Average Error: 16.6 → 3.3
Time: 29.6s
Precision: 64
Internal Precision: 128
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.999999999999997:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}} \cdot \sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}}{\frac{\sqrt{\left(\alpha + \beta\right) + 2.0}}{\sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}}} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\alpha \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + \left(-1.0\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (- beta alpha) (+ (+ alpha beta) 2.0)) < -0.999999999999997

    1. Initial program 60.5

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

    3. Applied div-sub60.5

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

    4. Applied associate-+l-58.6

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

    6. Applied add-sqr-sqrt58.6

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

    7. Applied associate-/r*58.6

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

    9. Applied add-cube-cbrt58.6

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

    10. Applied associate-/l*58.6

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

    11. Taylor expanded around -inf 10.9

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

    12. Simplified10.9

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

    if -0.999999999999997 < (/ (- beta alpha) (+ (+ alpha beta) 2.0))

    1. Initial program 0.5

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

    3. Applied div-sub0.5

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

    4. Applied associate-+l-0.5

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

    6. Applied div-inv0.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \left(\color{blue}{\alpha \cdot \frac{1}{\left(\alpha + \beta\right) + 2.0}} - 1.0\right)}{2.0}\]

    7. Applied fma-neg0.5

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - \color{blue}{(\alpha \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + \left(-1.0\right))_*}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} \le -0.999999999999997:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}} \cdot \sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}}{\frac{\sqrt{\left(\alpha + \beta\right) + 2.0}}{\sqrt[3]{\frac{\beta}{\sqrt{\left(\alpha + \beta\right) + 2.0}}}}} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\alpha \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + \left(-1.0\right))_*}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +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))