\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
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: 53.1
Output Error: 28.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((i * 2 + \alpha)_*\right) * \left((\beta * 2 + \left((i * 2 + \alpha)_*\right))_*\right) + \left({\beta}^2 - 1.0\right))_*}{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}} & \text{when } \beta \le 1.8414905412296409 \cdot 10^{+137} \\ (\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right)\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_* \cdot \frac{\frac{i}{\beta + (i * 2 + \alpha)_*} \cdot \frac{\left(\beta + i\right) + \alpha}{\beta + (i * 2 + \alpha)_*}}{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{\beta}\right) * 2 + \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right))_*\right) + \left(\frac{1}{\beta \cdot \beta} - 1.0\right))_*} & \text{otherwise} \end{cases}\)

    if beta < 1.8414905412296409e+137

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      50.8
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      35.0
    3. Applied taylor to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}\]
      35.0
    4. Taylor expanded around 0 to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0}\]
      35.0
    5. Applied simplify to get
      \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((i * 2 + \alpha)_*\right) * \left((\beta * 2 + \left((i * 2 + \alpha)_*\right))_*\right) + \left({\beta}^2 - 1.0\right))_*}{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}}}\]
      35.0

    if 1.8414905412296409e+137 < 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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      56.3
    3. Applied taylor to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}\]
      56.3
    4. Taylor expanded around 0 to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right)} - 1.0}\]
      56.3
    5. Applied simplify to get
      \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\left(2 \cdot \left(\beta \cdot (i * 2 + \alpha)_*\right) + \left({\left((i * 2 + \alpha)_*\right)}^2 + {\beta}^2\right)\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((i * 2 + \alpha)_*\right) * \left((\beta * 2 + \left((i * 2 + \alpha)_*\right))_*\right) + \left({\beta}^2 - 1.0\right))_*}{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}}}\]
      56.3
    6. Applied taylor to get
      \[\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((i * 2 + \alpha)_*\right) * \left((\beta * 2 + \left((i * 2 + \alpha)_*\right))_*\right) + \left({\beta}^2 - 1.0\right))_*}{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}} \leadsto \frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{\beta}\right) * 2 + \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right))_*\right) + \left(\frac{1}{{\beta}^2} - 1.0\right))_*}{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}}\]
      0.5
    7. Taylor expanded around -inf to get
      \[\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\color{red}{\frac{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{\beta}\right) * 2 + \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right))_*\right) + \left(\frac{1}{{\beta}^2} - 1.0\right))_*}{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}}} \leadsto \frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\color{blue}{\frac{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{\beta}\right) * 2 + \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right))_*\right) + \left(\frac{1}{{\beta}^2} - 1.0\right))_*}{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}}}\]
      0.5
    8. Applied simplify to get
      \[\frac{\frac{i}{(i * 2 + \alpha)_* + \beta} \cdot \frac{\left(\alpha + i\right) + \beta}{(i * 2 + \alpha)_* + \beta}}{\frac{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{\beta}\right) * 2 + \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right))_*\right) + \left(\frac{1}{{\beta}^2} - 1.0\right))_*}{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}} \leadsto (\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right)\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_* \cdot \frac{\frac{i}{\beta + (i * 2 + \alpha)_*} \cdot \frac{\left(\beta + i\right) + \alpha}{\beta + (i * 2 + \alpha)_*}}{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{\beta}\right) * 2 + \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right))_*\right) + \left(\frac{1}{\beta \cdot \beta} - 1.0\right))_*}\]
      1.5
    9. Applied final simplification
  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)))