\[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
Test:
Octave 3.8, jcobi/3
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Time: 40.3 s
Input Error: 3.2
Output Error: 0.5
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{\log \left(e^{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}\right)}{\left(\alpha + 1.0\right) + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)} & \text{when } \frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1} \le 1.0541412f0 \\ 0 & \text{otherwise} \end{cases}\)

    if (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) < 1.0541412f0

    1. Started with
      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
      0.2
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}} \leadsto \color{blue}{\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}}\]
      0.6
    3. Using strategy rm
      0.6
    4. Applied associate-/r* to get
      \[\color{red}{\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}} \leadsto \color{blue}{\frac{\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\alpha + 1.0\right) + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}}\]
      0.1
    5. Using strategy rm
      0.1
    6. Applied add-log-exp to get
      \[\frac{\frac{\color{red}{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}}{\left(\alpha + 1.0\right) + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)} \leadsto \frac{\frac{\color{blue}{\log \left(e^{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}\right)}}{\left(\alpha + 1.0\right) + \left(2 + \beta\right)}}{\alpha + \left(2 + \beta\right)}\]
      0.5

    if 1.0541412f0 < (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1)))

    1. Started with
      \[\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
      24.7
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\frac{\left(\left(\alpha + \beta\right) + \beta \cdot \alpha\right) + 1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}} \leadsto \color{blue}{\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}}\]
      22.2
    3. Applied taylor to get
      \[\frac{\frac{\left(\alpha + 1.0\right) + (\beta * \alpha + \beta)_*}{\alpha + \left(2 + \beta\right)}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)} \leadsto \frac{0}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\]
      0
    4. Taylor expanded around inf to get
      \[\frac{\color{red}{0}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)} \leadsto \frac{\color{blue}{0}}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}\]
      0
    5. Applied simplify to get
      \[\color{red}{\frac{0}{\left(\left(\alpha + 1.0\right) + \left(2 + \beta\right)\right) \cdot \left(\alpha + \left(2 + \beta\right)\right)}} \leadsto \color{blue}{0}\]
      0
  1. Removed slow pow expressions

Original test:


(lambda ((alpha default) (beta default))
  #:name "Octave 3.8, jcobi/3"
  (/ (/ (/ (+ (+ (+ alpha beta) (* beta alpha)) 1.0) (+ (+ alpha beta) (* 2 1))) (+ (+ alpha beta) (* 2 1))) (+ (+ (+ alpha beta) (* 2 1)) 1.0)))