Average Error: 16.4 → 1.8
Time: 1.7m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(1.0 - \frac{\alpha}{\beta + \left(2.0 + \alpha\right)}\right) + \frac{\beta}{\beta + \left(2.0 + \alpha\right)} \le 1.9291659072491616 \cdot 10^{-229}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \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}\\ \mathbf{if}\;\left(1.0 - \frac{\alpha}{\beta + \left(2.0 + \alpha\right)}\right) + \frac{\beta}{\beta + \left(2.0 + \alpha\right)} \le 2.6084727666159498 \cdot 10^{-21}:\\ \;\;\;\;\frac{\left(\left(\left(\left(\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 2}{\frac{\alpha}{\log 2}}}} + \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) + (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 \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)} \cdot \frac{-\log \alpha}{\alpha}\right) + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot \left(\left(-\frac{\log \beta}{\alpha}\right) \cdot 1.0\right)\right))_*\right) + \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 \alpha \cdot \log \alpha}{\alpha} \cdot 1.0\right) + (6.0 \cdot \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{(2 \cdot \left(\log 2 - \log \alpha\right) + \left(\log \beta \cdot 2\right))_*}{-\frac{\log \beta}{\alpha}}}\right) + \left(e^{\left(\log 2 - \log \alpha\right) + \log \beta}\right))_*\right) + (\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot \frac{\log \beta \cdot \log \beta}{\alpha}}{\left(\left(\log 2 - \log \alpha\right) + \log \beta\right) \cdot \left(\left(\log 2 - \log \alpha\right) + \log \beta\right)}\right) \cdot 1.0 + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 1.0}{\beta}\right))_*\right)\right) + \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{2.0 \cdot e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\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)\right) - \left(\left((\left(\left(-\frac{\log \beta}{\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)}\right) \cdot 1.0 + \left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta} \cdot 1.0}{\frac{\left(\log 2 - \log \alpha\right) + \log \beta}{\frac{\log 2}{\alpha}}}\right))_* + (\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{(2 \cdot \left(\log 2 - \log \alpha\right) + \left(\log \beta \cdot 2\right))_*}{\frac{-\log \alpha}{\alpha}}}\right) \cdot 6.0 + \left(\frac{1.0}{\alpha} \cdot \frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\left(\log 2 - \log \alpha\right) + \log \beta}\right))_*\right) + (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}}}\right) + \left((\left(\frac{e^{\left(\log 2 - \log \alpha\right) + \log \beta}}{\frac{(2 \cdot \left(\log 2 - \log \alpha\right) + \left(\log \beta \cdot 2\right))_*}{\frac{\log 2}{\alpha}}}\right) \cdot 6.0 + \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 \alpha}}}}\right))_*\right))_*\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2.0 \cdot 2.0}\right) \cdot \left(\left(\alpha + \beta\right) - 2.0\right) + \left(-1.0\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 3 regimes

  2. if (+ (- 1.0 (/ alpha (+ beta (+ 2.0 alpha)))) (/ beta (+ beta (+ 2.0 alpha)))) < 1.9291659072491616e-229

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-60.3

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

    5. 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 1.9291659072491616e-229 < (+ (- 1.0 (/ alpha (+ beta (+ 2.0 alpha)))) (/ beta (+ beta (+ 2.0 alpha)))) < 2.6084727666159498e-21

    1. Initial program 61.4

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

    3. Applied div-sub61.4

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

    4. Applied associate-+l-54.7

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

    6. Applied add-exp-log54.7

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

    8. Applied add-cbrt-cube54.7

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

    9. Applied simplify54.7

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

    10. Taylor expanded around inf 62.4

      \[\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 simplify6.8

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

    if 2.6084727666159498e-21 < (+ (- 1.0 (/ alpha (+ beta (+ 2.0 alpha)))) (/ beta (+ beta (+ 2.0 alpha))))

    1. Initial program 0.9

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

    3. Applied div-sub0.9

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

    4. Applied associate-+l-0.8

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

    6. Applied flip-+1.1

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

    7. Applied associate-/r/1.1

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

    8. Applied fma-neg1.1

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018201 +o rules:numerics
(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))