Average Error: 16.4 → 1.4
Time: 1.3m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}\right) \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} - 1.0\right)}{2.0} \le 9.645829536245808 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(\frac{1}{{\alpha}^{3}} \cdot 8.0 + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}\right) \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} - 1.0\right)}{2.0} \le 1.5435960758848893 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\beta} \cdot 1.0 + \frac{\left(1.0 \cdot \left(\log \beta \cdot \log \beta\right)\right) \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}\right) + \left(\frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot 2}{-\frac{\log \beta}{\alpha}}} + e^{\left(\log 2 - \log \alpha\right) + \log \beta}\right)\right) + \left(\left(1.0 \cdot \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\log 2 - \log \alpha\right) + \log \beta}{-\frac{\log \beta}{\alpha}}} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 1.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{\log \alpha \cdot \log \alpha}{\alpha}}}\right) + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\alpha}{-\log \alpha}} \cdot \frac{1.0}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} \cdot \left(1.0 \cdot \frac{\log 2}{\frac{\alpha}{\log 2}}\right)\right)\right)\right) + \left(\left(1.0 \cdot \frac{\log 2}{\alpha}\right) \cdot \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 2.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \alpha}{\frac{\alpha}{\log 2}}}}\right)\right) - \left(\left(\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot \left(\log \beta \cdot \left(-1.0\right)\right)}{\left(\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 2.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \beta}{\frac{\alpha}{\log 2}}}}\right) + \left(\frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot 2}{-\frac{\log \alpha}{\alpha}}} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\alpha}{\log 2}} \cdot \frac{1.0}{\left(\log 2 - \log \alpha\right) + \log \beta}\right)\right) + \left(1.0 \cdot \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\log 2 - \log \alpha\right) + \log \beta}{-\frac{\log \alpha}{\alpha}}} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha}\right) + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot \left(2.0 \cdot \left(\log \beta \cdot \log \alpha\right)\right)}{\left(\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot 2}{\frac{\log 2}{\alpha}}}\right)\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}\right) \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} - 1.0\right)}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* (* (cbrt (/ alpha (+ (+ alpha beta) 2.0))) (cbrt (/ alpha (+ (+ alpha beta) 2.0)))) (cbrt (/ alpha (+ (+ alpha beta) 2.0)))) 1.0)) 2.0) < 9.645829536245808e-230

    1. Initial program 60.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub60.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-60.2

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]

    5. Taylor expanded around inf 2.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{\left(4.0 \cdot \frac{1}{{\alpha}^{2}} - \left(8.0 \cdot \frac{1}{{\alpha}^{3}} + 2.0 \cdot \frac{1}{\alpha}\right)\right)}}{2.0}\]

    if 9.645829536245808e-230 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* (* (cbrt (/ alpha (+ (+ alpha beta) 2.0))) (cbrt (/ alpha (+ (+ alpha beta) 2.0)))) (cbrt (/ alpha (+ (+ alpha beta) 2.0)))) 1.0)) 2.0) < 1.5435960758848893e-12

    1. Initial program 60.2

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub60.2

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-53.8

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-exp-log53.8

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}}{2.0}\]
    7. Using strategy rm

    8. Applied add-cbrt-cube53.8

      \[\leadsto \frac{e^{\color{blue}{\sqrt[3]{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right) \cdot \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)\right) \cdot \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)\right)}}}}{2.0}\]

    9. Applied simplify53.8

      \[\leadsto \frac{e^{\sqrt[3]{\color{blue}{{\left(\log \left(\left(1.0 - \frac{\alpha}{\beta + \left(2.0 + \alpha\right)}\right) + \frac{\beta}{\beta + \left(2.0 + \alpha\right)}\right)\right)}^{3}}}}}{2.0}\]

    10. Taylor expanded around inf 62.3

      \[\leadsto \frac{\color{blue}{\left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log 2}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + \left(2.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \left(\log \left(\frac{1}{\alpha}\right) \cdot \log 2\right)}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot {\left(\log 2\right)}^{2}}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log \left(\frac{1}{\alpha}\right)}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log \left(\frac{1}{\beta}\right)}{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right) \cdot \alpha} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot {\left(\log \left(\frac{1}{\alpha}\right)\right)}^{2}}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + \left(6.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log \left(\frac{1}{\beta}\right)}{\alpha \cdot \left(\left(2 \cdot \log \left(\frac{1}{\alpha}\right) + 2 \cdot \log 2\right) - 2 \cdot \log \left(\frac{1}{\beta}\right)\right)} + \left(e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}}}{\beta} + 1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot {\left(\log \left(\frac{1}{\beta}\right)\right)}^{2}}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha}\right)\right)\right)\right)\right)\right)\right)\right)\right) - \left(2.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \left(\log 2 \cdot \log \left(\frac{1}{\beta}\right)\right)}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log \left(\frac{1}{\beta}\right)}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log 2}{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right) \cdot \alpha} + \left(6.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log \left(\frac{1}{\alpha}\right)}{\alpha \cdot \left(\left(2 \cdot \log \left(\frac{1}{\alpha}\right) + 2 \cdot \log 2\right) - 2 \cdot \log \left(\frac{1}{\beta}\right)\right)} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}}}{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right) \cdot \alpha} + \left(1.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log \left(\frac{1}{\alpha}\right)}{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right) \cdot \alpha} + \left(2.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \left(\log \left(\frac{1}{\alpha}\right) \cdot \log \left(\frac{1}{\beta}\right)\right)}{{\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{2} \cdot \alpha} + 6.0 \cdot \frac{e^{{\left({\left(\left(\log \left(\frac{1}{\alpha}\right) + \log 2\right) - \log \left(\frac{1}{\beta}\right)\right)}^{3}\right)}^{\frac{1}{3}}} \cdot \log 2}{\alpha \cdot \left(\left(2 \cdot \log \left(\frac{1}{\alpha}\right) + 2 \cdot \log 2\right) - 2 \cdot \log \left(\frac{1}{\beta}\right)\right)}\right)\right)\right)\right)\right)\right)\right)}}{2.0}\]

    11. Applied simplify8.2

      \[\leadsto \color{blue}{\frac{\left(\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} \cdot \left(\frac{\log 2}{\alpha} \cdot 1.0\right) + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 2.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \alpha}{\frac{\alpha}{\log 2}}}}\right) + \left(\left(\left(\frac{1.0}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} \cdot \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\alpha}{-\log \alpha}} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} \cdot \left(\frac{\log 2}{\frac{\alpha}{\log 2}} \cdot 1.0\right)\right) + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\log 2 - \log \alpha\right) + \log \beta}{\frac{-\log \beta}{\alpha}}} \cdot 1.0 + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 1.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{\log \alpha \cdot \log \alpha}{\alpha}}}\right)\right) + \left(\left(e^{\left(\log 2 - \log \alpha\right) + \log \beta} + \frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{2 \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \beta}{\alpha}}}\right) + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\beta} \cdot 1.0 + \frac{\left(1.0 \cdot \left(\log \beta \cdot \log \beta\right)\right) \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\alpha \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}\right)\right)\right)\right) - \left(\left(\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 2.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \beta}{\frac{\alpha}{\log 2}}}} + \frac{\left(1.0 \cdot \left(-\log \beta\right)\right) \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\alpha \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}\right) + \left(\frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{2 \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \alpha}{\alpha}}} + \frac{1.0}{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\alpha}{\log 2}}\right)\right) + \left(\left(\frac{\left(2.0 \cdot \left(\left(-\log \beta\right) \cdot \left(-\log \alpha\right)\right)\right) \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\alpha \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{2 \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{\log 2}{\alpha}}}\right) + 1.0 \cdot \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\log 2 - \log \alpha\right) + \log \beta}{\frac{-\log \alpha}{\alpha}}}\right)\right)\right)}{2.0}}\]

    if 1.5435960758848893e-12 < (/ (- (/ beta (+ (+ alpha beta) 2.0)) (- (* (* (cbrt (/ alpha (+ (+ alpha beta) 2.0))) (cbrt (/ alpha (+ (+ alpha beta) 2.0)))) (cbrt (/ alpha (+ (+ alpha beta) 2.0)))) 1.0)) 2.0)

    1. Initial program 0.3

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied div-sub0.3

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)} + 1.0}{2.0}\]

    4. Applied associate-+l-0.3

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}}{2.0}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{\left(\sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}} \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}\right) \cdot \sqrt[3]{\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}}} - 1.0\right)}{2.0}\]
  3. Recombined 3 regimes into one program.

  4. Applied simplify1.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}\right) \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} - 1.0\right)}{2.0} \le 9.645829536245808 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{1}{{\alpha}^{2}} \cdot 4.0 - \left(\frac{1}{{\alpha}^{3}} \cdot 8.0 + 2.0 \cdot \frac{1}{\alpha}\right)\right)}{2.0}\\ \mathbf{if}\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}\right) \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} - 1.0\right)}{2.0} \le 1.5435960758848893 \cdot 10^{-12}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\beta} \cdot 1.0 + \frac{\left(1.0 \cdot \left(\log \beta \cdot \log \beta\right)\right) \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}\right) + \left(\frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot 2}{-\frac{\log \beta}{\alpha}}} + e^{\left(\log 2 - \log \alpha\right) + \log \beta}\right)\right) + \left(\left(1.0 \cdot \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\log 2 - \log \alpha\right) + \log \beta}{-\frac{\log \beta}{\alpha}}} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 1.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{\log \alpha \cdot \log \alpha}{\alpha}}}\right) + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\alpha}{-\log \alpha}} \cdot \frac{1.0}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} \cdot \left(1.0 \cdot \frac{\log 2}{\frac{\alpha}{\log 2}}\right)\right)\right)\right) + \left(\left(1.0 \cdot \frac{\log 2}{\alpha}\right) \cdot \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 2.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \alpha}{\frac{\alpha}{\log 2}}}}\right)\right) - \left(\left(\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot \left(\log \beta \cdot \left(-1.0\right)\right)}{\left(\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 2.0}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}{\frac{-\log \beta}{\frac{\alpha}{\log 2}}}}\right) + \left(\frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot 2}{-\frac{\log \alpha}{\alpha}}} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\alpha}{\log 2}} \cdot \frac{1.0}{\left(\log 2 - \log \alpha\right) + \log \beta}\right)\right) + \left(1.0 \cdot \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\log 2 - \log \alpha\right) + \log \beta}{-\frac{\log \alpha}{\alpha}}} + \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha}\right) + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot \left(2.0 \cdot \left(\log \beta \cdot \log \alpha\right)\right)}{\left(\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \alpha\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} + \frac{6.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot 2}{\frac{\log 2}{\alpha}}}\right)\right)\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}}\right) \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\alpha + \beta\right)}} - 1.0\right)}{2.0}\\ \end{array}}\]

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018201 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))