\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 1.5 m
Input Error: 52.8
Output Error: 36.9
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} & \text{when } \beta \le 4.205781944509689 \cdot 10^{+135} \\ \frac{\left(\frac{i}{\beta} \cdot \frac{1.0}{\beta} + i\right) \cdot i}{{\left(\left(\beta + \alpha\right) + i \cdot 2\right)}^2} & \text{otherwise} \end{cases}\)

    if beta < 4.205781944509689e+135

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      50.6
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      35.4

    if 4.205781944509689e+135 < beta

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      62.4
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      54.3
    3. Using strategy rm
      54.3
    4. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{blue}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}\]
      54.3
    5. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{red}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{blue}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      54.3
    6. Applied square-mult to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{blue}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      54.3
    7. Applied times-frac to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1} \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      54.3
    8. Applied times-frac to get
      \[\frac{\color{red}{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1} \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{\color{blue}{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      54.3
    9. Applied times-frac to get
      \[\color{red}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}} \leadsto \color{blue}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{1} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      54.3
    10. Applied simplify to get
      \[\color{red}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{1}} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \color{blue}{\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)}} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      54.3
    11. Using strategy rm
      54.3
    12. Applied *-un-lft-identity to get
      \[\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{blue}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}\]
      54.3
    13. Applied div-inv to get
      \[\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\color{red}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\color{blue}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      54.3
    14. Applied *-un-lft-identity to get
      \[\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{\color{red}{\beta + \left(i + \alpha\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{\color{blue}{1 \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      54.3
    15. Applied times-frac to get
      \[\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\color{red}{\frac{1 \cdot \left(\beta + \left(i + \alpha\right)\right)}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\color{blue}{\frac{1}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      62.2
    16. Applied times-frac to get
      \[\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \color{red}{\frac{\frac{1}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}} \leadsto \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \color{blue}{\left(\frac{\frac{1}{\left(\beta + \alpha\right) + 2 \cdot i}}{1} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}\]
      62.2
    17. Applied associate-*r* to get
      \[\color{red}{\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \left(\frac{\frac{1}{\left(\beta + \alpha\right) + 2 \cdot i}}{1} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)} \leadsto \color{blue}{\left(\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\left(\beta + \alpha\right) + 2 \cdot i}}{1}\right) \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      62.2
    18. Applied simplify to get
      \[\color{red}{\left(\frac{i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{\frac{1}{\left(\beta + \alpha\right) + 2 \cdot i}}{1}\right)} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \color{blue}{\frac{\frac{i}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i}} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      62.2
    19. Applied taylor to get
      \[\frac{\frac{i}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(1.0 \cdot \frac{i}{{\beta}^2} + i\right)\]
      12.6
    20. Taylor expanded around inf to get
      \[\frac{\frac{i}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \color{red}{\left(1.0 \cdot \frac{i}{{\beta}^2} + i\right)} \leadsto \frac{\frac{i}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \color{blue}{\left(1.0 \cdot \frac{i}{{\beta}^2} + i\right)}\]
      12.6
    21. Applied simplify to get
      \[\frac{\frac{i}{\left(\beta + \alpha\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(1.0 \cdot \frac{i}{{\beta}^2} + i\right) \leadsto \frac{\frac{1.0}{\beta} \cdot \frac{i}{\beta} + i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)}\]
      0.3
    22. Applied final simplification
    23. Applied simplify to get
      \[\color{red}{\frac{\frac{1.0}{\beta} \cdot \frac{i}{\beta} + i}{i \cdot 2 + \left(\alpha + \beta\right)} \cdot \frac{i}{i \cdot 2 + \left(\alpha + \beta\right)}} \leadsto \color{blue}{\frac{\left(\frac{i}{\beta} \cdot \frac{1.0}{\beta} + i\right) \cdot i}{{\left(\left(\beta + \alpha\right) + i \cdot 2\right)}^2}}\]
      43.2
  1. Removed slow pow expressions

Original test:


(lambda ((alpha default) (beta default) (i default))
  #:name "Octave 3.8, jcobi/4"
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))