\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 2.1 m
Input Error: 52.9
Output Error: 9.7
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\frac{\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\right)}^2 & \text{when } \frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \le 1.8136140911964688 \cdot 10^{+258} \\ \frac{\frac{i}{8}}{i \cdot 2 + \left(\alpha + \beta\right)} & \text{otherwise} \end{cases}\)

    if (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) < 1.8136140911964688e+258

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      0.3
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      0.2
    3. Using strategy rm
      0.2
    4. Applied add-sqr-sqrt to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{blue}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}}\]
      0.2
    5. Applied add-sqr-sqrt to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}\]
      0.2
    6. Applied add-sqr-sqrt to get
      \[\frac{\frac{\color{red}{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\frac{\color{blue}{{\left(\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}^2}}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}\]
      0.3
    7. Applied square-undiv to get
      \[\frac{\color{red}{\frac{{\left(\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}^2}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\color{blue}{{\left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}\right)}^2}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}\]
      0.2
    8. Applied square-undiv to get
      \[\color{red}{\frac{{\left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}\right)}^2}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\right)}^2}\]
      0.3

    if 1.8136140911964688e+258 < (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      62.2
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      45.8
    3. Using strategy rm
      45.8
    4. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{red}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{blue}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      45.8
    5. Applied square-mult to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{blue}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      45.8
    6. Applied times-frac to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{\left(\left(\beta + \alpha\right) + 2 \cdot i\right) \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1} \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      45.8
    7. Applied times-frac to get
      \[\frac{\color{red}{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1} \cdot \frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\color{blue}{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      45.8
    8. Applied associate-/l* to get
      \[\color{red}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}}\]
      45.8
    9. Applied taylor to get
      \[\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}} \leadsto \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{8}\]
      11.3
    10. Taylor expanded around 0 to get
      \[\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\color{red}{8}} \leadsto \frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{\color{blue}{8}}\]
      11.3
    11. Applied simplify to get
      \[\color{red}{\frac{\frac{i}{\frac{\left(\beta + \alpha\right) + 2 \cdot i}{1}}}{8}} \leadsto \color{blue}{\frac{\frac{i}{8}}{i \cdot 2 + \left(\alpha + \beta\right)}}\]
      11.3
  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)))