Average Error: 15.7 → 6.0
Time: 3.4m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 21106279031.22786:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - \left(\log \left(\frac{\sqrt[3]{e^{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}} \cdot \sqrt[3]{e^{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}}}{\sqrt{e^{{1.0}^{3}}}}\right) + \log \left(\frac{\sqrt[3]{e^{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}}}{\sqrt{e^{{1.0}^{3}}}}\right)\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)}{\left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 21106279031.22786

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied flip3--0.1

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

    7. Applied frac-sub0.2

      \[\leadsto \frac{\color{blue}{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {1.0}^{3}\right)}{\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + \left(1.0 \cdot 1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot 1.0\right)\right)}}}{2.0}\]
    8. Using strategy rm

    9. Applied add-log-exp0.2

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

    10. Applied add-log-exp0.2

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

    11. Applied diff-log0.2

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

    13. Applied add-sqr-sqrt1.1

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

    14. Applied add-cube-cbrt1.1

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

    15. Applied times-frac0.2

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

    16. Applied log-prod0.2

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

    if 21106279031.22786 < alpha

    1. Initial program 49.4

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

    3. Applied div-sub49.4

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

    4. Applied associate-+l-47.9

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

    5. Taylor expanded around -inf 18.4

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

    6. Simplified18.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{(\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 21106279031.22786:\\ \;\;\;\;\frac{\frac{\beta \cdot \left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - \left(\log \left(\frac{\sqrt[3]{e^{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}} \cdot \sqrt[3]{e^{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}}}{\sqrt{e^{{1.0}^{3}}}}\right) + \log \left(\frac{\sqrt[3]{e^{{\left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right)}^{3}}}}{\sqrt{e^{{1.0}^{3}}}}\right)\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)}{\left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

herbie shell --seed 2018249 +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))