Average Error: 16.3 → 0.2
Time: 3.0m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 2.031575370521667 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \beta - (\beta \cdot \left(\frac{4.0}{\alpha} - 2.0\right) + \left(-4.0\right))_*}{(\left(\beta + \left(\alpha + 2.0\right)\right) \cdot 1.0 + \alpha)_*}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \beta - (\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + \left(\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(-1.0\right) \cdot 1.0\right)\right))_*}{(\left(\beta + \left(\alpha + 2.0\right)\right) \cdot 1.0 + \alpha)_*}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 2.031575370521667e-09

    1. Initial program 60.1

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

    3. Applied div-sub60.1

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

    4. Applied associate-+l-58.2

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

    6. Applied flip--58.2

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

    7. Applied frac-sub58.2

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

    8. Simplified58.2

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot 1.0\right)}{\color{blue}{(\left(\left(2.0 + \alpha\right) + \beta\right) \cdot 1.0 + \left(\frac{\alpha}{1}\right))_*}}}{2.0}\]
    9. Using strategy rm

    10. Applied sub-neg58.2

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

    11. Applied distribute-lft-in58.0

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

    12. Taylor expanded around inf 0.3

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0\right) - \color{blue}{\left(4.0 \cdot \frac{\beta}{\alpha} - \left(2.0 \cdot \beta + 4.0\right)\right)}}{(\left(\left(2.0 + \alpha\right) + \beta\right) \cdot 1.0 + \left(\frac{\alpha}{1}\right))_*}}{2.0}\]

    13. Simplified0.3

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0\right) - \color{blue}{(\beta \cdot \left(\frac{4.0}{\alpha} - 2.0\right) + \left(-4.0\right))_*}}{(\left(\left(2.0 + \alpha\right) + \beta\right) \cdot 1.0 + \left(\frac{\alpha}{1}\right))_*}}{2.0}\]

    if 2.031575370521667e-09 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

    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 flip--0.2

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

    7. Applied frac-sub0.2

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

    8. Simplified0.2

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0\right) - \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0 \cdot 1.0\right)}{\color{blue}{(\left(\left(2.0 + \alpha\right) + \beta\right) \cdot 1.0 + \left(\frac{\alpha}{1}\right))_*}}}{2.0}\]
    9. Using strategy rm

    10. Applied sub-neg0.2

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

    11. Applied distribute-lft-in0.2

      \[\leadsto \frac{\frac{\beta \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0\right) - \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + \left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(-1.0 \cdot 1.0\right)\right)}}{(\left(\left(2.0 + \alpha\right) + \beta\right) \cdot 1.0 + \left(\frac{\alpha}{1}\right))_*}}{2.0}\]
    12. Using strategy rm

    13. Applied fma-def0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 2.031575370521667 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{\left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \beta - (\beta \cdot \left(\frac{4.0}{\alpha} - 2.0\right) + \left(-4.0\right))_*}{(\left(\beta + \left(\alpha + 2.0\right)\right) \cdot 1.0 + \alpha)_*}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(1.0 + \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) \cdot \beta - (\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} \cdot \frac{\alpha}{\left(\alpha + \beta\right) + 2.0}\right) + \left(\left(\left(\alpha + \beta\right) + 2.0\right) \cdot \left(\left(-1.0\right) \cdot 1.0\right)\right))_*}{(\left(\beta + \left(\alpha + 2.0\right)\right) \cdot 1.0 + \alpha)_*}}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018225 +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))