\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 47.6 s
Input Error: 11.1
Output Error: 1.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\left(\frac{8.0}{{\alpha}^3} - \frac{\frac{4.0}{\alpha}}{\alpha}\right) + \frac{2.0}{\alpha}}{2.0} & \text{when } \frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \le -2.109379f+16 \\ \frac{\frac{\frac{\alpha + \beta}{{\left({\left(\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)}^3\right)}^{1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} & \text{otherwise} \end{cases}\)

    if (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) < -2.109379f+16

    1. Started with
      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      31.0
    2. Applied taylor to get
      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}{2.0}\]
      1.2
    3. Taylor expanded around inf to get
      \[\frac{\color{red}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}}{2.0} \leadsto \frac{\color{blue}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}}{2.0}\]
      1.2
    4. Applied simplify to get
      \[\color{red}{\frac{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}{2.0}} \leadsto \color{blue}{\frac{\frac{2.0}{\alpha} + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}}\]
      1.2
    5. Applied simplify to get
      \[\frac{\color{red}{\frac{2.0}{\alpha} + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0} \leadsto \frac{\color{blue}{\left(\frac{8.0}{{\alpha}^3} - \frac{\frac{4.0}{\alpha}}{\alpha}\right) + \frac{2.0}{\alpha}}}{2.0}\]
      1.2

    if -2.109379f+16 < (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i)))

    1. Started with
      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      7.0
    2. Using strategy rm
      7.0
    3. Applied associate-/l* to get
      \[\frac{\frac{\color{red}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      1.4
    4. Using strategy rm
      1.4
    5. Applied pow1 to get
      \[\frac{\frac{\frac{\alpha + \beta}{\color{red}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{\color{blue}{{\left(\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}\right)}^{1}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      1.2
    6. Using strategy rm
      1.2
    7. Applied add-cube-cbrt to get
      \[\frac{\frac{\frac{\alpha + \beta}{{\color{red}{\left(\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}\right)}}^{1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{{\color{blue}{\left({\left(\sqrt[3]{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}\right)}^3\right)}}^{1}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      1.2
  1. Removed slow pow expressions

Original test:


(lambda ((alpha default) (beta default) (i default))
  #:name "Octave 3.8, jcobi/2"
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))