}}\right)}^2 - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \leadsto \left(\frac{\frac{\beta}{2.0}}{\left(\beta + \alpha\right) + 2.0} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right) + \frac{\frac{2.0}{\alpha} + \frac{8.0}{{\alpha}^3}}{2.0}\]
      0.1
    10. Applied final simplification

    if -0.9960165f0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      0.1
    2. Using strategy rm
      0.1
    3. Applied add-exp-log to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}}{2.0} \leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0\right)}}}{2.0}\]
      0.1
  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))