Average Error: 16.0 → 5.8
Time: 37.4s
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 6861464139492653.0:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta}{\sqrt{2.0 + \left(\alpha + \beta\right)}}}{\sqrt{2.0 + \left(\alpha + \beta\right)}} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if alpha < 6861464139492653.0

    1. Initial program 0.4

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

    3. Applied div-sub0.4

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

    4. Applied associate-+l-0.4

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

    6. Applied add-sqr-sqrt0.5

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

    7. Applied associate-/r*0.5

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

    9. Applied *-un-lft-identity0.5

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

    10. Applied sqrt-prod0.5

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

    11. Applied *-un-lft-identity0.5

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

    12. Applied times-frac0.5

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

    13. Simplified0.5

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

    14. Simplified0.4

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

    if 6861464139492653.0 < alpha

    1. Initial program 50.5

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

    3. Applied div-sub50.4

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

    4. Applied associate-+l-49.0

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

    6. Applied add-sqr-sqrt49.0

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

    7. Applied associate-/r*49.0

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

    8. Taylor expanded around -inf 17.5

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

    9. Simplified17.5

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6861464139492653.0:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)} - \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\beta}{\sqrt{2.0 + \left(\alpha + \beta\right)}}}{\sqrt{2.0 + \left(\alpha + \beta\right)}} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{2.0}{\alpha}\right))_*}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 37.4s)Debug logProfile

herbie shell --seed 2018235 +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))