Average Error: 16.5 → 3.3
Time: 1.4m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0} \le 2.7355289247521508 \cdot 10^{-05}:\\ \;\;\;\;\frac{\left((\left(\frac{4.0}{\alpha}\right) \cdot \left(\frac{-1}{\alpha}\right) + \left(\frac{2.0}{\alpha}\right))_* + \frac{\frac{8.0}{\alpha}}{\alpha \cdot \alpha}\right) + \frac{\beta}{\left(2.0 + \alpha\right) + \beta}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\beta - \alpha\right) \cdot \left(\frac{1}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0)_*}{2.0}\\ \end{array}\]

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) < 2.7355289247521508e-05

    1. Initial program 59.0

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

    3. Applied div-sub59.0

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

    4. Applied associate-+l-57.2

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

    5. Taylor expanded around inf 11.5

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

    6. Applied simplify11.5

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

    if 2.7355289247521508e-05 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0)

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    4. Applied fma-def0.0

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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))