\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 2.0 m
Input Error: 53.1
Output Error: 28.6
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\sqrt{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} & \text{when } \alpha \le 2.069742248392879 \cdot 10^{+108} \\ \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right) \cdot \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right)} \cdot \frac{\frac{\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)}{i}}{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) + \left(\frac{\frac{1}{\beta}}{\beta}\right))_* - (\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\beta}\right) * 2 + 1.0)_*} & \text{otherwise} \end{cases}\)

    if alpha < 2.069742248392879e+108

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      50.5
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      35.4
    3. Using strategy rm
      35.4
    4. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{\color{blue}{{\left(\sqrt{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\]
      35.4

    if 2.069742248392879e+108 < 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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      56.8
    3. Using strategy rm
      56.8
    4. Applied add-cbrt-cube to get
      \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \color{blue}{\sqrt[3]{{\left(\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^3}}\]
      55.6
    5. Applied taylor to get
      \[\sqrt[3]{{\left(\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^3} \leadsto \sqrt[3]{{\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)}{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right)}^2 \cdot \left(i \cdot \left(\left(\frac{1}{{\beta}^2} + {\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2\right) - \left(2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\beta} + 1.0\right)\right)\right)}\right)}^3}\]
      26.4
    6. Taylor expanded around -inf to get
      \[\sqrt[3]{\color{red}{{\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)}{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right)}^2 \cdot \left(i \cdot \left(\left(\frac{1}{{\beta}^2} + {\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2\right) - \left(2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\beta} + 1.0\right)\right)\right)}\right)}^3}} \leadsto \sqrt[3]{\color{blue}{{\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)}{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right)}^2 \cdot \left(i \cdot \left(\left(\frac{1}{{\beta}^2} + {\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2\right) - \left(2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\beta} + 1.0\right)\right)\right)}\right)}^3}}\]
      26.4
    7. Applied simplify to get
      \[\sqrt[3]{{\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)}{{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right)}^2 \cdot \left(i \cdot \left(\left(\frac{1}{{\beta}^2} + {\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right)}^2\right) - \left(2 \cdot \frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\beta} + 1.0\right)\right)\right)}\right)}^3} \leadsto \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\alpha \cdot \beta}\right))_*}{\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right) \cdot \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_* - \frac{1}{\beta}\right)} \cdot \frac{\frac{\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)}{i}}{(\left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) * \left((\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*\right) + \left(\frac{\frac{1}{\beta}}{\beta}\right))_* - (\left(\frac{(\left(\frac{-1}{i}\right) * 2 + \left(\frac{-1}{\alpha}\right))_*}{\beta}\right) * 2 + 1.0)_*}\]
      3.0
    8. 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)))