\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/1
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 20.2 s
Input Error: 6.9
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \left(\frac{\frac{\beta}{2.0}}{\left(\alpha + 2.0\right) + \beta} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right) + \frac{\frac{2.0}{\alpha} + \frac{8.0}{{\alpha}^3}}{2.0} & \text{when } \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.9999955f0 \\ (\left(\frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*}\right) * 0.5 + 0.5)_* - \frac{0.5 \cdot \alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*} & \text{otherwise} \end{cases}\)

    if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.9999955f0

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      27.7
    2. Using strategy rm
      27.7
    3. Applied div-sub to get
      \[\frac{\color{red}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]
      27.7
    4. Applied associate-+l- to get
      \[\frac{\color{red}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}}{2.0} \leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
      24.0
    5. Applied taylor to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\]
      0.1
    6. Taylor expanded around inf to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{red}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0} \leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0}\]
      0.1
    7. Applied simplify to get
      \[\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(4.0 \cdot \frac{1}{{\alpha}^2} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0} \leadsto \left(\frac{\frac{\beta}{2.0}}{\left(\alpha + 2.0\right) + \beta} - \frac{\frac{4.0}{\alpha \cdot \alpha}}{2.0}\right) + \frac{\frac{2.0}{\alpha} + \frac{8.0}{{\alpha}^3}}{2.0}\]
      0.1
    8. Applied final simplification

    if -0.9999955f0 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Started with
      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
      0.5
    2. Using strategy rm
      0.5
    3. Applied *-un-lft-identity to get
      \[\frac{\frac{\beta - \alpha}{\color{red}{\left(\alpha + \beta\right)} + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\alpha + \beta\right)} + 2.0} + 1.0}{2.0}\]
      0.5
    4. Applied fma-def to get
      \[\frac{\frac{\beta - \alpha}{\color{red}{1 \cdot \left(\alpha + \beta\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\frac{\beta - \alpha}{\color{blue}{(1 * \left(\alpha + \beta\right) + 2.0)_*}} + 1.0}{2.0}\]
      0.2
    5. Applied taylor to get
      \[\frac{\frac{\beta - \alpha}{(1 * \left(\alpha + \beta\right) + 2.0)_*} + 1.0}{2.0} \leadsto \left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}\]
      0.2
    6. Taylor expanded around 0 to get
      \[\color{red}{\left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}} \leadsto \color{blue}{\left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}}\]
      0.2
    7. Applied simplify to get
      \[\color{red}{\left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}} \leadsto \color{blue}{\frac{0.5}{(1 * \left(\beta + \alpha\right) + 2.0)_*} \cdot \left(\beta - \alpha\right) + 0.5}\]
      0.2
    8. Applied taylor to get
      \[\frac{0.5}{(1 * \left(\beta + \alpha\right) + 2.0)_*} \cdot \left(\beta - \alpha\right) + 0.5 \leadsto \left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}\]
      0.2
    9. Taylor expanded around 0 to get
      \[\color{red}{\left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}} \leadsto \color{blue}{\left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}}\]
      0.2
    10. Applied simplify to get
      \[\left(0.5 \cdot \frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*} + 0.5\right) - 0.5 \cdot \frac{\alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*} \leadsto (\left(\frac{\beta}{(1 * \left(\beta + \alpha\right) + 2.0)_*}\right) * 0.5 + 0.5)_* - \frac{0.5 \cdot \alpha}{(1 * \left(\beta + \alpha\right) + 2.0)_*}\]
      0.3
    11. Applied final simplification
  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))