\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
Test:
Octave 3.8, jcobi/2
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 49.6 s
Input Error: 11.1
Output Error: 0.7
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\left(\frac{8.0}{{\alpha}^3} - \frac{\frac{4.0}{\alpha}}{\alpha}\right) + \frac{2.0}{\alpha}}{2.0} & \text{when } \frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \le -7.8873644f+11 \\ \frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{{\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}^3}} + 1.0}{2.0} & \text{otherwise} \end{cases}\)

    if (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) < -7.8873644f+11

    1. Started with
      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      30.7
    2. Applied taylor to get
      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}{2.0}\]
      0.4
    3. Taylor expanded around inf to get
      \[\frac{\color{red}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}}{2.0} \leadsto \frac{\color{blue}{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}}{2.0}\]
      0.4
    4. Applied simplify to get
      \[\color{red}{\frac{\left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right) - 4.0 \cdot \frac{1}{{\alpha}^2}}{2.0}} \leadsto \color{blue}{\frac{\frac{2.0}{\alpha} + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}}\]
      0.4
    5. Applied simplify to get
      \[\frac{\color{red}{\frac{2.0}{\alpha} + \left(\frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0} \leadsto \frac{\color{blue}{\left(\frac{8.0}{{\alpha}^3} - \frac{\frac{4.0}{\alpha}}{\alpha}\right) + \frac{2.0}{\alpha}}}{2.0}\]
      0.4

    if -7.8873644f+11 < (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i)))

    1. Started with
      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      6.5
    2. Using strategy rm
      6.5
    3. Applied *-un-lft-identity to get
      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{red}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      6.5
    4. Applied times-frac to get
      \[\frac{\frac{\color{red}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0} \leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
      0.9
    5. Applied associate-/l* to get
      \[\frac{\color{red}{\frac{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0} \leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}} + 1.0}{2.0}\]
      0.9
    6. Using strategy rm
      0.9
    7. Applied add-cube-cbrt to get
      \[\frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\frac{\beta - \alpha}{\color{red}{\left(\alpha + \beta\right) + 2 \cdot i}}}} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\frac{\beta - \alpha}{\color{blue}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}}} + 1.0}{2.0}\]
      1.0
    8. Applied add-cube-cbrt to get
      \[\frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\frac{\color{red}{\beta - \alpha}}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\frac{\color{blue}{{\left(\sqrt[3]{\beta - \alpha}\right)}^3}}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}} + 1.0}{2.0}\]
      0.7
    9. Applied cube-undiv to get
      \[\frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\color{red}{\frac{{\left(\sqrt[3]{\beta - \alpha}\right)}^3}{{\left(\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}\right)}^3}}}} + 1.0}{2.0} \leadsto \frac{\frac{\frac{\alpha + \beta}{1}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\color{blue}{{\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}^3}}} + 1.0}{2.0}\]
      0.7
  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))