\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 3.5 m
Input Error: 52.5
Output Error: 34.5
Log: ⚲
Profile: 🕒
\(\begin{cases} e^{\log \left(\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)} & \text{when } \alpha \le 2.6756344892722176 \cdot 10^{+155} \\ (\left(\frac{{\beta}^3}{i}\right) * \left((\left(\frac{1}{i}\right) * \left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_*\right) + \left(\frac{(\left(\frac{1}{i}\right) * \left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_*}{\frac{\beta}{\frac{{\beta}^3}{i}}} \cdot \left(\frac{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_* \cdot 10}{\frac{\beta}{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*}} + \frac{1.0}{\beta}\right)\right))_* - \frac{(\left(\frac{1}{i}\right) * \left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_* \cdot \left(\left((\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_* \cdot {\beta}^3\right) \cdot 4\right)}{\beta \cdot i} & \text{otherwise} \end{cases}\)

    if alpha < 2.6756344892722176e+155

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      50.7
    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-exp-log to get
      \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \color{blue}{e^{\log \left(\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}}\]
      35.8

    if 2.6756344892722176e+155 < alpha

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      62.5
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}}\]
      62.3
    3. Using strategy rm
      62.3
    4. Applied add-exp-log to get
      \[\color{red}{\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}} \leadsto \color{blue}{e^{\log \left(\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)}}\]
      62.3
    5. Applied taylor to get
      \[e^{\log \left(\frac{\left(\frac{\left(\beta + \alpha\right) + i}{\beta + (i * 2 + \alpha)_*} \cdot (i * \left(\left(\beta + \alpha\right) + i\right) + \left(\alpha \cdot \beta\right))_*\right) \cdot \frac{i}{\beta + (i * 2 + \alpha)_*}}{{\left(\beta + (i * 2 + \alpha)_*\right)}^2 - 1.0}\right)} \leadsto \left(1.0 \cdot \frac{e^{\left(\log \left((\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*\right) + 3 \cdot \log \beta\right) - \log i}}{{\beta}^2} + \left(10 \cdot \frac{e^{\left(\log \left((\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot {\left((\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*\right)}^2}{{\beta}^2} + e^{\left(\log \left((\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*\right) + 3 \cdot \log \beta\right) - \log i}\right)\right) - 4 \cdot \frac{e^{\left(\log \left((\left(\frac{1}{i}\right) * \left(\frac{1}{\beta} + \left(\frac{1}{i} + \frac{1}{\alpha}\right)\right) + \left(\frac{1}{\beta \cdot \alpha}\right))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot (\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*}{\beta}\]
      43.3
    6. Taylor expanded around inf to get
      \[\color{red}{\left(1.0 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i}}{{\beta}^2} + \left(10 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot {\left((\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*\right)}^2}{{\beta}^2} + e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i}\right)\right) - 4 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot (\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*}{\beta}} \leadsto \color{blue}{\left(1.0 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i}}{{\beta}^2} + \left(10 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot {\left((\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*\right)}^2}{{\beta}^2} + e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i}\right)\right) - 4 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot (\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*}{\beta}}\]
      43.3
    7. Applied simplify to get
      \[\color{red}{\left(1.0 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i}}{{\beta}^2} + \left(10 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot {\left((\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*\right)}^2}{{\beta}^2} + e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i}\right)\right) - 4 \cdot \frac{e^{\left(\log \left((\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))_*\right) + 3 \cdot \log \beta\right) - \log i} \cdot (\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*}{\beta}} \leadsto \color{blue}{(\left(\frac{{\beta}^3}{i}\right) * \left((\left(\frac{1}{i}\right) * \left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_*\right) + \left(\frac{(\left(\frac{1}{i}\right) * \left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_*}{\frac{\beta}{\frac{{\beta}^3}{i}}} \cdot \left(\frac{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_* \cdot 10}{\frac{\beta}{(\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_*}} + \frac{1.0}{\beta}\right)\right))_* - \frac{(\left(\frac{1}{i}\right) * \left(\left(\frac{1}{i} + \frac{1}{\alpha}\right) + \frac{1}{\beta}\right) + \left(\frac{\frac{1}{\beta}}{\alpha}\right))_* \cdot \left(\left((\left(\frac{1}{i}\right) * 2 + \left(\frac{1}{\alpha}\right))_* \cdot {\beta}^3\right) \cdot 4\right)}{\beta \cdot i}}\]
      27.7
  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)))