\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 1.2 m
Input Error: 52.8
Output Error: 30.8
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} & \text{when } \alpha \le 2.303838227591108 \cdot 10^{+161} \\ \frac{\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} & \text{otherwise} \end{cases}\)

    if alpha < 2.303838227591108e+161

    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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      36.1
    3. Using strategy rm
      36.1
    4. Applied *-un-lft-identity to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{1 \cdot \left({\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0\right)}}\]
      36.1
    5. Applied associate-/r* to get
      \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{1 \cdot \left({\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0\right)}} \leadsto \color{blue}{\frac{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{1}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      36.1
    6. Applied simplify to get
      \[\frac{\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{1}}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\color{blue}{\frac{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\]
      36.1

    if 2.303838227591108e+161 < alpha

    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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      52.3
    3. Using strategy rm
      52.3
    4. Applied *-un-lft-identity to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{1 \cdot \left({\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0\right)}}\]
      52.3
    5. Applied associate-/r* to get
      \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{1 \cdot \left({\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0\right)}} \leadsto \color{blue}{\frac{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{1}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      52.3
    6. Applied simplify to get
      \[\frac{\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{1}}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\color{blue}{\frac{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\]
      52.3
    7. Applied taylor to get
      \[\frac{\frac{(i * \left(\left(\alpha + i\right) + \beta\right) + \left(\alpha \cdot \beta\right))_*}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\]
      1.6
    8. Taylor expanded around -inf to get
      \[\frac{\frac{\color{red}{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\frac{\color{blue}{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}}{\frac{(i * 2 + \alpha)_* + \beta}{\left(\alpha + i\right) + \beta}} \cdot \frac{i}{(i * 2 + \alpha)_* + \beta}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\]
      1.6
  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)))