\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
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: 25.5
Output Error: 4.2
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{i}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1} \cdot \left(\frac{\left(\beta + i\right) + \alpha}{{\left(2 \cdot i + \left(\beta + \alpha\right)\right)}^2 - 1.0} \cdot \left(i \cdot \left(\left(\beta + i\right) + \alpha\right) + \alpha \cdot \beta\right)\right) & \text{when } \frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \le 4.2838985f+28 \\ \frac{1}{16} \cdot e^{\frac{\frac{0.25}{i}}{i}} & \text{otherwise} \end{cases}\)

    if (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) < 4.2838985f+28

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      0.5
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      0.4
    3. Using strategy rm
      0.4
    4. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{red}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{\color{blue}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}}\]
      0.4
    5. Applied div-inv to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 \cdot \frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      0.5
    6. Applied times-frac to get
      \[\frac{\color{red}{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 \cdot \frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)} \leadsto \frac{\color{blue}{\frac{i}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}\]
      0.7
    7. Applied times-frac to get
      \[\color{red}{\frac{\frac{i}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{1 \cdot \left({\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0\right)}} \leadsto \color{blue}{\frac{\frac{i}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1} \cdot \frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      0.7
    8. Applied simplify to get
      \[\frac{\frac{i}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1} \cdot \color{red}{\frac{\frac{\beta + \left(i + \alpha\right)}{\frac{1}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \frac{\frac{i}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1} \cdot \color{blue}{\left(\frac{\left(\beta + i\right) + \alpha}{{\left(2 \cdot i + \left(\beta + \alpha\right)\right)}^2 - 1.0} \cdot \left(i \cdot \left(\left(\beta + i\right) + \alpha\right) + \alpha \cdot \beta\right)\right)}\]
      0.7

    if 4.2838985f+28 < (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      30.2
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      22.9
    3. Using strategy rm
      22.9
    4. Applied add-exp-log to get
      \[\color{red}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \color{blue}{e^{\log \left(\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)}}\]
      22.9
    5. Applied taylor to get
      \[e^{\log \left(\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\right)} \leadsto e^{0.25 \cdot \frac{1}{{i}^2} + \log \frac{1}{16}}\]
      4.9
    6. Taylor expanded around inf to get
      \[e^{\color{red}{0.25 \cdot \frac{1}{{i}^2} + \log \frac{1}{16}}} \leadsto e^{\color{blue}{0.25 \cdot \frac{1}{{i}^2} + \log \frac{1}{16}}}\]
      4.9
    7. Applied simplify to get
      \[e^{0.25 \cdot \frac{1}{{i}^2} + \log \frac{1}{16}} \leadsto \frac{1}{16} \cdot e^{\frac{\frac{0.25}{i}}{i}}\]
      4.9
    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)))