\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 17.9 s
Input Error: 6.6
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{\frac{\beta}{2.0}}{\left(2.0 + \alpha\right) + \beta} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right) + \frac{\frac{8.0}{{\alpha}^3} + \frac{2.0}{\alpha}}{2.0} & \text{when } \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.999999f0 \\ \frac{\frac{\beta - \alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} + 1.0}{2.0} & \text{otherwise} \end{cases}\)

    if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.999999f0

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      27.9
    2. Using strategy rm
      27.9
    3. Applied pow1 to get
      \[\frac{\frac{\beta - \alpha}{\color{red}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}} + 1.0}{2.0}\]
      27.3
    4. Using strategy rm
      27.3
    5. Applied div-sub to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}\right)} + 1.0}{2.0}\]
      25.6
    6. Applied associate-+l- to get
      \[\frac{\color{red}{\left(\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \left(\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0\right)}}{2.0}\]
      23.0
    7. Applied simplify to get
      \[\frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \color{red}{\left(\frac{\alpha}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - 1.0\right)}}{2.0} \leadsto \frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \color{blue}{\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
      23.7
    8. Applied taylor to get
      \[\frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0} \leadsto \frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\]
      0.1
    9. Taylor expanded around inf to get
      \[\frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \color{red}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0} \leadsto \frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0}\]
      0.1
    10. Applied simplify to get
      \[\frac{\frac{\beta}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}} - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \leadsto \left(\frac{\frac{\beta}{2.0}}{\left(2.0 + \alpha\right) + \beta} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right) + \frac{\frac{8.0}{{\alpha}^3} + \frac{2.0}{\alpha}}{2.0}\]
      0.1
    11. Applied final simplification

    if -0.999999f0 < (/ (- 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 pow1 to get
      \[\frac{\frac{\beta - \alpha}{\color{red}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{\color{blue}{{\left(\left(\alpha + \beta\right) + 2.0\right)}^{1}}} + 1.0}{2.0}\]
      0.3
  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))