\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 1.6 m
Input Error: 52.8
Output Error: 31.1
Log: ⚲
Profile: 🕒
\(\begin{cases} {\left(\frac{\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}}{\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\right)}^2 & \text{when } \beta \le 3.3268046412700747 \cdot 10^{+165} \\ (\left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\alpha} + \left(\frac{1}{\beta} + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\beta}\right) * \left(\frac{i}{\beta \cdot \beta}\right) + \left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\alpha} + \left(\frac{1}{\beta} + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\frac{{\beta}^{4}}{i}} \cdot \left(i + \alpha\right)\right))_* & \text{otherwise} \end{cases}\)

    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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      35.5
    3. Using strategy rm
      35.5
    4. Applied add-sqr-sqrt to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{red}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{\color{blue}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}}\]
      35.5
    5. Applied add-sqr-sqrt to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \color{red}{\frac{i}{\beta + (i * 2 + \alpha)_*}}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \color{blue}{{\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}\]
      35.9
    6. Applied add-sqr-sqrt to get
      \[\frac{\color{red}{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right)} \cdot {\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*}\right)}^2} \cdot {\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}\]
      35.9
    7. Applied square-unprod to get
      \[\frac{\color{red}{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*}\right)}^2 \cdot {\left(\sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2} \leadsto \frac{\color{blue}{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}\]
      35.7
    8. Applied square-undiv to get
      \[\color{red}{\frac{{\left(\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}\right)}^2}{{\left(\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}^2}} \leadsto \color{blue}{{\left(\frac{\sqrt{\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*} \cdot \sqrt{\frac{i}{\beta + (i * 2 + \alpha)_*}}}{\sqrt{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\right)}^2}\]
      35.8

    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
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      52.9
    3. Applied taylor to get
      \[\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0} \leadsto \frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}} + \frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}}\right)\]
      39.9
    4. Taylor expanded around inf to get
      \[\color{red}{\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}} + \frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}}\right)} \leadsto \color{blue}{\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}} + \frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}}\right)}\]
      39.9
    5. Applied simplify to get
      \[\color{red}{\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot i}{{\beta}^{3}} + \left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot \left(\alpha \cdot i\right)}{{\beta}^{4}} + \frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_* \cdot {i}^2}{{\beta}^{4}}\right)} \leadsto \color{blue}{(\left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\alpha} + \left(\frac{1}{\beta} + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\beta}\right) * \left(\frac{i}{\beta \cdot \beta}\right) + \left(\frac{(\left(\frac{1}{i}\right) * \left(\frac{1}{\alpha} + \left(\frac{1}{\beta} + \frac{1}{i}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*}{\frac{{\beta}^{4}}{i}} \cdot \left(i + \alpha\right)\right))_*}\]
      4.9
  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)))