\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 26.6 s
Input Error: 16.2
Output Error: 0.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{2.0 + \frac{8.0}{\alpha \cdot \alpha}}{2.0 \cdot \alpha} + \left(\frac{\frac{\beta}{2.0}}{\left(\alpha + 2.0\right) + \beta} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right) & \text{when } \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9996956325017868 \\ \frac{\frac{{\left(\frac{\beta}{\left(\alpha + 2.0\right) + \beta}\right)}^3 - {\left(\frac{\alpha}{\left(\alpha + 2.0\right) + \beta} - 1.0\right)}^3}{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^2 + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0} & \text{otherwise} \end{cases}\)

    if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9996956325017868

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      59.6
    2. Using strategy rm
      59.6
    3. Applied div-sub to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]
      59.6
    4. Applied associate-+l- to get
      \[\frac{\color{red}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
      48.2
    5. Applied taylor to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\]
      0.0
    6. Taylor expanded around inf to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{red}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0}\]
      0.0
    7. Applied simplify to get
      \[\color{red}{\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}} \leadsto \color{blue}{\frac{2.0 + \frac{8.0}{\alpha \cdot \alpha}}{2.0 \cdot \alpha} + \left(\frac{\frac{\beta}{2.0}}{\left(\alpha + 2.0\right) + \beta} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right)}\]
      0.1

    if -0.9996956325017868 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      0.0
    2. Using strategy rm
      0.0
    3. Applied div-sub to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]
      0.0
    4. Applied associate-+l- to get
      \[\frac{\color{red}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
      0.0
    5. Using strategy rm
      0.0
    6. Applied flip3-- to get
      \[\frac{\color{red}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0} \leadsto \frac{\color{blue}{\frac{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}}{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^2 + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}}{2.0}\]
      0.0
    7. Applied simplify to get
      \[\frac{\frac{\color{red}{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}}}{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^2 + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0} \leadsto \frac{\frac{\color{blue}{{\left(\frac{\beta}{\left(\alpha + 2.0\right) + \beta}\right)}^3 - {\left(\frac{\alpha}{\left(\alpha + 2.0\right) + \beta} - 1.0\right)}^3}}{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^2 + \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^2 + \frac{\beta}{\left(\alpha + \beta\right) + 2.0} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}{2.0}\]
      0.0
  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))