\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 1.7 m
Input Error: 52.9
Output Error: 30.4
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{{\left(\sqrt{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} & \text{when } \beta \le 7.181513688320944 \cdot 10^{+151} \\ (\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\beta \cdot \beta}\right) * \left(\frac{i}{\beta}\right) + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\frac{{\beta}^{4}}{i}} \cdot \left(\alpha + i\right)\right))_* & \text{otherwise} \end{cases}\)

    if beta < 7.181513688320944e+151

    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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      35.4
    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 7.181513688320944e+151 < 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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      52.0
    3. Applied taylor to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}}\right)\]
      39.8
    4. Taylor expanded around -inf to get
      \[\color{red}{\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}}\right)} \leadsto \color{blue}{\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}}\right)}\]
      39.8
    5. Applied simplify to get
      \[\color{red}{\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}} + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{\alpha} + \frac{1}{i}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \frac{(\left(\frac{-1}{i}\right) * \left(-\left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}}\right)} \leadsto \color{blue}{(\left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\beta \cdot \beta}\right) * \left(\frac{i}{\beta}\right) + \left(\frac{(\left(\frac{-1}{i}\right) * \left(-\left(\left(\frac{1}{\beta} + \frac{1}{\alpha}\right) + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\frac{{\beta}^{4}}{i}} \cdot \left(\alpha + i\right)\right))_*}\]
      6.8
  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)))