Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 83.4%
Time: 19.6s
Alternatives: 13
Speedup: TODO×

Specification

?
\[\begin{array}{l} t_0 := \alpha + \beta\\ t_1 := i \cdot \left(t_0 + i\right)\\ t_2 := t_0 + 2 \cdot i\\ t_3 := t_2 \cdot t_2\\ \frac{\frac{t_1 \cdot \left(\beta \cdot \alpha + t_1\right)}{t_3}}{t_3 - 1} \end{array} \]

Your Program's Arguments

Results

Enter valid numbers for all inputs

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 13 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Alternative 1?

\[\begin{array}{l} t_0 := 0.0625 + \frac{\beta}{i} \cdot 0.125\\ t_1 := -0.125 \cdot \frac{\beta}{i}\\ t_2 := \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.8 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{t_2}{\beta}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+175}:\\ \;\;\;\;t_0 + \log \left(e^{t_1}\right)\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+198}:\\ \;\;\;\;\frac{t_2}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.6 \cdot 10^{+215}:\\ \;\;\;\;t_1 + t_0\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{i}{\beta}\\ \end{array} \]
Derivation
  1. Split input into 6 regimes

  2. if beta < 4.29999999999999998e117

    1. Initial program 17.5%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/15.8%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*15.7%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac24.8%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified44.3%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 82.6%

      \[\leadsto \color{blue}{0.0625} \]

    if 4.29999999999999998e117 < beta < 6.7999999999999998e158

    1. Initial program 2.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac13.9%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified75.0%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in beta around inf 63.8%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
    5. Step-by-step derivation

      1. *-commutative63.8%

        \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

      2. associate-/l*64.0%

        \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative64.0%

        \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow264.0%

        \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

    6. Simplified64.0%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
    7. Step-by-step derivation

      1. *-un-lft-identity64.0%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l*64.2%

        \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

    8. Applied egg-rr64.2%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
    9. Step-by-step derivation

      1. *-un-lft-identity64.2%

        \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      2. div-inv64.2%

        \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

      3. +-commutative64.2%

        \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

      4. div-inv64.2%

        \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

      5. clear-num64.0%

        \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

    10. Applied egg-rr64.0%

      \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
    11. Step-by-step derivation

      1. associate-*r/64.0%

        \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

      2. *-rgt-identity64.0%

        \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

      3. associate-/r*64.4%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

    12. Simplified64.4%

      \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]
    13. Step-by-step derivation

      1. associate-/r/64.2%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

    14. Applied egg-rr64.2%

      \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

    if 6.7999999999999998e158 < beta < 3.7999999999999997e175

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified0.0%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 94.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
    5. Step-by-step derivation

      1. cancel-sign-sub-inv94.0%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      2. distribute-lft-out94.0%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

      3. metadata-eval94.0%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

    6. Simplified94.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

    7. Taylor expanded in alpha around 0 94.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right)} \]
    8. Step-by-step derivation

      1. add-log-exp94.0%

        \[\leadsto \color{blue}{\log \left(e^{-0.125 \cdot \frac{\beta}{i}}\right)} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) \]

    9. Applied egg-rr94.0%

      \[\leadsto \color{blue}{\log \left(e^{-0.125 \cdot \frac{\beta}{i}}\right)} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) \]

    if 3.7999999999999997e175 < beta < 1.89999999999999994e198

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified3.5%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in beta around inf 5.0%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
    5. Step-by-step derivation

      1. *-commutative5.0%

        \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

      2. associate-/l*5.7%

        \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative5.7%

        \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow25.7%

        \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

    6. Simplified5.7%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
    7. Step-by-step derivation

      1. *-un-lft-identity5.7%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l*60.5%

        \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

    8. Applied egg-rr60.5%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
    9. Step-by-step derivation

      1. *-un-lft-identity60.5%

        \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      2. div-inv60.5%

        \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

      3. +-commutative60.5%

        \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

      4. div-inv60.5%

        \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

      5. clear-num60.5%

        \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

    10. Applied egg-rr60.5%

      \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
    11. Step-by-step derivation

      1. associate-*r/60.6%

        \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

      2. *-rgt-identity60.6%

        \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

      3. associate-/r*69.5%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

    12. Simplified69.5%

      \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

    if 1.89999999999999994e198 < beta < 6.5999999999999997e215

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified50.0%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 100.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
    5. Step-by-step derivation

      1. cancel-sign-sub-inv100.0%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      2. distribute-lft-out100.0%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

      3. metadata-eval100.0%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

    6. Simplified100.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

    7. Taylor expanded in alpha around 0 100.0%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right)} \]

    if 6.5999999999999997e215 < beta

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified13.4%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in beta around inf 31.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
    5. Step-by-step derivation

      1. *-commutative31.2%

        \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

      2. associate-/l*33.6%

        \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative33.6%

        \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow233.6%

        \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

    6. Simplified33.6%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
    7. Step-by-step derivation

      1. *-un-lft-identity33.6%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l*50.4%

        \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

    8. Applied egg-rr50.4%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]

    9. Taylor expanded in beta around 0 31.2%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
    10. Step-by-step derivation

      1. unpow231.2%

        \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

      2. times-frac70.7%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

    11. Simplified70.7%

      \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]
  3. Recombined 6 regimes into one program.

  4. Final simplification80.8%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.8 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{\frac{i + \alpha}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+175}:\\ \;\;\;\;\left(0.0625 + \frac{\beta}{i} \cdot 0.125\right) + \log \left(e^{-0.125 \cdot \frac{\beta}{i}}\right)\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+198}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.6 \cdot 10^{+215}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 2?

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := \beta + \left(i + \alpha\right)\\ t_5 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t_4, \alpha \cdot \beta\right)}{t_5} \cdot \frac{i}{\frac{t_5}{t_4}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

    1. Initial program 39.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. times-frac99.7%

        \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. +-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. +-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. *-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      5. fma-udef99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      6. +-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\color{blue}{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \beta \cdot \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      7. +-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)} + \beta \cdot \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      8. *-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right) + \color{blue}{\alpha \cdot \beta}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      9. fma-udef99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      10. +-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\color{blue}{2 \cdot i + \left(\alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      11. *-commutative99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\color{blue}{i \cdot 2} + \left(\alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      12. fma-udef99.7%

        \[\leadsto \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\color{blue}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    3. Applied egg-rr99.7%

      \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    4. Step-by-step derivation

      1. *-commutative99.7%

        \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. associate-+r+99.7%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \color{blue}{\left(i + \alpha\right) + \beta}, \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. +-commutative99.7%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \color{blue}{\left(\alpha + i\right)} + \beta, \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. *-commutative99.7%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \color{blue}{\beta \cdot \alpha}\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      5. +-commutative99.7%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \color{blue}{\beta + \alpha}\right)} \cdot \frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      6. associate-/l*99.9%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      7. +-commutative99.9%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \color{blue}{\beta + \alpha}\right)}{i + \left(\alpha + \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      8. associate-+r+99.9%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\color{blue}{\left(i + \alpha\right) + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      9. +-commutative99.9%

        \[\leadsto \frac{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\color{blue}{\left(\alpha + i\right)} + \beta}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    5. Simplified99.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\alpha + i\right) + \beta}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified5.9%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 78.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
    5. Step-by-step derivation

      1. cancel-sign-sub-inv78.8%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      2. distribute-lft-out78.8%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

      3. metadata-eval78.8%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

    6. Simplified78.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

    7. Taylor expanded in alpha around 0 74.9%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification84.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\beta + \left(i + \alpha\right)}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]

Alternative 3?

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ t_4 := \beta + \left(i + \alpha\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{i \cdot \frac{t_4}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, t_4, \alpha \cdot \beta\right)}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

    1. Initial program 39.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. expm1-log1p-u36.2%

        \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-udef36.2%

        \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    3. Applied egg-rr90.7%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)} - 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    4. Step-by-step derivation

      1. expm1-def90.7%

        \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. expm1-log1p99.8%

        \[\leadsto \frac{\color{blue}{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. associate-*r/99.8%

        \[\leadsto \frac{\color{blue}{i \cdot \frac{i + \left(\alpha + \beta\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. associate-+r+99.8%

        \[\leadsto \frac{i \cdot \frac{\color{blue}{\left(i + \alpha\right) + \beta}}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      5. +-commutative99.8%

        \[\leadsto \frac{i \cdot \frac{\color{blue}{\left(\alpha + i\right)} + \beta}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      6. +-commutative99.8%

        \[\leadsto \frac{i \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{{\left(\mathsf{fma}\left(i, 2, \color{blue}{\beta + \alpha}\right)\right)}^{2}}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      7. associate-+r+99.8%

        \[\leadsto \frac{i \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\mathsf{fma}\left(i, \color{blue}{\left(i + \alpha\right) + \beta}, \alpha \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      8. +-commutative99.8%

        \[\leadsto \frac{i \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\mathsf{fma}\left(i, \color{blue}{\left(\alpha + i\right)} + \beta, \alpha \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      9. *-commutative99.8%

        \[\leadsto \frac{i \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \color{blue}{\beta \cdot \alpha}\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    5. Simplified99.8%

      \[\leadsto \frac{\color{blue}{i \cdot \frac{\left(\alpha + i\right) + \beta}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)}^{2}}{\mathsf{fma}\left(i, \left(\alpha + i\right) + \beta, \beta \cdot \alpha\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified5.9%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 78.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
    5. Step-by-step derivation

      1. cancel-sign-sub-inv78.8%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      2. distribute-lft-out78.8%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

      3. metadata-eval78.8%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

    6. Simplified78.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

    7. Taylor expanded in alpha around 0 74.9%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification84.0%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{i \cdot \frac{\beta + \left(i + \alpha\right)}{\frac{{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}^{2}}{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \alpha \cdot \beta\right)}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]

Alternative 4?

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + i \cdot 2\\ t_1 := t_0 \cdot t_0\\ t_2 := t_1 + -1\\ t_3 := i \cdot \left(i + \left(\alpha + \beta\right)\right)\\ \mathbf{if}\;\frac{\frac{t_3 \cdot \left(t_3 + \alpha \cdot \beta\right)}{t_1}}{t_2} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{{\left(i + \beta\right)}^{2}}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1)) < +inf.0

    1. Initial program 39.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    2. Taylor expanded in alpha around 0 34.5%

      \[\leadsto \frac{\color{blue}{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    3. Step-by-step derivation

      1. associate-/l*90.5%

        \[\leadsto \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      2. unpow290.5%

        \[\leadsto \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      3. +-commutative90.5%

        \[\leadsto \frac{\frac{i \cdot i}{\frac{{\color{blue}{\left(2 \cdot i + \beta\right)}}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      4. *-commutative90.5%

        \[\leadsto \frac{\frac{i \cdot i}{\frac{{\left(\color{blue}{i \cdot 2} + \beta\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

      5. fma-udef90.5%

        \[\leadsto \frac{\frac{i \cdot i}{\frac{{\color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    4. Simplified90.5%

      \[\leadsto \frac{\color{blue}{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

    if +inf.0 < (/.f64 (/.f64 (*.f64 (*.f64 i (+.f64 (+.f64 alpha beta) i)) (+.f64 (*.f64 beta alpha) (*.f64 i (+.f64 (+.f64 alpha beta) i)))) (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (-.f64 (*.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 1))

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified5.9%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 78.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
    5. Step-by-step derivation

      1. cancel-sign-sub-inv78.8%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      2. distribute-lft-out78.8%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

      3. metadata-eval78.8%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

    6. Simplified78.8%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

    7. Taylor expanded in alpha around 0 74.9%

      \[\leadsto \color{blue}{-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification80.6%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1} \leq \infty:\\ \;\;\;\;\frac{\frac{i \cdot i}{\frac{{\left(\mathsf{fma}\left(i, 2, \beta\right)\right)}^{2}}{{\left(i + \beta\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) \cdot \left(\left(\alpha + \beta\right) + i \cdot 2\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \end{array} \]

Alternative 5?

\[\begin{array}{l} t_0 := \frac{0.125}{\frac{i}{\alpha + \beta}}\\ t_1 := 0.0625 + t_0\\ t_2 := \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{t_2}{\beta}\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t_1, t_1, \left(\frac{\beta}{i} \cdot \frac{\beta}{i}\right) \cdot \left(-0.015625\right)\right)}{0.0625 + \left(t_0 - -0.125 \cdot \frac{\alpha + \beta}{i}\right)}\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+198}:\\ \;\;\;\;\frac{t_2}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.8 \cdot 10^{+214}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{i}{\beta}\\ \end{array} \]
Derivation
  1. Split input into 6 regimes

  2. if beta < 2.5999999999999999e117

    1. Initial program 17.5%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/15.8%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*15.7%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac24.8%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified44.3%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 82.6%

      \[\leadsto \color{blue}{0.0625} \]

    if 2.5999999999999999e117 < beta < 3.3999999999999999e158

    1. Initial program 2.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac13.9%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified75.0%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in beta around inf 63.8%

      \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
    5. Step-by-step derivation

      1. *-commutative63.8%

        \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

      2. associate-/l*64.0%

        \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative64.0%

        \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow264.0%

        \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

    6. Simplified64.0%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
    7. Step-by-step derivation

      1. *-un-lft-identity64.0%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l*64.2%

        \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

    8. Applied egg-rr64.2%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
    9. Step-by-step derivation

      1. *-un-lft-identity64.2%

        \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      2. div-inv64.2%

        \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

      3. +-commutative64.2%

        \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

      4. div-inv64.2%

        \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

      5. clear-num64.0%

        \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

    10. Applied egg-rr64.0%

      \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
    11. Step-by-step derivation

      1. associate-*r/64.0%

        \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

      2. *-rgt-identity64.0%

        \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

      3. associate-/r*64.4%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

    12. Simplified64.4%

      \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]
    13. Step-by-step derivation

      1. associate-/r/64.2%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

    14. Applied egg-rr64.2%

      \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

    if 3.3999999999999999e158 < beta < 3.8999999999999999e177

    1. Initial program 0.0%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/0.0%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*0.0%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac0.0%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified0.0%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 94.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
    5. Step-by-step derivation

      1. cancel-sign-sub-inv94.0%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

      2. distribute-lft-out94.0%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

      3. metadata-eval94.0%

        \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

    6. Simplified94.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]
    7. Step-by-step derivation

      1. flip-+94.0%

        \[\leadsto \color{blue}{\frac{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) \cdot \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) - \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \cdot \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) - -0.125 \cdot \frac{\beta + \alpha}{i}}} \]

      2. associate-/l*94.0%

        \[\leadsto \frac{\left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2}{\frac{i}{\beta + \alpha}}}\right) \cdot \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) - \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \cdot \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) - -0.125 \cdot \frac{\beta + \alpha}{i}} \]

      3. associate-/l*94.0%

        \[\leadsto \frac{\left(0.0625 + 0.0625 \cdot \frac{2}{\frac{i}{\beta + \alpha}}\right) \cdot \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2}{\frac{i}{\beta + \alpha}}}\right) - \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \cdot \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) - -0.125 \cdot \frac{\beta + \alpha}{i}} \]

      4. associate-/l*94.0%

        \[\leadsto \frac{\left(0.0625 + 0.0625 \cdot \frac{2}{\frac{i}{\beta + \alpha}}\right) \cdot \left(0.0625 + 0.0625 \cdot \frac{2}{\frac{i}{\beta + \alpha}}\right) - \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \cdot \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)}{\left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2}{\frac{i}{\beta + \alpha}}}\right) - -0.125 \cdot \frac{\beta + \alpha}{i}} \]

    8. Applied egg-rr94.0%

      \[\leadsto \color{blue}{\frac{\left(0.0625 + 0.0625 \cdot \frac{2}{\frac{i}{\beta + \alpha}}\right) \cdot \left(0.0625 + 0.0625 \cdot \frac{2}{\frac{i}{\beta + \alpha}}\right) - \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right) \cdot \left(-0.125 \cdot \frac{\beta + \alpha}{i}\right)}{\left(0.0625 + 0.0625 \cdot \frac{2}{\frac{i}{\beta + \alpha}}\right) - -0.125 \cdot \frac{\beta + \alpha}{i}}} \]
    9. Step-by-step derivation

      1. Simplified94.0%

        \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, 0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, -0.015625 \cdot \left(\frac{\alpha + \beta}{i} \cdot \frac{\alpha + \beta}{i}\right)\right)}{0.0625 + \left(\frac{0.125}{\frac{i}{\alpha + \beta}} - -0.125 \cdot \frac{\alpha + \beta}{i}\right)}} \]

      2. Taylor expanded in alpha around 0 0.0%

        \[\leadsto \frac{\mathsf{fma}\left(0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, 0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, -0.015625 \cdot \color{blue}{\frac{{\beta}^{2}}{{i}^{2}}}\right)}{0.0625 + \left(\frac{0.125}{\frac{i}{\alpha + \beta}} - -0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]
      3. Step-by-step derivation

        1. unpow20.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, 0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, -0.015625 \cdot \frac{\color{blue}{\beta \cdot \beta}}{{i}^{2}}\right)}{0.0625 + \left(\frac{0.125}{\frac{i}{\alpha + \beta}} - -0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

        2. unpow20.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, 0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, -0.015625 \cdot \frac{\beta \cdot \beta}{\color{blue}{i \cdot i}}\right)}{0.0625 + \left(\frac{0.125}{\frac{i}{\alpha + \beta}} - -0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

        3. times-frac94.0%

          \[\leadsto \frac{\mathsf{fma}\left(0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, 0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, -0.015625 \cdot \color{blue}{\left(\frac{\beta}{i} \cdot \frac{\beta}{i}\right)}\right)}{0.0625 + \left(\frac{0.125}{\frac{i}{\alpha + \beta}} - -0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

      4. Simplified94.0%

        \[\leadsto \frac{\mathsf{fma}\left(0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, 0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, -0.015625 \cdot \color{blue}{\left(\frac{\beta}{i} \cdot \frac{\beta}{i}\right)}\right)}{0.0625 + \left(\frac{0.125}{\frac{i}{\alpha + \beta}} - -0.125 \cdot \frac{\alpha + \beta}{i}\right)} \]

      if 3.8999999999999999e177 < beta < 1.75000000000000006e198

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified3.5%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 5.0%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative5.0%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*5.7%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative5.7%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow25.7%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified5.7%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity5.7%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*60.5%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr60.5%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
      9. Step-by-step derivation

        1. *-un-lft-identity60.5%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

        2. div-inv60.5%

          \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

        3. +-commutative60.5%

          \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

        4. div-inv60.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

        5. clear-num60.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

      10. Applied egg-rr60.5%

        \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
      11. Step-by-step derivation

        1. associate-*r/60.6%

          \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

        2. *-rgt-identity60.6%

          \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

        3. associate-/r*69.5%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      12. Simplified69.5%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      if 1.75000000000000006e198 < beta < 6.7999999999999996e214

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified50.0%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 100.0%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
      5. Step-by-step derivation

        1. cancel-sign-sub-inv100.0%

          \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

        2. distribute-lft-out100.0%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

        3. metadata-eval100.0%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

      6. Simplified100.0%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

      7. Taylor expanded in alpha around 0 100.0%

        \[\leadsto \color{blue}{-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right)} \]

      if 6.7999999999999996e214 < beta

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified13.4%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 31.2%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative31.2%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*33.6%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative33.6%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow233.6%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified33.6%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity33.6%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*50.4%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr50.4%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]

      9. Taylor expanded in beta around 0 31.2%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      10. Step-by-step derivation

        1. unpow231.2%

          \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

        2. times-frac70.7%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

      11. Simplified70.7%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]
    10. Recombined 6 regimes into one program.

    11. Final simplification80.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{\frac{i + \alpha}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+177}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, 0.0625 + \frac{0.125}{\frac{i}{\alpha + \beta}}, \left(\frac{\beta}{i} \cdot \frac{\beta}{i}\right) \cdot \left(-0.015625\right)\right)}{0.0625 + \left(\frac{0.125}{\frac{i}{\alpha + \beta}} - -0.125 \cdot \frac{\alpha + \beta}{i}\right)}\\ \mathbf{elif}\;\beta \leq 1.75 \cdot 10^{+198}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.8 \cdot 10^{+214}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

    Alternative 6?

    \[\begin{array}{l} t_0 := -0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ t_1 := \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{t_1}{\beta}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{t_1}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+216}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{i}{\beta}\\ \end{array} \]
    Derivation
    1. Split input into 5 regimes

    2. if beta < 4.2000000000000002e117

      1. Initial program 17.5%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/15.8%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*15.7%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac24.8%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified44.3%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 82.6%

        \[\leadsto \color{blue}{0.0625} \]

      if 4.2000000000000002e117 < beta < 3.30000000000000017e158

      1. Initial program 2.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac13.9%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified75.0%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 63.8%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative63.8%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*64.0%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative64.0%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow264.0%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified64.0%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity64.0%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*64.2%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr64.2%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
      9. Step-by-step derivation

        1. *-un-lft-identity64.2%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

        2. div-inv64.2%

          \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

        3. +-commutative64.2%

          \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

        4. div-inv64.2%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

        5. clear-num64.0%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

      10. Applied egg-rr64.0%

        \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
      11. Step-by-step derivation

        1. associate-*r/64.0%

          \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

        2. *-rgt-identity64.0%

          \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

        3. associate-/r*64.4%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      12. Simplified64.4%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]
      13. Step-by-step derivation

        1. associate-/r/64.2%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

      14. Applied egg-rr64.2%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

      if 3.30000000000000017e158 < beta < 1.5e176 or 3.1999999999999998e198 < beta < 2.7000000000000001e216

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified14.3%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 95.7%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
      5. Step-by-step derivation

        1. cancel-sign-sub-inv95.7%

          \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

        2. distribute-lft-out95.7%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

        3. metadata-eval95.7%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

      6. Simplified95.7%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

      7. Taylor expanded in alpha around 0 95.7%

        \[\leadsto \color{blue}{-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right)} \]

      if 1.5e176 < beta < 3.1999999999999998e198

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified3.5%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 5.0%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative5.0%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*5.7%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative5.7%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow25.7%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified5.7%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity5.7%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*60.5%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr60.5%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
      9. Step-by-step derivation

        1. *-un-lft-identity60.5%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

        2. div-inv60.5%

          \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

        3. +-commutative60.5%

          \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

        4. div-inv60.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

        5. clear-num60.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

      10. Applied egg-rr60.5%

        \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
      11. Step-by-step derivation

        1. associate-*r/60.6%

          \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

        2. *-rgt-identity60.6%

          \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

        3. associate-/r*69.5%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      12. Simplified69.5%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      if 2.7000000000000001e216 < beta

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified13.4%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 31.2%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative31.2%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*33.6%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative33.6%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow233.6%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified33.6%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity33.6%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*50.4%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr50.4%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]

      9. Taylor expanded in beta around 0 31.2%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      10. Step-by-step derivation

        1. unpow231.2%

          \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

        2. times-frac70.7%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

      11. Simplified70.7%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]
    3. Recombined 5 regimes into one program.

    4. Final simplification80.8%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.3 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{\frac{i + \alpha}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+176}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+198}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 2.7 \cdot 10^{+216}:\\ \;\;\;\;-0.125 \cdot \frac{\beta}{i} + \left(0.0625 + \frac{\beta}{i} \cdot 0.125\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

    Alternative 7?

    \[\begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+158} \lor \neg \left(\beta \leq 1.6 \cdot 10^{+176} \lor \neg \left(\beta \leq 3.5 \cdot 10^{+198}\right) \land \beta \leq 6.8 \cdot 10^{+214}\right):\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
    Derivation
    1. Split input into 2 regimes

    2. if beta < 4.29999999999999998e117 or 6.5000000000000001e158 < beta < 1.5999999999999999e176 or 3.50000000000000013e198 < beta < 6.7999999999999996e214

      1. Initial program 16.9%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/15.3%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*15.2%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac24.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified43.3%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 82.5%

        \[\leadsto \color{blue}{0.0625} \]

      if 4.29999999999999998e117 < beta < 6.5000000000000001e158 or 1.5999999999999999e176 < beta < 3.50000000000000013e198 or 6.7999999999999996e214 < beta

      1. Initial program 0.4%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac2.8%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified23.5%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 31.8%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative31.8%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*33.4%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative33.4%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow233.4%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified33.4%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity33.4%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*55.4%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr55.4%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]

      9. Taylor expanded in beta around 0 31.8%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      10. Step-by-step derivation

        1. unpow231.8%

          \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

        2. times-frac68.9%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

      11. Simplified68.9%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]
    3. Recombined 2 regimes into one program.

    4. Final simplification80.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.5 \cdot 10^{+158} \lor \neg \left(\beta \leq 1.6 \cdot 10^{+176} \lor \neg \left(\beta \leq 3.5 \cdot 10^{+198}\right) \land \beta \leq 6.8 \cdot 10^{+214}\right):\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

    Alternative 8?

    \[\begin{array}{l} t_0 := \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{t_0}{\beta}\\ \mathbf{elif}\;\beta \leq 7.4 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+198} \lor \neg \left(\beta \leq 6.8 \cdot 10^{+214}\right):\\ \;\;\;\;t_0 \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
    Derivation
    1. Split input into 3 regimes

    2. if beta < 4.29999999999999998e117 or 4.9999999999999996e158 < beta < 7.4000000000000004e174 or 3.4e198 < beta < 6.7999999999999996e214

      1. Initial program 16.9%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/15.3%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*15.2%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac24.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified43.3%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 82.5%

        \[\leadsto \color{blue}{0.0625} \]

      if 4.29999999999999998e117 < beta < 4.9999999999999996e158

      1. Initial program 2.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac13.9%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified75.0%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 63.8%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative63.8%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*64.0%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative64.0%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow264.0%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified64.0%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity64.0%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*64.2%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr64.2%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
      9. Step-by-step derivation

        1. *-un-lft-identity64.2%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

        2. div-inv64.2%

          \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

        3. +-commutative64.2%

          \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

        4. div-inv64.2%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

        5. clear-num64.0%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

      10. Applied egg-rr64.0%

        \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
      11. Step-by-step derivation

        1. associate-*r/64.0%

          \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

        2. *-rgt-identity64.0%

          \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

        3. associate-/r*64.4%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      12. Simplified64.4%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]
      13. Step-by-step derivation

        1. associate-/r/64.2%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

      14. Applied egg-rr64.2%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

      if 7.4000000000000004e174 < beta < 3.4e198 or 6.7999999999999996e214 < beta

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified10.6%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 23.8%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative23.8%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*25.8%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative25.8%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow225.8%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified25.8%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity25.8%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*53.2%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr53.2%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]

      9. Taylor expanded in beta around 0 23.8%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      10. Step-by-step derivation

        1. unpow223.8%

          \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

        2. times-frac70.2%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

      11. Simplified70.2%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]
    3. Recombined 3 regimes into one program.

    4. Final simplification80.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 5 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{\frac{i + \alpha}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 7.4 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+198} \lor \neg \left(\beta \leq 6.8 \cdot 10^{+214}\right):\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

    Alternative 9?

    \[\begin{array}{l} t_0 := \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.4 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{t_0}{\beta}\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+198}:\\ \;\;\;\;\frac{t_0}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.2 \cdot 10^{+214}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{i}{\beta}\\ \end{array} \]
    Derivation
    1. Split input into 4 regimes

    2. if beta < 4.29999999999999998e117 or 4.4000000000000002e158 < beta < 1.55e174 or 3.4e198 < beta < 8.2e214

      1. Initial program 16.9%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/15.3%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*15.2%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac24.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified43.3%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 82.5%

        \[\leadsto \color{blue}{0.0625} \]

      if 4.29999999999999998e117 < beta < 4.4000000000000002e158

      1. Initial program 2.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac13.9%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified75.0%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 63.8%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative63.8%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*64.0%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative64.0%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow264.0%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified64.0%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity64.0%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*64.2%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr64.2%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
      9. Step-by-step derivation

        1. *-un-lft-identity64.2%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

        2. div-inv64.2%

          \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

        3. +-commutative64.2%

          \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

        4. div-inv64.2%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

        5. clear-num64.0%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

      10. Applied egg-rr64.0%

        \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
      11. Step-by-step derivation

        1. associate-*r/64.0%

          \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

        2. *-rgt-identity64.0%

          \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

        3. associate-/r*64.4%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      12. Simplified64.4%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]
      13. Step-by-step derivation

        1. associate-/r/64.2%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

      14. Applied egg-rr64.2%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\beta} \cdot i} \]

      if 1.55e174 < beta < 3.4e198

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified3.5%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 5.0%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative5.0%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*5.7%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative5.7%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow25.7%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified5.7%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity5.7%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*60.5%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr60.5%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
      9. Step-by-step derivation

        1. *-un-lft-identity60.5%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

        2. div-inv60.5%

          \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

        3. +-commutative60.5%

          \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

        4. div-inv60.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

        5. clear-num60.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

      10. Applied egg-rr60.5%

        \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
      11. Step-by-step derivation

        1. associate-*r/60.6%

          \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

        2. *-rgt-identity60.6%

          \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

        3. associate-/r*69.5%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      12. Simplified69.5%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      if 8.2e214 < beta

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified13.4%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 31.2%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative31.2%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*33.6%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative33.6%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow233.6%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified33.6%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity33.6%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*50.4%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr50.4%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]

      9. Taylor expanded in beta around 0 31.2%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      10. Step-by-step derivation

        1. unpow231.2%

          \[\leadsto \frac{i \cdot \left(i + \alpha\right)}{\color{blue}{\beta \cdot \beta}} \]

        2. times-frac70.7%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

      11. Simplified70.7%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]
    3. Recombined 4 regimes into one program.

    4. Final simplification80.4%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.4 \cdot 10^{+158}:\\ \;\;\;\;i \cdot \frac{\frac{i + \alpha}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 1.55 \cdot 10^{+174}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.4 \cdot 10^{+198}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 8.2 \cdot 10^{+214}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

    Alternative 10?

    \[\begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.95 \cdot 10^{+158} \lor \neg \left(\beta \leq 6.8 \cdot 10^{+214}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]
    Derivation
    1. Split input into 2 regimes

    2. if beta < 4.29999999999999998e117 or 3.95000000000000019e158 < beta < 6.7999999999999996e214

      1. Initial program 16.2%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/14.7%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*14.6%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac23.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified41.7%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 80.3%

        \[\leadsto \color{blue}{0.0625} \]

      if 4.29999999999999998e117 < beta < 3.95000000000000019e158 or 6.7999999999999996e214 < beta

      1. Initial program 0.5%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac3.6%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified29.3%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 39.6%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative39.6%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*41.5%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative41.5%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow241.5%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified41.5%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity41.5%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*53.9%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr53.9%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]

      9. Taylor expanded in alpha around 0 40.0%

        \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
      10. Step-by-step derivation

        1. unpow240.0%

          \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

        2. unpow240.0%

          \[\leadsto \frac{i \cdot i}{\color{blue}{\beta \cdot \beta}} \]

        3. times-frac65.8%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]

      11. Simplified65.8%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{i}{\beta}} \]
    3. Recombined 2 regimes into one program.

    4. Final simplification78.5%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.95 \cdot 10^{+158} \lor \neg \left(\beta \leq 6.8 \cdot 10^{+214}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

    Alternative 11?

    \[\begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+227}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]
    Derivation
    1. Split input into 2 regimes

    2. if beta < 3.19999999999999988e227

      1. Initial program 15.4%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/13.9%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*13.8%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac22.2%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified42.0%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 78.1%

        \[\leadsto \color{blue}{0.0625} \]

      if 3.19999999999999988e227 < beta

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified16.7%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in beta around inf 39.4%

        \[\leadsto \color{blue}{\frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}}} \]
      5. Step-by-step derivation

        1. *-commutative39.4%

          \[\leadsto \frac{\color{blue}{\left(i + \alpha\right) \cdot i}}{{\beta}^{2}} \]

        2. associate-/l*41.7%

          \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

        3. +-commutative41.7%

          \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

        4. unpow241.7%

          \[\leadsto \frac{\alpha + i}{\frac{\color{blue}{\beta \cdot \beta}}{i}} \]

      6. Simplified41.7%

        \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
      7. Step-by-step derivation

        1. *-un-lft-identity41.7%

          \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

        2. associate-/l*54.5%

          \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

      8. Applied egg-rr54.5%

        \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
      9. Step-by-step derivation

        1. *-un-lft-identity54.5%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

        2. div-inv54.5%

          \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}}} \]

        3. +-commutative54.5%

          \[\leadsto \color{blue}{\left(i + \alpha\right)} \cdot \frac{1}{\frac{\beta}{\frac{i}{\beta}}} \]

        4. div-inv54.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\color{blue}{\beta \cdot \frac{1}{\frac{i}{\beta}}}} \]

        5. clear-num54.5%

          \[\leadsto \left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \color{blue}{\frac{\beta}{i}}} \]

      10. Applied egg-rr54.5%

        \[\leadsto \color{blue}{\left(i + \alpha\right) \cdot \frac{1}{\beta \cdot \frac{\beta}{i}}} \]
      11. Step-by-step derivation

        1. associate-*r/54.5%

          \[\leadsto \color{blue}{\frac{\left(i + \alpha\right) \cdot 1}{\beta \cdot \frac{\beta}{i}}} \]

        2. *-rgt-identity54.5%

          \[\leadsto \frac{\color{blue}{i + \alpha}}{\beta \cdot \frac{\beta}{i}} \]

        3. associate-/r*75.2%

          \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      12. Simplified75.2%

        \[\leadsto \color{blue}{\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}} \]

      13. Taylor expanded in i around 0 40.7%

        \[\leadsto \color{blue}{\frac{i \cdot \alpha}{{\beta}^{2}}} \]
      14. Step-by-step derivation

        1. unpow240.7%

          \[\leadsto \frac{i \cdot \alpha}{\color{blue}{\beta \cdot \beta}} \]

        2. times-frac47.2%

          \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha}{\beta}} \]

      15. Simplified47.2%

        \[\leadsto \color{blue}{\frac{i}{\beta} \cdot \frac{\alpha}{\beta}} \]
    3. Recombined 2 regimes into one program.

    4. Final simplification76.0%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+227}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

    Alternative 12?

    \[\begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+227}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]
    Derivation
    1. Split input into 2 regimes

    2. if beta < 2.65000000000000004e227

      1. Initial program 15.4%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/13.9%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*13.8%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac22.2%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified42.0%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 78.1%

        \[\leadsto \color{blue}{0.0625} \]

      if 2.65000000000000004e227 < beta

      1. Initial program 0.0%

        \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      2. Step-by-step derivation

        1. associate-/l/0.0%

          \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

        2. associate-*l*0.0%

          \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

        3. times-frac0.0%

          \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

      3. Simplified16.7%

        \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

      4. Taylor expanded in i around inf 56.1%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]
      5. Step-by-step derivation

        1. cancel-sign-sub-inv56.1%

          \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i}} \]

        2. distribute-lft-out56.1%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{2 \cdot \left(\beta + \alpha\right)}}{i}\right) + \left(-0.125\right) \cdot \frac{\beta + \alpha}{i} \]

        3. metadata-eval56.1%

          \[\leadsto \left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + \color{blue}{-0.125} \cdot \frac{\beta + \alpha}{i} \]

      6. Simplified56.1%

        \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \left(\beta + \alpha\right)}{i}\right) + -0.125 \cdot \frac{\beta + \alpha}{i}} \]

      7. Taylor expanded in i around 0 41.7%

        \[\leadsto \color{blue}{\frac{-0.125 \cdot \left(\beta + \alpha\right) + 0.125 \cdot \left(\beta + \alpha\right)}{i}} \]
      8. Step-by-step derivation

        1. distribute-rgt-out41.7%

          \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right) \cdot \left(-0.125 + 0.125\right)}}{i} \]

        2. metadata-eval41.7%

          \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{0}}{i} \]

        3. mul0-rgt41.7%

          \[\leadsto \frac{\color{blue}{0}}{i} \]

      9. Simplified41.7%

        \[\leadsto \color{blue}{\frac{0}{i}} \]
    3. Recombined 2 regimes into one program.

    4. Final simplification75.6%

      \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+227}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{i}\\ \end{array} \]

    Alternative 13?

    \[0.0625 \]
    Derivation

    1. Initial program 14.3%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
    2. Step-by-step derivation

      1. associate-/l/12.9%

        \[\leadsto \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]

      2. associate-*l*12.8%

        \[\leadsto \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

      3. times-frac20.6%

        \[\leadsto \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

    3. Simplified40.2%

      \[\leadsto \color{blue}{\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \left(\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right)} \]

    4. Taylor expanded in i around inf 73.9%

      \[\leadsto \color{blue}{0.0625} \]

    5. Final simplification73.9%

      \[\leadsto 0.0625 \]

    Reproduce

    ?
    herbie shell --seed 2023166 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))