\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 1.4 m
Input Error: 52.8
Output Error: 30.1
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\frac{\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\right)}^2 & \text{when } \beta \le 3.3268046412700747 \cdot 10^{+165} \\ \frac{1}{{\left(\beta \cdot i\right)}^2} - \frac{6}{{i}^3 \cdot {\beta}^3} & \text{otherwise} \end{cases}\)

    if beta < 3.3268046412700747e+165

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      50.9
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      35.6
    3. Using strategy rm
      35.6
    4. Applied add-sqr-sqrt to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{blue}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}}\]
      35.6
    5. Applied add-sqr-sqrt to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}\]
      35.6
    6. Applied add-sqr-sqrt to get
      \[\frac{\frac{\color{red}{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\frac{\color{blue}{{\left(\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}^2}}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}\]
      35.6
    7. Applied square-undiv to get
      \[\frac{\color{red}{\frac{{\left(\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}\right)}^2}{{\left(\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}\right)}^2}}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\color{blue}{{\left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}\right)}^2}}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}\]
      35.6
    8. Applied square-undiv to get
      \[\color{red}{\frac{{\left(\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}\right)}^2}{{\left(\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\frac{\sqrt{i \cdot \left(\beta + \left(i + \alpha\right)\right)}}{\sqrt{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{\sqrt{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\right)}^2}\]
      35.6

    if 3.3268046412700747e+165 < beta

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      63.0
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      51.8
    3. Using strategy rm
      51.8
    4. Applied add-cbrt-cube to get
      \[\color{red}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^3}}\]
      51.8
    5. Applied taylor to get
      \[\sqrt[3]{{\left(\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}^3} \leadsto \frac{1}{{\beta}^2 \cdot {i}^2} - 6 \cdot \frac{1}{{\beta}^{3} \cdot {i}^{3}}\]
      0
    6. Taylor expanded around inf to get
      \[\color{red}{\frac{1}{{\beta}^2 \cdot {i}^2} - 6 \cdot \frac{1}{{\beta}^{3} \cdot {i}^{3}}} \leadsto \color{blue}{\frac{1}{{\beta}^2 \cdot {i}^2} - 6 \cdot \frac{1}{{\beta}^{3} \cdot {i}^{3}}}\]
      0
    7. Applied simplify to get
      \[\frac{1}{{\beta}^2 \cdot {i}^2} - 6 \cdot \frac{1}{{\beta}^{3} \cdot {i}^{3}} \leadsto \frac{1}{\left(i \cdot i\right) \cdot \left(\beta \cdot \beta\right)} - \frac{\frac{6}{{\beta}^3}}{{i}^3}\]
      0
    8. Applied final simplification
    9. Applied simplify to get
      \[\color{red}{\frac{1}{\left(i \cdot i\right) \cdot \left(\beta \cdot \beta\right)} - \frac{\frac{6}{{\beta}^3}}{{i}^3}} \leadsto \color{blue}{\frac{1}{{\left(\beta \cdot i\right)}^2} - \frac{6}{{i}^3 \cdot {\beta}^3}}\]
      0
  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)))