Average Error: 15.9 → 3.4
Time: 49.5s
Precision: 64
Internal Precision: 1408
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0} \le 3.539000204000331 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\beta}{2.0 + \left(\alpha + \beta\right)}}{2.0} - (\left(\frac{\frac{1}{\alpha}}{2.0 \cdot \alpha}\right) \cdot \left(4.0 - \frac{8.0}{\alpha}\right) + \left(-\frac{1}{\alpha}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2.0}\right)}^{3} - {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2.0} - 1.0\right)}^{3}}{(\left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)}\right) \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) + \left((\left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) \cdot \left(\frac{\alpha}{2.0 + \left(\alpha + \beta\right)} - 1.0\right) + \left(\frac{\beta}{2.0 + \left(\alpha + \beta\right)} \cdot \frac{\beta}{2.0 + \left(\alpha + \beta\right)}\right))_*\right))_*}}{2.0}\\ \end{array}\]

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0) < 3.539000204000331e-10

    1. Initial program 59.9

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

    3. Applied div-sub59.9

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

    4. Applied associate-+l-57.8

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

    6. Applied add-cube-cbrt57.8

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

    7. Applied associate-/r*57.8

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

    8. Taylor expanded around inf 12.5

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

    9. Applied simplify12.5

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

    if 3.539000204000331e-10 < (/ (+ (/ (- 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 flip3--0.2

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

    7. Applied simplify0.2

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

Runtime

Time bar (total: 49.5s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))