Average Error: 16.2 → 3.3
Time: 1.2m
Precision: 64
Internal Precision: 1344
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 1.641367264033235 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 0.5000001989457653:\\ \;\;\;\;\frac{(\left(\frac{\beta - \alpha}{{2.0}^{3} + {\left(\alpha + \beta\right)}^{3}}\right) \cdot \left(\left(2.0 \cdot 2.0 - \left(\alpha + \beta\right) \cdot 2.0\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right) + 1.0)_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 3 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 1.641367264033235e-13

    1. Initial program 60.4

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

    3. Applied div-sub60.4

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

    4. Applied associate-+l-58.5

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

    5. Taylor expanded around inf 11.0

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

    6. Simplified11.0

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

    if 1.641367264033235e-13 < (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 0.5000001989457653

    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 flip3-+0.7

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

    4. Applied associate-/r/0.7

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

    5. Applied fma-def0.7

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

    if 0.5000001989457653 < (/ (+ (/ (- 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-sub0.0

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

    4. Applied associate-+l-0.0

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

    6. Applied add-exp-log0.4

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 1.641367264033235 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2.0} - (\left(\frac{1}{\alpha \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(\frac{-2.0}{\alpha}\right))_*}{2.0}\\ \mathbf{elif}\;\frac{1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}}{2.0} \le 0.5000001989457653:\\ \;\;\;\;\frac{(\left(\frac{\beta - \alpha}{{2.0}^{3} + {\left(\alpha + \beta\right)}^{3}}\right) \cdot \left(\left(2.0 \cdot 2.0 - \left(\alpha + \beta\right) \cdot 2.0\right) + \left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right) + 1.0)_*}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}{2.0}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018216 +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))