\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
Test:
Octave 3.8, jcobi/4
Bits:
128 bits
Bits error versus alpha
Bits error versus beta
Bits error versus i
Time: 5.7 m
Input Error: 52.7
Output Error: 8.1
Log: ⚲
Profile: 🕒
\(\begin{cases} \frac{\frac{i}{\frac{{1}^2}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}} & \text{when } \frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \le 3.1746031384160272 \cdot 10^{+218} \\ \frac{1}{16} & \text{otherwise} \end{cases}\)

    if (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) < 3.1746031384160272e+218

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      0.3
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      0.2
    3. Using strategy rm
      0.2
    4. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{red}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{blue}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      0.2
    5. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\color{red}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}^2}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\color{blue}{\left(1 \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right)}}^2}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      0.2
    6. Applied square-prod to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{red}{{\left(1 \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right)}^2}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{blue}{{1}^2 \cdot {\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      0.2
    7. Applied times-frac to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{{1}^2 \cdot {\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{\frac{{1}^2}{1} \cdot \frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      0.2
    8. Applied times-frac to get
      \[\frac{\color{red}{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{1}^2}{1} \cdot \frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\color{blue}{\frac{i}{\frac{{1}^2}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      0.2
    9. Applied associate-/l* to get
      \[\color{red}{\frac{\frac{i}{\frac{{1}^2}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{\frac{{1}^2}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}}\]
      0.2

    if 3.1746031384160272e+218 < (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))))

    1. Started with
      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
      62.1
    2. Applied simplify to get
      \[\color{red}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}} \leadsto \color{blue}{\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}}\]
      46.1
    3. Using strategy rm
      46.1
    4. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{red}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\color{blue}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      46.1
    5. Applied *-un-lft-identity to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\color{red}{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}^2}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{\color{blue}{\left(1 \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right)}}^2}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      46.1
    6. Applied square-prod to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{red}{{\left(1 \cdot \left(\left(\beta + \alpha\right) + 2 \cdot i\right)\right)}^2}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{\color{blue}{{1}^2 \cdot {\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      46.1
    7. Applied times-frac to get
      \[\frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{red}{\frac{{1}^2 \cdot {\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{1 \cdot \left(\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\color{blue}{\frac{{1}^2}{1} \cdot \frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      46.1
    8. Applied times-frac to get
      \[\frac{\color{red}{\frac{i \cdot \left(\beta + \left(i + \alpha\right)\right)}{\frac{{1}^2}{1} \cdot \frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0} \leadsto \frac{\color{blue}{\frac{i}{\frac{{1}^2}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}\]
      46.1
    9. Applied associate-/l* to get
      \[\color{red}{\frac{\frac{i}{\frac{{1}^2}{1}} \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}} \leadsto \color{blue}{\frac{\frac{i}{\frac{{1}^2}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}}}\]
      46.1
    10. Applied taylor to get
      \[\frac{\frac{i}{\frac{{1}^2}{1}}}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2 - 1.0}{\frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\left(\beta + \alpha\right) + 2 \cdot i\right)}^2}{\alpha \cdot \beta + i \cdot \left(\beta + \left(i + \alpha\right)\right)}}}} \leadsto \frac{\frac{i}{\frac{{1}^2}{1}}}{16 \cdot i}\]
      9.7
    11. Taylor expanded around 0 to get
      \[\frac{\frac{i}{\frac{{1}^2}{1}}}{\color{red}{16 \cdot i}} \leadsto \frac{\frac{i}{\frac{{1}^2}{1}}}{\color{blue}{16 \cdot i}}\]
      9.7
    12. Applied simplify to get
      \[\color{red}{\frac{\frac{i}{\frac{{1}^2}{1}}}{16 \cdot i}} \leadsto \color{blue}{\frac{1}{16}}\]
      9.4
  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)))