Average Error: 16.6 → 6.1
Time: 3.7m
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 11476007602.97136:\\ \;\;\;\;\frac{\beta \cdot \left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - \left(2.0 + \left(\beta + \alpha\right)\right) \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - {1.0}^{3}} \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - {1.0}^{3}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - {1.0}^{3}}\right)\right)}{\left(\left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 11476007602.97136

    1. Initial program 0.2

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

    3. Applied div-sub0.2

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

    4. Applied associate-+l-0.2

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

    6. Applied flip3--0.2

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

    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. Applied associate-/l/0.2

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

    10. Applied unpow30.2

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

    12. Applied add-cube-cbrt0.2

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

    if 11476007602.97136 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-48.9

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

    5. Taylor expanded around -inf 18.3

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

    6. Simplified18.3

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 11476007602.97136:\\ \;\;\;\;\frac{\beta \cdot \left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - \left(2.0 + \left(\beta + \alpha\right)\right) \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - {1.0}^{3}} \cdot \left(\sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - {1.0}^{3}} \cdot \sqrt[3]{\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \left(\frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) - {1.0}^{3}}\right)\right)}{\left(\left(\left(1.0 \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} + 1.0 \cdot 1.0\right) + \frac{\alpha}{2.0 + \left(\beta + \alpha\right)} \cdot \frac{\alpha}{2.0 + \left(\beta + \alpha\right)}\right) \cdot \left(2.0 + \left(\beta + \alpha\right)\right)\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\beta + \alpha\right)} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 3.7m)Debug logProfile

herbie shell --seed 2018336 +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))